Quantum computers (QCs) operate totally differently than classical computers. Due to the quantum effects known as superposition and entanglement, quantum bits (called qubits) can take on nonbinary states represented by complex numbers. This facilitates computational solutions to mathematical problems that cannot be solved by classical computers because they require sequentially computing an astronomical number of combinations or permutations.
This ability of QCs mean that they particularly excel at optimization problems, where the optimal combination is only found after trying out an enormous number of possible combinations. Several important problems in finance are in essence optimization problems which meet this description. The portfoliooptimization problem in finance is one good example of such a problem. Asset pricing, creditscoring, and Monte Carlotype risk analysis are other examples. For example, it is estimated that running a risk assessment of a large portfolio which needs to be done overnight or can even take days with classical computers, could one day be done in realtime by a fullscale QC. That explains the keen interest of the finance industry in quantum solutions.
The calculating
power of a QC grows exponentially with the number of qubits. Quantumcomputing
roadmaps cite the number of qubits or competing metrics to indicate the rising
power of these machines, with some setting thresholds for socalled quantum supremacy,
the point at which QCs will surpass classical supercomputers. But there are
still enormous technical challenges to solve before atscale QCs can be
commercialized, most notably the challenges of stability and error correction.
However, quantuminspired software – which is software running on classical computers but based on novel algorithms that reframe mathematical problems in terms of quantum principles – is already here. Several quantuminspired solutions are currently focused on portfoliooptimization problems, and seem well positioned for their nearfuture adoption by the financial assetmanagement industry.
Even the limitedsize, noisy QCs currently available lend themselves to portfoliooptimization solutions. Early proofsofconcept (POCs) of hybrid or full quantum solutions to assetportfolio optimizations such as stock selection have already been demonstrated with encouraging results. Many of the largest names in finance are already investing in quantum, or at least partnering with technology providers to explore finance applications. Financial services companies who wait too long to gain experience in the field run the risk of getting left behind.
Introduction
Quantum computing exploits quantum mechanics, the properties and behavior of fundamental particles at the subatomic level, as predicted by our best current understanding of quantum physics. The goal of quantum computing is to build hardware and develop suitable algorithms that process information in ways that are superior to socalled classical computers, i.e. the ubiquitous digital computers that the Information Age was built on.
The essential elements of a QC were postulated in the early 1980s, but of late work in this area has accelerated with several large established companies and startups building quantumcomputing hardware. An even larger ecosystem of software platforms and solution providers exist around the hardware providers. Collaboration models such as alliances and partnerships are common. Many universities are involved, while governments are also supporting quantumcomputing research.
Typical of a new industry, standards and metrics are still in flux, and competing architectures, which leverage different mechanisms and implementations of quantum principles, vie for technical supremacy and investment dollars. Announcements of new breakthroughs are made almost daily, which makes it important to distinguish the hype from real progress.
This paper attempts to demystify the technology, by explaining the basic principles of quantum computing and the competing technologies vying for quantum supremacy. An overview of the current quantum computing industry and the main players is provided, as well as a look at the first applications and the different industries that could benefit. The focus then turns to the finance industry, with an overview of the most important computational problems in finance that lends themselves to quantum computing, with a deeper dive into portfolio optimization. Notable recent case studies and their participants are reviewed. The paper concludes with an assessment of the current state of quantum computing and the business impact that can be expected in the short and medium term.
A Brief Primer on the Technology
How It Works and Where It Holds Advantages or Not
What we now call classical (or conventional) digital computers perform all their calculations in an aggregate of individual bits that are either 0 or 1 in value, because they are implemented by transistors that are each either switched completely on or off. This is called binary logic[1], which is the essence of any digital computer, and implemented in a longstanding computerscience paradigm originating with Turing and Von Neumann. Conventional computers operate by switching billions of little transistors on and off, with all state changes governed by the computer’s clock cycle. With n transistors, there are 2^{n }possible states for the computer to be in at any given time. Importantly, the computer can only be in one of these states at a time. Digital computers are highly complex with typical computer chips holding 20x10^{19} bits, yet incredibly reliable at the semiconductor level with fewer than one error in 10^{24} operations. (Software and mechanicalrelated errors are far more common in computers.)
Analog computers precede digital computers. In contrast to digital computers, classical analog computers perform calculations with electrical parameters (voltage or current) that take a full range of values along a continuous linear scale. Analog computers do not necessarily need to be electrical – they can be mechanical too, such as the first ones built by the ancient Greeks[2] – but the most sophisticated ones from the 20^{th} century ones were electrical. Unlike digital computers, analog computers do not need a clock cycle, and all values change continuously. Before the digital revolution was enabled through the mass integration of transistors on chips, analog computers were used in several applications, for example, to calculate flight trajectories or in early autopilot systems. But since the 1960s analog computers have largely fallen into disuse due to the dominance of digital computers over the last few decades.
Both classical digital and analog computers are at their core electrical devices, in the sense that they perform logic operations that are reflected by the electrical state of devices, typically semiconductor devices such as transistors (or vacuum tubes for mid20^{th} century analog computers), which comes about because of voltage differences and current flow. Current flow is physically manifested in terms of the flow of electrons in an electrical circuit.[3]
Quantum computers (QCs), on the other hand, directly exploit the strange and counterintuitive behavior of subatomic particles (electrons, nuclei or photons) as predicted by quantum theory to implement a new type of mathematics. In a QC, quantum bits called qubits can be measured as 0> or 1>, which are the quantum equivalents of the binary 0 and 1 in classical computers. However, due to a quantum property called superposition, qubits can be nonbinary in a superposition state and interact with one another in that state during processing. It is this special property that allows QCs to theoretically offer exponentially more processing power than classical computers in some applications. Once the processing is complete, the result can only be measured in the binary states, 0> or 1>, because superpositioning is always collapsed by the measurement process.
Because of another curious quantum property called entanglement, the behavior of two or more quantum objects is correlated even if they are physically separated. According to the laws of quantum mechanics, this pattern is consistent whether a millimeter or kilometer or an astronomical distance separates them.[4] While one qubit is situated in a superposition between two basis states, 10 qubits utilizing entanglement, could be in a superposition of 1,024 basis states.
Unlike the linearity of classical computers, the calculating power of a QC grows exponentially with the number of qubits. It is this ability that gives QCs the extraordinary power of processing a huge number of possible outcomes simultaneously. When in the unobserved state of superposition, n qubits can contain the same amount of information as 2^{n} classical bits. So, four qubits are equivalent to 16 classical bits, which might not sound like a big improvement. But 16 qubits are equivalent to 85,536 classical bits, and 300 qubits can contain more states than all the atoms estimated to be in the universe. That is not only an astronomical number; it is beyond astronomical. This exponential effect is why there is so much hope for the future of quantum computing. With single or doubledigit numbers of qubits, the advantage over classical computing is not immediately clear, but the power of quantum computing scales exponentially beyond that in ways that are truly hard to imagine. This explains why there is so much anticipation about the technology exploding once a certain number of qubits have been reached in a reliable QC.
However, to reliably encode information and expect it to be returned upon measurement, there are only two acceptable states for a qubit: ∣0⟩ and ∣1⟩. This means a qubit can only store 1 bit of information at a time. Even with many qubits, the scaling of information storage doesn't improve beyond what you'd get classically: ten qubits can store 10 bits of information and one thousand qubits can store 1,000 bits. Because a qubit can only be measured in one of these two states, qubits cannot store any more data than conventional computer bits. There is thus no quantum advantage in data storage. The advantage is in information processing, and that advantage comes from the special quantum properties of a qubit – that it can occupy a superposition of states when not being measured.
Another point to keep in mind is that due to probabilistic waveform properties of qubits, QCs do not typically deliver one answer, but rather a narrow range of possible answers. Multiple runs of the same calculation can further narrow the range, but at the expense of lessening speed gains.
Classical computers will not be replaced by QCs. A primary reason for this is that QCs cannot run the “if/then/else” logic functions that are a cornerstone of the classical Von Neumann computer architecture. Instead QCs will be used alongside classical computers to solve those problems that they are particularly good at, such as optimization problems.
The strengths of QCs in simultaneous calculations mean that they excel at finding optimal solutions to problems with a large number of variables, where the optimal combination is only found after trying out an enormous number of possible combinations or permutations. Such problems are found, for example, in optimizing any portfolio composition, or trying out millions of possible new molecular combinations for drugs, or in routing many aircraft between many hubs. In such problems there are typically 2^{n} possibilities and they all have to be tried out to find an optimal solution. If there are 100 elements to combine, it becomes a 2^{100} computation, which is almost impossible to solve with a classical computer but a 100qubit computer could solve it in one operation.
Quite a few hard problems in finance are in essence optimization problems and therefore meet the description of problems that can be solved by QCs. The portfoliooptimization problem in finance is one good example of such a problem. Asset pricing, creditscoring, and Monte Carlotype risk analysis are other examples. That explains the keen interest of the finance industry in quantum solutions. The finance industry is also well positioned to be an early adopter, because financial algorithms are much quicker to deploy than algorithms that drive industrial or other physical processes.
Making a Quantum Computer
A QC architecture can be seen as a stack with the following typical layers:
· At the bottom is the actual quantum hardware (usually held at nearabsolute zero temperatures to minimize thermal noise, and/or in a vacuum)
· The next level up comprises the control systems that regulate the quantum hardware and enable the calculation
· Above those comes the software layer that implements the algorithms (and in future, also will do the error correction). It includes a quantumclassical interface that compiles source code into executable programs
· The top of the stack comprises the wider variety of services to utilize the QC, e.g. the operating systems and software platforms that help translate reallife problems into a format suitable for quantum computing
There are many different ways to physically realize qubits— from using trapped calcium ions to superconducting structures.[5] In each case, quantum states are being manipulated to perform calculations. Quantum computers can entangle qubits by passing them through quantum logic gates. For example, a “CNOT” (conditional NOT) gate flips—or doesn’t flip—a qubit based on the state of another qubit. Stringing multiple quantum logic gates together creates a quantum circuit.
The designers of QCs need to master and control both superposition and entanglement:
Without superposition, qubits would behave like classical bits, and would not be in the multiple states that allow quantum programmers to run the equivalent of many calculations at once. Without entanglement, the qubits would sit in superposition without generating additional insight by interacting. No calculation would take place because the state of each qubit would remain independent from the others. The key to creating business value from qubits is to manage superposition and entanglement effectively.[i]
The simplest and most typical physical properties that can serve as a qubit is the electron’s internal angular momentum, spin for short. It has the quantum property of having only two possible projections on any coordinate axis, +1/2 or 1/2 in units of the Planck constant. For any chosen axis the two basic quantum states of the electron’s spin can be denoted as ↑ (up) or ↓ (down). But these are not the only states possible for a quantum bit, because the spin state of an electron is described by a quantummechanical wave function. That function includes two complex[6] numbers, called quantum amplitudes, α and β, each with its own magnitude. The rules of quantum mechanics dictate that QUOTE <?msoapplication progid="Word.Document"?> 16Î±2+Î²2=1"> <?msoapplication progid="Word.Document"?> 16Î±2+Î²2=1"> . Both α and β have real and imaginary parts. The squared magnitudes α^{2} and β^{2} correspond to the probabilities of the spin of the electron to be in the basic states ↑ or ↓ when they are measured. Since those are the only two outcomes possible, their squared magnitudes must equal 1. In contrast to a classical bit, which can only be in one of its two binary states, a qubit can be in any continuum of possible states, as defined by the quantum amplitudes α and β. In the popular press this is often explained by the oversimplified, and somewhat mystical, statement that a qubit can exist simultaneously in both its ↑ or ↓ states. That is analogous to saying that a plane flying northwest is simultaneously flying both west and north, which is not incorrect strictly speaking, but not a particularly helpful mental model either.
Because a qubit can only be measured in one of these two states, qubits cannot store any more data than conventional computer bits. There is thus no quantum advantage in data storage. The advantage is in information processing, and that advantage comes from the special quantum properties of a qubit meaning it can occupy a superposition of states when not being measured. During computation, qubits can interact with one another while in their superposition state. For example, a set of 6 qubits can occupy any linear combination of all the 2^{6} = 64 different length 6bit strings. With 64 continuous variables describing this state, the space of configurations available to a QC during a calculation is much greater than a classical one. The measurement limitations of storing information do not apply during the runtime execution of a quantum algorithm: During processing every qubit in a quantum algorithm can occupy a superposition. Thus, in a superposition state, every possible bit string (in this example, 2^{6} = 64 different strings)) can be combined. Each bit string in the superposition has an independent complex number coefficient with a magnitude (A) and a phase (θ):
α_{i} = A_{i}e^{iθ}^{i}
A modern digital computer, with billions of transistors in its processors, typically has 64 bits, not 6 as in our quantum example above. This allows it to consider 64 bits at once, which allows for 2^{64} states. While 2^{64} is a large number, equal to approximately 2 x 10^{19}, quantum computing can offer much more. The space of continuous states of QCs is much larger than the space of classical bit states. That is because the possibility of many particles interacting at the quantum level to form a common wave function, allowing changes in one particle to affect all others instantaneously and in a wellordered manner. That is akin to massive parallel computing, which can beat classical multicore systems.
Quantum computing operations can mostly be handled according to the standard rules of linear algebra, in particular matrix multiplication. The quantum state is represented by a state vector[7] written in matrix form, and the gates in the quantum circuit (whereby the calculations are executed) are represented as matrices too. Multiplying a state vector by a gate matrix yields another state vector. Recent progress has been made to use quantum algorithms to crack nonlinear equations, by using techniques that disguise nonlinear systems as linear ones.[ii]
Key QuantumComputerScience Concepts
The possibility of quantum computing was raised by Caltech physicist, Richard Feynman, in 1981. The person considered by most to be the founder of quantum computing, David Deutsch, first defined a QC in a seminal paper in 1985.[iii]
In 1994, a Bell Labs mathematician, Peter Shor, developed a quantum computing algorithm that can efficiently decompose any integer number into its prime factors.[iv] It has since become known as the Shor algorithm and has great significance for quantum computing. Shor’s algorithm was a purely theoretical exercise at the time, but it anticipated that a hypothetical QC could one day solve NPhard problems of the type used as the basis for modern cryptography. Shor’s algorithm relies on the special properties of a quantum machine. While the most efficient classical factoring algorithm, known as the general number field sieve, uses an exponential function of a constant x d^{1/3 }to factor an integer with d digits; Shor’s algorithm can do that by executing a runtime function that is only a polynomial function, namely a constant x d^{3}. Accordingly, classical computers are limited to factoring integers with only a few hundred digits, which is why using integers in the thousands in cryptography keys is considered to make for practically unbreakable codes. But a QC using the Kitaev version of Shor’s algorithm only needs 10d qubits, and will have a runtime roughly equal to d^{3}.[v]
In summary, the Shor algorithm means that a QC can solve an NPhard mathematical problem in polynomial time that classical computers can only solve in exponential time.[8] Therefore, Shor’s algorithm can demonstrate by how much quantum computing can improve processing time over classical computing. While a fullscale QC with the thousands of qubits needed to employ Shor’s algorithm in practice to crack codes is not yet available, many players are working towards machines of that size.
Another important early QC algorithm is Grover’s algorithm, a search algorithm which finds a particular register in an unordered database. This problem can be visualized as a phonebook with N names arranged in completely random order. In order to find someone's phone number with a probability of ½, any classical algorithm (whether deterministic or probabilistic) will need to look at a minimum of N/2 names. But the quantum algorithm needs only QUOTE <?msoapplication progid="Word.Document"?> 16ON"> <?msoapplication progid="Word.Document"?> 16ON"> steps.[vi] This algorithm can also be adapted for optimization problems.
Most quantum calculations are performed in what is called a quantum circuit. The quantum circuit is a series of quantum gates that operate on a system of qubits. Each quantum gate has inputs and outputs and operates akin to the hardware logic gates in classical digital computers. Like digital logic gates, the quantum gates are connected sequentially to implement quantum algorithms.
Quantum algorithms are algorithms that run on QCs, and which are structured to use the unique properties of quantum mechanics, such as superposition or quantum entanglement, to solve particular problem statements. Major quantum algorithms include the quantum evolutionary algorithm (QEA), the quantum particle swarm optimization algorithm (QPSO), the quantum annealing algorithm (QAA), the quantum neural network (QNN), the quantum Bayesian network (QBN), the quantum wavelet transform (QWT), and the quantum clustering algorithm (QC).[vii] A comprehensive catalog of quantum algorithms can be found online in the Quantum Algorithm Zoo.[viii]
Quantum software is the umbrella term used to describe the full collection of QC instructions, from hardwarerelated code, to compilers, to circuits, all algorithms and workflow software.
Quantum annealing[9] is an alternative model to circuitbased algorithms, as it is not built up out of gates. Quantum annealing naturally returns lowenergy solutions by utilizing a fundamental law of physics that any system will tend to seek its minimum state. In the case of optimization problems, quantum annealing uses quantum physics to find the minimum energy state of the problem, which equates to the optimal or nearoptimal combination of its constituent elements.[ix]
An Ising machine is a noncircuit alternative that works for optimization problems specifically. In the Ising model, the energy from interactions between the spins of every pair of electrons in a collection of atoms is summed. Since the amount of energy depends on whether spins are aligned or not, the total energy of the collection depends on the direction in which each spin in the system points. The general Ising optimization problem is determining in which state the spins should be so that the total energy of the system is minimized. To use the Ising model for optimization requires mapping parameters of the original optimization problem, such as an optimal route for the Traveling Salesman,[10] into a representative set of spins, and to define how the spins influence one another.[x]
Hybrid computing typically entails transferring the problem (say optimization) into a quantum algorithm, of which the first iteration is run on a QC. This provides a very fast answer, but only a rough assessment of the valid total solution space. The refined answer is then found with a powerful classical computer, which only has to examine a subset of the original solution space.[xi]
The Fundamental Challenges
The Achilles heel of the QC is the loss of coherence, or decoherence, caused by mechanical (vibration), thermal (temperature fluctuations), or electromagnetic disturbance of the subatomic particles used as qubits. Until the technology improves, various workarounds are needed. Commonly algorithms are designed to reduce the number of gates in an attempt to finish execution before decoherence and other sources of errors can corrupt the results.[xii] This often entails a hybrid computing scheme which moves as much work as possible from the QC to classical computers.
Current guestimates by experts are that truly useful QCs would need to be between 1,000 and 100,000 qubits. However, quantumcomputing skeptics such as Mikhail Dyakonov, a noted quantum physicist, point out that the enormous number of continuous parameters that would describe the state of a useful QC might also be its Achilles heel. Taking the low end of a 1,000 qubits machine, would imply a QC with 2^{1,000} parameters describing its state at any moment. That is roughly 10^{300}, a number greater than the number of subatomic particles in the universe: “A useful QC needs to process a set of continuous parameters that is larger than the number of subatomic particles in the observable universe.”[xiii] How would error control be done for 10^{300 }continuous parameters? According to quantumcomputing theorists the threshold theorem proves that it can be done. Their argument is that once the error per qubit per quantum gate is below a certain threshold value, indefinitely long quantum computation becomes possible, at a cost of substantially increasing the number of qubits needed. The extra qubits are needed to handle errors by forming logical qubits using multiple physical qubits. (This is a bit like error correction in current telecom systems, which use extra bits to validate data.) But that greatly increases the number of physical qubits to handle, which as we have seen, are already more than astronomical. At the very least, this brings into perspective the magnitude of the technological problems that scientists and engineers will have to overcome.
To put the comparative size of the QC errorcorrection problem in practical terms: For a typical 3Volt CMOS logic circuit used in classical digital computers, a binary 0 would be any voltage measured between 0V and 1V, while a binary 1 would be any voltage measured between 2V and 3V. Thus when e.g. 0.5V of noise is added to the signal for binary 0, the measurement would be 0.5V which would still correctly indicate a binary value of 0. For this reason, digital computers are very robust to noise. However, for a typical qubit, the difference in energy between a zero and a one is just 10^{24} Joules—one tentrillionth as much energy as an Xray photon. Error correction is one of the biggest hurdles to overcome in quantum computing, the concern being that it will impose such a huge overhead, in terms of auxiliary calculations, that it will make it very hard to scale QCs.
After Dyakonov published the skeptic’s viewpoint two years ago, a vigorous debate followed.[xiv] A typical response to the skeptic’s case comes from an industryinsider, Richard Versluis, systems architect at QuTech, a Dutch QC collaboration. Versluis acknowledges the engineering challenges to control a QC and to make sure its state is not affected. However, he states that the challenge is to make sure that the control signals and qubits perform as desired. Major sources of potential errors are quantum rotations that are not perfectly accurate, and decoherence as qubits lose their entanglement and the information they contain. Versluis goes on to define a fivelayered QC architecture that he believes will be up to the task. From top to bottom, the layers are 1. Application layer, 2. Classical processing, 3. Digital processing, 4. Analog processing, and 5. Quantum processing. Together the digital, analog, and quantumprocessing layers comprise the quantum processing unit (QPU). But Versluis also has to acknowledge that quantum error correction could solve the fundamental problem of decoherence only at the expense of 100 to 10,000 errorcorrecting physical qubits per logical (calculating) qubit. Furthermore, each of these millions of qubits will need to be controlled by continuous analog signals. And the biggest challenge of all is doing the thousands of measurements per second in a way that they do not disturb quantum information (which must remain unknown until the end of the calculation), while catching and correcting errors. The current paradigm of measuring all qubits with analog signals will not scale up to larger machines, and a major advance in the technology will be required.[xv]
Most experts agree that we will have to live with QCs over the next few years that will have high levels of errors that go uncorrected. There is even an accepted industry term and acronym for such QCs: NISQ (Noisy IntermediateScale Quantum) devices. The NISQ era is expected to last for the next five years at least, bar any major breakthroughs that might shorten that timeline.
Current Status of Essential Building Blocks and Systems
Once critical technical breakthroughs are made, QC adoption may happen faster than expected due to the prevalence of cloud computing. Making QC services easily accessible over the cloud speeds both adoption and learning. It has the added advantage that it forces hardware makers to focus on building QCs with a high percentage of uptime, so as to ensure continued availability over the cloud.
Most QC makers already offer cloud access to their latest QCs. There are programming environments – software development kits (SDKs) that facilitate the building of quantum circuits – available over the cloud for QC programmers to learn how to write the software that unleashes the magic of quantum computing, and to experiment with it. As more functionality is added to the hardware, these SDKs are continually updated.
The implication is that a whole ecosystem is being brought up to speed on how to make the best use of a quantum capability that does not quite exist yet. An analogy would be having had flight simulators to train future pilots while the Wright brothers were still figuring out how to keep their plane in the air for more than a few hundred feet. The upside of this approach is that any real advances in making reliable QCs with capabilities superior to classical computers will be very quickly exploited by realworld applications. This situation is in contrast to most major technological breakthroughs we have seen in the past. For example, it took a generation or two for industrial engineers to learn how to properly use electrical power in the place of steam power in factories. More recently, it took a generation to fully exploit the capabilities of digital computing in business and elsewhere. But in the case of quantum computing, all the knowledge building in anticipation of a successful QC could be rapidly translated into applications by a corps of developers who are all trained up and ready to “fly the plane” once it is finally built. That is the optimistic perspective.
Quantum circuits are already being developed using quantum programming languages and socalled quantum development kits (QDKs)[11], such as Qiskit by IBM and Google Cirq based on Python; and Q# by Microsoft based on the C# language. The next step is to develop libraries and workflows for different application domains. Examples of the former are IBM’s Aqua and Q# libraries. Examples of the latter are DWave’s Ocean development tool kit for hybrid quantumclassical applications and to translate quantum optimization problems into quantum circuits; or Zapata’s Orquestra to compose, run and analyze quantum workflows. On top of the circuits and libraries come the domainspecific application platforms. “Orchestrating and integrating classical and quantum workflows to solve real problems with hybrid quantumclassical algorithms is the name of the game for the next few years.”[xvi]
Quantuminspired software is already in operation, because these applications run on classical computers and not on quantum machines. A major example is Fujitsu QuantumInspired Digital Annealer Services.[xvii] Even on a theoretical level, quantum ideas have already been fruitful in several problem areas, where restructuring problems using quantum principles have resulted in improved algorithms, proofs, and refuting erroneous old algorithms.[xviii] Quantuminspired software is closely related to quantumready software, which can be run on suitable QCs once they are available.
How Quantum Computing Could Change the World
The industrialization of QCs has entered a critical period. Major countries and leading enterprises in the world are investing huge human and material resources to advance research in quantum computing.
The Race for Quantum Supremacy
Google perhaps prematurely used the term quantum supremacy in October 2019 when it announced the results of its “quantum supremacy experiment” in a blog[xix] and an article in Nature.[xx] The experiment used Google’s 54qubit processor, named “Sycamore,” to perform a contrived benchmark test in 200 seconds that would take the fastest supercomputer 10,000 years to do. But at some point in the future, true quantum supremacy may indeed be achieved.
Quantum supremacy was originally defined by Caltech’s John Preskill[xxi] as the point at which the capabilities of a QC exceed those of any available classical computer; the latter is usually understood to be the most advanced supercomputer built on classical architecture. At one point this was estimated be when a QC with 50 or more qubits could be demonstrated. But some experts say it depends more on how many logical operations (gates) can be implemented in a system of qubits before their coherence decays, at which point errors proliferate and further computation becomes impossible. How the qubits are connected also matters.[xxii]
This led IBMresearchers to formulate the concept of quantum volume (QV) in 2017. More QV means a more powerful computer, but QV cannot be increased by increasing only the number of qubits. QV is a hardwareagnostic performance measurement for gatebased QCs that considers a number of elements including the number of qubits, connectivity of the qubits, gate fidelity, cross talk, and circuit compiler efficiency. In late 2020, IonQ announced that it has calculated a QV of 4 million for its 5^{th} generation QC. Before this announcement, Honeywell's 7qubit iontrap QC had the industry's highest published quantum volume of 128, and IBM had the next highest QV of 64 with its 27qubit superconducting quantum machine.[xxiii] In early March 2021, Honeywell claimed to have regained the lead by achieving a QV of 512 with an updated version of System Model H1 QC.[xxiv] Alternating announcements like these from the major QC developers are likely to continue for the time being, as each compete for the title of most powerful QC.
Rather than thinking about quantum supremacy as an absolute threshold or milestone, it is wiser to think about socalled quantum supremacy experiments as benchmarking experiments for the new technology, perhaps similar to the way we came to express automobile engine power in measures of horse power. There is also an intriguing question lingering over the whole concept of quantum supremacy, which is: “How could anyone know… that a quantum computer is genuinely doing something that is impossible for a classical one to do – rather than that they just haven’t yet found a classical algorithm that is clever enough to do the job?”[xxv] It may be that the advent of quantum computing will force and inspire new developments in classical computing algorithms, something we are already seeing in the concept of quantuminspired computing software, which will be discussed further in a later section.
There is a difference between quantum advantage and quantum supremacy. Quantum supremacy is when it can be demonstrated that a QC can do something that cannot be done on a classical computer. Quantum advantage is that a quantum solution can provide a realworld advantage over using the classical approach. (It does not imply that a classical computer could not do it at all.)
There is a second meaning one could attach to quantum supremacy, which is to mean which nation will hold the technological advantage to this technology of the future. The current list of Top 500 (classical) supercomputers[xxvi] provides a good indication of where the hot spots of quantum computing will likely be, since no country or region will want to cede a hardgained advantage in classical computing. Currently, 43 percent of supercomputers are in China, 23 percent in the United States, 7 percent in Japan, and about 19 percent in Europe (including the United Kingdom but excluding Russia).
In the European Union, the European Commission founded the Quantum Flagship as a tenyear coordinated research initiative which will have at least €1 billion in funding. The longterm vision is the creation of a “Quantum Web,” defined as “quantum computers, simulators and sensors interconnected via quantum networks distributing information and quantum resources such as coherence and entanglement.”[xxvii]
The equivalent U.S. initiative is known as the National Quantum Initiative (NQI), and the $1.2 billion of U.S. government funds are going to the National Institute of Standards and Technology (NIST), National Science Foundation (NSF) Multidisciplinary Centers for Quantum Research and Education and to the Department of Energy Research and National Quantum Information Science Research Centers.[xxviii] NIST partners with the University of Colorado Boulder on quantum computing research through JILA’s Quantum Information Science&Technology (QIST).[xxix] NIST, the Laboratory for Physical Sciences (LPS), and the University of Maryland have formed the Joint Quantum Institute (JQI)[xxx] to conduct fundamental quantum research. The Joint Center for Quantum Information and Computer Science (QuICS)[xxxi] was founded in another partnership between NIST and the University of Maryland to specifically advance advances research QC science and quantum information theory.
The Chinese government is investing upwards of $10bn in quantum computing, an order of magnitude greater than the respective investments of $1.2bn by the U.S. government and the E.U. The U.K. and Japanese governments are each investing in the order of $300m, with Canada and South Korea investing about $40m each.[xxxii]
China’s multibillion quantum computing initiative aims to achieve significant breakthroughs by 2030. President Xi has committed billions to establish the Chinese National Laboratory for Quantum Information Sciences.
The implication of the difference in funding with China, is that United States is mostly relying on private investments by its tech giants to remain competitive. Time will tell if that is a wise strategy. It is not as if large tech companies in China are not investing in quantum computing too – Alibaba, Tencent, and Baidu are all known to be heavily investing in the technology. According to some metrics, China has already gained an early advantage by accumulating more quantum computingrelated patents than the United States.[xxxiii] In 2019 Google announced that its a QC performed a particular computation in 200 seconds that would take today’s fastest supercomputers 10,000 years. But in December 2000, Chinese researchers at the University of Science and Technology in China (USTC) claimed that their prototype QC (based on photons) is 10 billion times faster than Google’s.[xxxiv]
The Chinese desire to lead the world on quantum computing is not purely motivated by a desire for industrial competitiveness and economic power. Threat assessments[xxxv] point to Chinese quantum research and experiments in defense applications such as:
· Using entanglement for secure longdistance military communications, e.g. between satellites and earth stations
· Quantum radar that could nullify current U.S. advantages in stealth technology against conventional radars
· Quantum submarine detection to ranges of over five kilometers that would limit the operations of U.S. nuclear submarines
Competing Architectures
Quantum computers are very hard to build. They require intricate manipulations of subatomic particles, and operating in a vacuum environment or at cryogenic temperatures.
The state of quantum computing resembles the early days of the aircraft and automobile industries, when there was a similar proliferation of diverse architectures and exotic designs. Eventually, as quantum technology matures, a convergence can be expected similar to what we have seen in those industries. In fact, the arrival of such a technological convergence would be a good measure of a growing maturity of quantum computing technology.
There are a number of technical criteria[xxxvi] for making a good QC:
· Qubit must stay coherent for long enough to allow the computing to be completed in the state of superposition. That requires isolation because decoherence occurs when qubits interact with the outside world
· Qubits must be highly connected. This occurs through entanglement and is needed for operations to act on multiple qubits
· Highfidelity operations are needed. As pointed out above, classical digital computers rely on the digital nature of signals for noise resistance. However, since qubits need to precisely represent numbers that are not just zero and one during the computation state, digital noise reduction is not possible and the noise problem is more analogous to that in an oldfashioned analog computer. Since noise cannot be easily prevented and must therefore be mitigated, the focus of current research is on noisecorrection techniques
· Gate operations must be fast. In practice, this is a tradeoff between maintaining coherence and highfidelity
· High scalability. It should be obvious that QCs will only be useful when they can be scaled large enough to solve valuable problems
Currently, the two quantum technologies showing the greatest promise and attracting the most interest and investment dollars are superconducting qubits and trapped ions. These and other more nascent or theoretical technologies are presented in Table 1, along with the main proponents in each technology.
Table SEQ Table \* ARABIC1. Qubit Technologies and Main Proponents
Technology 
Main Proponents 
Superconducting qubits (called transmons by some) are realized by using a microwave signal to put a resistancefree current in a superposition state. This technology has fast gate times and the advantage of more proven technologies – superconducting circuits are based on wellknown complementary metaloxide semiconductor technology (CMOS) used in digital computers. But superconducting qubits have fast decoherence times and require more error correction. Superconduction requires cooling to a temperature very close to absolute zero. The technology is considered to be highly scalable. 
· IBM · Rigetti · DWave · Alibaba · Intel · NEC · Quantum Circuits · Oxford Quantum Circuits 
Ion Trap QCs work by trapping ions[12] electric fields and holding them in place. The outermost electron orbiting the nucleus is put in different states and used as a qubit. Ion Trap qubits have longer coherence times and can operate with minor cooling, but do require a high vacuum. Thought the first quantum logic gate was demonstrated in 1995 using trapped atomic ions, at a system level this technology is less mature and require progress in multiple domains including vacuum, laser, and optical systems, radio frequency and microwave technology, and coherent electronic controllers 
· IonQ · Honeywell · Alpine Quantum Technologies

Photonic qubits are photons (light particles) that operate on silicon chip pathways. Such qubits do no require extreme cooling, and silicon chip fabrication techniques are wellestablished, making this technology highly scalable. 
· PsiQuantum · Xanadu 
Neutral atoms are similar to Ion Traps but do not use ionized charges to keep the qubits in place, but laser “tweezers” instead. The technology would share the advantage of longer coherence times with Ion Traps but also the same challenges to scaling up. The technology is considered unproven and highly nascent. 
· Atom Computing · PASQAL · QuEra 
Silicon qubits entails making ions by adding an electron to silicon. The electron’s state is then controlled using microwaves. If proven this approach could support longer coherence times than the superconducting approach. It has the advantage of working with silicon and building on decades of semiconductor industry experience. The technology is still nascent. 
· Intel · Silicon Quantum Computing 
Topological qubits would operate on different principles by utilizing exotic new quasiparticles such as Majorana Fermions and Anyons. The hope is for long coherence times and higher fidelity based on the theory. However, the existence of these particles has not been experimentally confirmed, so this technology is not even nascent, but purely theoretical. 
· Microsoft 
Note: There is not yet a universallyaccepted measure to compare the computing power of the different technologies. That is because obvious measures such as calculation cycles (qubit lifetime / gate operation time) is skewed by the current infidelities of gate operations, and the varying overheads imposed by errorcorrection schemes. For all gatebased technologies, clock speeds will also be limited for the foreseeable future due to the need for fault tolerance.[xxxvii]
Industry Structure and Major Players
The QC hardware covered above is of course only one layer of the quantumcomputing stack. Immediately above the hardware is the systems layer, and on top of the systems layer are the software and applications layers. At the very top is the systems layer, which is most commonly cloudenabled these days.[xxxviii]
Only a relatively small proportion of firms in the quantum computing ecosystem actually build working QCs, since that requires major resources and highlyspecialized skills in quantum physics and hardware engineering. Many of the companies that selfidentify as quantumcomputing companies are actually on the software and services side. It is more common for providers of QC hardware to move upwards in the ecosystem by adding software and services, than it is for software and services players to attempt to move downward by developing their own quantum computing hardware.
Hardware makers typically enable access to their QCs over the internet and through the cloud, often through subscription plans, sometimes for free. The cloudbased offering is typically a hybrid quantumclassical computing system, which breaks up the problem to be solved into parts that can be solved by conventional computers and parts that are best solved by the QC. This situation resembles the early days of classical computing when only a few computers were available, these computers filled whole rooms, and they had to be shared among many users.
Below are short profiles of some of the major players in quantum computing, divided between mainly hardware providers versus software platform and solution providers.
Hardware and Systems
The major competitors in the QC hardware space each make their own QCs with competing architectures and specifications. The most significant of these general systems are made by the large companies IBM, Google, and Honeywell, and the startups Rigetti, and IonQ. DWave, another startup, makes and sells hybrid QCs that are specialized for quantum annealing and particularly wellsuited to solving optimization problems, such as those of interest to the finance industry.
Regardless of how this major setback is resolved by Microsoft, it draws attention to the extremely high technical risks in trying to build computers upon technologies that are not only unproven, but for which the fundamental physics are not even wellestablished yet.
DWave.[xxxix] DWave is Canadian startup and a foremost proponent of quantum annealing, which includes optimization for finance applications. It is a pioneer in selling QCs to other organizations.[13] These QCs are packaged as complete systems within physical enclosures measuring 10' x 7' x 10' (L x W x H) that each houses the complete cryogenic refrigeration, shielding, and I/O systems to support a single thumbnailsized QPU (quantum processing unit). Its latest model, the DWave Advantage QC, has 5,000 qubits up from the 2,000 qubits of the previous model. The company claims that It can solve problems with up to 10,000 variables.[xl] DWave also lets customer access its quantum hardware through its “hybrid solver,” which breaks up the computing task into components, some of which is solved by its QC, and the rest that is solved by traditional computers over a cloudbased interface. In October 2020, The Globe and Mail reported that DWave was forced to undertake a costly refinancing round that devalued the shares of several longterm investors. DWave has struggled to generate revenue with only a handful over buyers for its machines.[xli]
Google.[xlii] Google (Alphabet Inc.) showed their strategic commitment to quantum computing with their fall 2019 announcement that their QC had achieved socalled quantum supremacy, surpassing classical supercomputers at a particular task. Their current QC, “Sycamore,” has a 54qubit processor with fast, highfidelity quantum logic gates. Already a leader on AI, Google is at the forefront of exploring AIrelated applications of quantum technology. Google researchers have published a great number of articles on quantum computing.
Honeywell Quantum Solutions.[xliii] In October 2020, Honeywell announced its next generation QC, System Model H1, using Honeywell’s differentiated quantum chargecoupled device1 (QCCD) trappedion technology.[xliv] This QC initially has 10 connected physical qubits for a quantum volume of 128 (claimed to be the highest in the industry). System Model H1 may be directly accessed via a cloud application programming interface (API), as well as through Microsoft Azure Quantum, and through certain channel partners including Zapata Computing and Cambridge Quantum Computing.[xlv] Honeywell has a unique midcircuit measurement capability and reset function.[xlvi]
IBM Quantum.[xlvii] IBM is one of few remaining manufacturers of classical mainframe computers. IBM Q systems is based around the transmon qubit. IBM prefers the term quantum advantage over quantum supremacy, and they specific the power of their QCs as quantum volume (QV) not qubits. (Quantum advantage is where a problem can be solved faster on a QC than on a classical supercomputer so it makes sense to use over classical computers.) QV takes into account the number of qubits, connectivity, and gate and measurement errors. The latest QV to be announced by IBM was 32. While they have made a 53qubit system, IBM also operates 16 and 20qubit systems.[xlviii] IBM is intent on building a cloudenabled ecosystems of quantum partners, and are actively promoting coding skills for quantum computing through annual coding challenges.[xlix] This is consistent with IBM’s longstanding strategy of supporting opensource software tools. JPMorgan Chase and Barclays were charter members of the IBM quantum computing network.[l]
Intel. Intel is a component maker for QCs, not a system builder. It is determined to continue its longstanding leadership in the silicon processor market with quantum computing chips. Intel Labs have developed “Horse Ridge,” a firstofitskind cryogenic control chip (meaning it operates close to the qubits inside the cryogenic freezer but at a slightly higher temperature), which will enable the control of multiple qubits. Horse Ridge brings the qubit controls into the quantum refrigerator — as close as possible to the qubits themselves and reduces the complexity of quantum control engineering from hundreds of cables running into and out of a refrigerator to a single, unified package operating near the quantum device.[li] Intel’s research partner is QuTech at TUDelft. It recently announced the ability to control multiple qubits with Horse Ridge.[lii]
IonQ.[liii] IonQ is a startup, which introduced the first commercial trappedion QC. The company has a fiveyear roadmap and plans to deploy rackmounted modular QCs small enough to be networked together in a datacenter by 2023. That will result in a quantum advantage in building for machine learning, the company expects. IonQ then plans to achieve broad quantum advantage by 2025. In late 2020, IonQ announced a new 32qubit QC available in private beta, and two nextgen computers also in the works. IonQ has made progress towards error correction, having incorporated a new error correction code that only uses 13 qubits, and is also working on a capability to do midcircuit measurement in future.[liv] IonQ counts AWS, Samsung, Lockheed Martin, HPE, Hyundai Motor, and others as investors. The Wall Street Journal recently reported that IonQ planned to go public via a merger with a specialpurpose acquisition company in a deal worth about $2 billion.[lv]
NEC.[lvi] This major IT company with a strong history in classical mainframes and supercomputers is developing quantum annealing machines with grants from the Japanese government. In 2020 it began a joint QCdevelopment project with DWave. Recently it entered a partnership with a ParityQC, an Austrian QC startup, to develop quantum annealing solutions. NEC will be combining this technology with its own superconducting parametron quantum devices, with the aim of building quantum annealers by 2023. Major applications targeted by NEC are financial portfolio optimization and manufacturing logistics and planning.
PsiQuantum.[lvii] This SiliconValley startup is developing a largescale linear optical QC (LOQC), with a target of 1 million qubits. The company believes that photonic technology is the only route to the large number of qubits needed to fully scale up an errorcorrected QC. PsiQuantum’s QC is based on photonic technology that incorporates extensive error correction and can be manufactured in a standard semiconductor wafer fab. Prominent investors are Microsoft and Blackrock.
Rigetti Computing.[lviii] This Berkeley, California based startup presents itself as an integrated systems company. It builds QCs and the superconducting quantum processors that power them. Through their Quantum Cloud Services (QCS) platform, their QCs can be integrated into public, private, or hybrid clouds.
Silicon Quantum Computing (SQC).[lix] SQC is an Australian startup launched in 2017 as a spinoff from the University of New South Wales (UNSW) Sydney to siliconbased QC. It received funding from both the federal and provincial governments as well as the Commonwealth Bank of Australia (CBA) and Telstra. SQC is building a QC based on donor spin qubit technology, which is a phosphorus donor embedded in a silicon structure originally conceived at UNSW. Potential advantages of donor qubits are high fidelities (more than 99%) with long coherence times measured in seconds for the electron spin states.[lx] CBA invested over $14 million in the startup.[lxi]
Xanadu.[lxii] Xanadu is a Canadian quantum startup funded by SeriesA financing and government grants from the U.S. DARPA and Canada’s SDTC agency. Xanadu is betting that photonics offers the most viable approach towards universal faulttolerant quantum computing. It claims to develop photonic chips that will result in nearterm QCs. Xseries chips are made from silicon and silicon nitride. It is the first company to offer cloud access to photonic computers. Currently, cloud customers may access its 8 or 12qubit photonic QCs, but a 24qubit processor is in the pipeline already. The aim is to double the number of qubits available every six months.[lxiii] Early clients include Creative Destruction Lab, Bank of Nova Scotia, Bank of Montreal, and the U.S. Oak Ridge National Laboratory.[lxiv]
Addendum on QC Hardware:
In December 2020, Bloomberg reported that Amazon was laying the groundwork to build its own QC and had started to hire a hardware team.[lxv]
Cisco seems to be getting ready to enter the QC field. The company is making strategic hires and looking to collaborate with quantum researchers at universities.[lxvi]
Microsoft planned its own QC, based on the Majorana fermion (a class of elementary particle), which would make for ideal qubits because they would be longer lived and less prone to noise. By placing its hope on the Majorana fermion, Microsoft hoped to leapfrog rivals IBM and Google, whose QCs are based on more established technology. However, this endeavor was thrown into turmoil in early 2021, when a 2018 paper in Nature confirming the existence of the Majorana by Dutch researchers was retracted and corrected.[lxvii] (The new article[lxviii] containing the retraction was posted in January 2021.) Microsoft is expected to regroup around betterknown technologies such as superconducting qubits or trappedion systems. This is, however, a major setback for Microsoft, who was trying to beat the competition by betting on a more esoteric computer model. But now quantum computing experts suggest that a computer using Majorana particles may be as much as 30 years on the future.[lxix]
However, QCs based on iontrap providers, superconducting qubits, photonics, neutral atoms, siliconbased and annealing are left in the game.
Software Platforms and Solutions
The two main competing platforms in North America for general cloudbased quantum computing solutions are those from Microsoft and Amazon, with the platform from the CanadianSpanish startup, Multiverse, being particularly relevant to the financial services sector due to its focus on financial applications. Superconducting, trapped ion, and quantum annealing (not gatebased) are offered through the Amazon Braket and Microsoft Azure cloud platforms. For now, photonic QCs are provided only on Xanadu's cloud.
1Qbit.[lxx] This Vancouverbased startup provides hardwareagnostic QC platforms. It has applications for material science, optimization and market sentiment measurement. RBS was a major early investor in 2015 and has contributed to a new round in 2020.[lxxi] Allianz is another a major investor.[lxxii]
Alibaba Cloud. Alibaba Cloud, the cloud computing arm of Alibaba Group and Chinese Academy of Sciences (CAS) have partnered to launch a quantumcomputing cloud service. The CAS QC is accessible through Alibaba’s cloud service. The first quantum laboratory in China was established in 2015 as a joint venture between Alibaba and CAS.[lxxiii] Alibaba Cloud Quantum Development Platform (ACQDP) is their simulatordriven development tool for quantum algorithms and QCs. Damo, Alibaba’s Quantum Lab, also conducts hardware research into quantum processors, quantum memory, and quantum computing systems.[lxxiv]
Amazon Braket.[lxxv] Amazon Braket is a fullymanaged cloudbased (over AWS) quantum computing service that offers access to quantum computing hardware from DWave, IonQ, and Rigetti. Amazon focuses on three main application areas: molecular simulation, optimization, and quantum machine learning. Collaboration with other organizations are facilitated through the Amazon Quantum Solutions Lab. Amazon Braket provides access to several QCs from other companies, including hardware based on superconducting qubits from Rigetti, iontrap QCs from IonQ, and quantum annealing technology based on superconducting qubits from DWave.
Baidu.[lxxvi] Baidu established the Institute for Quantum Computing early in 2018, and focused on building a bridge between AI and quantum computing. Paddle Quantum is a quantum machinelearning toolkit to facilitate the rapid building and training of neural network models. It is based on Baidu’s deeplearning platform PaddlePaddle.[lxxvii]
Cambridge Quantum Computing (CQC).[lxxviii] CQC builds architectureagnostics QC solutions focusing in chemistry, machine learning, cybersecurity and finance. It recently announced version 0.7 of its tket> software platform, that removes all license restrictions for use of the tket’s Python module (known as pytket) making that software free to use.
CogniFrame.[lxxix] This startup operates at the intersection between machine learning and quantum computing. It solves NP hard and other complex optimization problems. Through partnerships with Toshiba and DWave it offers optimization and simulation solutions that range from quantuminspired to hybrid to pure quantum. It caters to the financial industry with a “Financial Services Operation Layer” that sits on top of the quantum cloud.
Chicago Quantum.[lxxx] The company is very active in researching the application of quantum algorithms to optimize financial portfolios. It has developed a quantum algorithm, loosely based on the Sharpe ratio,[14] that picks attractive stock portfolios based on one year of historical pricing data. The algorithm is run on the DWave quantum annealer and on Chicago Quantum’s own classical computers. The company occasionally publishes stocks picks selected with its algorithms. It recently published an efficient portfolio of 128 stocks selected from 3,514 stocks, as well as smaller portfolios and stocks with positive momentum and low risk.[lxxxi]
Microsoft Quantum.[lxxxii] Microsoft’s Azure Quantum, is a full set of publiccloud ecosystem quantum solutions, which has recently been opened for public review. It is intended for learning and solution building by developers, researchers, systems integrators, and customers. The ecosystem gives customers access to diverse quantum software and hardware solutions, a network of leading quantum researchers and developers, a robust resource library, and flexible selfservice or tailored development programs.[lxxxiii] Microsoft’s opensource quantum development kit (QDK), a software development kit that allows development of new algorithms with Q#, a quantumfocused highlevel programming language. QDK has a GitHub repository with opensource Q# libraries and sample programs.[lxxxiv] Microsoft’s main partners are Honeywell, IonQ, QCI, Toshiba, an 1Qbit.
Multiverse.[lxxxv] This quantum computing startup with offices in San Sebastian, Spain and Toronto, Canada focusses on quantumcomputing software solutions for the financial industry. They claim that their software runs on all quantum hardware technologies. In addition, Multiverse offers quantuminspired solutions such as tensor networks, digital annealing, qi optimization, and artificial intelligence.[lxxxvi]
QC Ware.[lxxxvii] This enterprisesoftware startup has a large team of quantum algorithm experts. It aims to provide solutions that can run on nearterm quantum hardware. It partners with hardware providers DWave, IBM, IonQ, and Rigetti.
Quantum Computing Inc (QCI).[lxxxviii] QCI is a startup developing platformagnostic software for quantum computing. Its platform, named “Mukai,” enables users to use quantuminspired methods on classical computers and quantumready methods on QCs. The Mukai platform supports developing applications to address complex optimization problems that are NPhard, often involving multidimensional solution spaces with thousands if not hundreds of thousands of variables. Its Quantum Asset Allocator (QAA) enables fund managers to use quantuminspired techniques to solve the NPhard problems standing in the way of optimal portfolio allocation. QCI claims that QAA can quickly calculate optimal or nearoptimal interactive solutions for complex financial asset allocation problems on classical computers.[lxxxix]
Toshiba. Toshiba is a leader in quantuminspired computing, using a deeper understanding of quantum mechanics to run optimizers on classical hardware. Toshiba’s claims that its Simulated Bifurcation Machine (SBM), which is derived from research on quantum bifurcation machines, is a readytouse Ising[xc] machine which can solve largescale combinatorial optimization problems at high speed. In September 2020, Toshiba announced that it was joining Microsoft’s Azure Quantum ecosystem. As a consequence, the SBM is accessible over the Azure Quantum cloud service, allowing it to harness the GPU resources in the Azure cloud.[xci] The SBM is also available over Amazon’s AWS platform.[xcii] Toshiba is aiming to be a large player in the nascent global quantum encryption market, and is planning to launch a quantum cryptography service by 2025. Conventional cryptography, which relies on the impracticality of factoring very large numbers, is at a real risk of getting cracked when sufficiently large QCs, which will far outperform classical supercomputers in such calculations, become available.[xciii]
Zapata Computing.[xciv] Zapata’s quantumcomputingpowered workflow solution, Orquestra,[xcv] that automates the workflow management of supply chain optimization, materials discovery, and asset allocation optimization. The software solution is designed to be hardwareagnostic, so that it will work with any major QC. Bosch, a large German Tier1 automotive supplier, is an investor in Zapata.
Academic and Other Research Centers
The U.S. National Institute of Standards and Technology (NIST) has been at the center of quantum computing research since the early 1990s. Partnerships between NIST and public universities have created research institutes such JILA[xcvi] (with the University of Colorado Boulder) and the Joint Quantum Institute (JQI)[xcvii], formed as partnership between NIST, the University of Maryland, and the Laboratory for Physical Sciences. There is a long and growing list of universities that have quantum computing research groups.[xcviii] According to a recent list compiled by the Quantum Daily,[xcix] the top 12 universitybased QCresearch organizations are:
1. The Institute for Quantum Computing[c] at the University of Waterloo
2. Oxford Quantum[ci] at the University of Oxford
3. The Harvard Quantum Initiative[cii]
4. The Center for Theoretical Physics[ciii] at MIT
5. The Centre for Quantum Technologies[civ] at the National University of Singapore and Nanyang Technological University
6. The Berkeley Center for Quantum Information and Computation[cv] at the University of California Berkeley
7. The Joint Quantum Institute (JQI) [cvi] at the University of Maryland
8. The Division of Quantum Physics and Quantum Information[cvii] at the University of Science and Technology of China (USTC)
9. The Chicago Quantum Exchange (CQE)[cviii] at the University of Chicago
10. The Quantum Science Group[cix] at the University of Sydney, Australia
11. The Quantum Applications and Research Laboratory (QARLab)[cx] at LMU Munich
12. Quantum Information & Computation[cxi] at the University of Innsbruck
Major Alliances
Google operates its own complete QC stack, but in most other cases QC companies partner to complement one another’s capabilities.
Some major quantum computing partnerships and alliances are:
· Microsoft Quantum Network – Microsoft (software and Azure cloud services) partnering with Honeywell Quantum Solutions (hardware) and IonQ (hardware), as well as Toshiba (software – quantuminspired), 1Qbit (software), and QCI (i.e. Quantum Computing Inc. – software)
· IBM Quantum Network – IBM hardware (20 QCs, of which 10 are available free) partnering with 140 participating organizations that include Samsung, JPMorgan, Barclays, and Daimler
· Honeywell – Honeywell has invested in both Zapata and CQC
· DWave with Multiverse, and NEC
· 1Qbit with Azure Quantum (Microsoft)
· NEASQC (NExt ApplicationS of Quantum Computing),[cxii] a European consortium for NISQ quantum computing. HSBC recently joined as the first financial services organization.[cxiii]
Potential FirstGeneration Applications of Quantum Computing
In advance of a fullscale QC being available, there may be QC applications for and QC influence on the financial sector in the near term. There are a few paths for these. The most prominent one is the combination of smallscale quantum computing with classical computing in socalled hybrid quantum computing. Another is the potential implementation of quantuminspired computing algorithms on classical computer hardware. Quantuminspired computing is based on the idea that a problem that is hard to solve on a classical computer may become easier to solve it is reframed in a way that is inspired by quantum physics. But the execution is still classical.
A good practical definition to distinguish between classical and quantum computing is:
If a solution leverages the quantum mechanical principles of superposition and entanglement it can be called a quantum solution, or at least a hybrid classical/quantum solution. If the solution does not utilize these phenomena, we will call it a classical solution even though it may not look like a normal classical computing solution.[cxiv]
Microsoft’s quantuminspired algorithms are designed to run on classical computers, and they have already had success with use cases such as improved cancer detection in radiology scans. Microsoft claims that its quantuminspired algorithms are “particularly useful for optimization problems — which involve sifting through a vast number of possibilities to find an optimal or efficient solution — that are so complex and require so much computing power that current technologies struggle to solve them.”[cxv] Microsoft also claims orders of magnitude of performance acceleration in Azure from recasting hard computational problems into quantuminspired solutions. They have been collaborating with Willis Towers Watson (a global advisory, broking and solutions firm) to explore how such algorithms may help in the areas of risk management, financial services, and investing.[cxvi]
Quantum Computing Inc. (QCI) is another example of a standardcomputerhardware implementation – they provide a software platform, called Mukai,[cxvii] that enables users to leverage the latest breakthroughs in quantum computing by running quantuminspired software on classical computer hardware. In fact, one of the applications is a quantum asset allocator (QAA) that uses quantuminspired techniques to solve NPhard problems standing in the way of optimal portfolio allocation. The company claims that QAA can solve NPhard problems including cardinality constraints and minimum buyin constraints. Mukai software is also quantumready and can thus be used on true QCs as these become available.
Toshiba’s SBM is also an example of a standardcomputerhardware implementation, as it runs on generalpurpose classical computers, claiming to largescale combinatorial optimization problems at high speed, 100 times faster than simulated annealing methods.[cxviii]
Fujitsu’s Digital Annealer is an example of a specialpurposecomputerhardware implementation. Its hardware was “purposefully designed for more efficiently solving larger and more complex combinatorial optimization (CO) problems.”[cxix]
Industries that Could Be Disrupted
A BCG analysis[cxx] from 2018 identified future use cases of quantum computing across five major sectors:
· Hightech use cases include AI and machine learning, cybersecurity, search, and bidding strategies for online ads
· Industrial goods use cases include logistics scheduling, product distribution, autonomous driving, traffic distribution, semiconductor chip layout optimization, aerospace fault analysis, and materials science applications
· Chemistry and pharma use cases include faster drug discovery, genomics, catalyst and enzyme design, and improved diagnostics capability (e.g. MRI)
· Finance use cases include trading strategies, portfolio optimization, asset pricing and risk analysis (Finance use cases will be covered in more detail in the next section)
· Energy use cases include energy distribution, network design, and oilwell optimization
A McKinsey analysis[cxxi] of 100 use cases by industry had the following distribution, which is a proxy for the impact of quantum computing and for which industries will be most affected:
· Finance, in first place with 28 use cases
· Global energy and materials, in second place with 16 use cases
· Advanced industries (incl. aerospace and automotive) in third place with 11 use cases
In early 2021, Forbes reported that the leading quantum computing proofs of concept (POCs) were in AI/machine language, financial services, molecular simulation, material science, oil/gas, security, manufacturing, transportation/logistics, IT, and healthcare (pharmaceuticals). Many of the AIrelated applications also apply to financial services, with applications in “trading strategies / treasury & asset management, option pricing, new financial models, portfolio optimization, predicting risk and uncertainty, customer product targeting from behaviors in realtime.” If financial services companies are currently too risk averse to grow their customer base because of limitations in computation ability, quantum computing could enable the industry to reach two billion unbanked people by reducing the $40 billion lost annually from fraud/poor data analysis and reducing 80 percent false positives. The conclusion is that the “largest near and farterm benefits” of quantum computing will be in financial services, which explains why industry heavyweights such as JP Morgan, BBVA, and Goldman Sachs are actively exploring quantum computing.[cxxii]
How Quantum Computing Could Change Finance
There are two ways to think about the influence of quantum computing on finance. The first, and most obvious, is that the special abilities of quantum computing will enable solving certain types of problems that even the most powerful classical computers cannot do in the time needed. Where companies currently run largescale analytics computations for risk management, forecasting, planning, and optimization, quantum computing could change future operations and even strategy. If an operation is not just run faster, but a million times faster, executives should ask themselves what fundamental changes in business operations become possible.[cxxiii]
However, there is also a second, and perhaps more important way the quantum computing can influence finance and economics over time. And that is to change the way problems are shaped, structured, and modeled. We often forget that economics in its infancy as a social science was heavily influenced by the prevailing physics of its time, which was thermodynamics. The paradigms of partial and general equilibrium in economics were borrowed from thermodynamics, the science that made the steam power of the first industrial revolution possible. And so, classical and neoclassical economics theories were based on the paradigm that everything will always want to go back to an equilibrium, and any departures from that equilibrium are only temporary. Much of the ongoing dissatisfaction with neoclassical economics theory comes from the fact that the real world does not seem to behave that way.
David Orrell has coined the term Quantum Economics in an eponymous book, where he expounds on his idea that quantum theory holds great promise as a new and better way of modeling, for example, stock price movements. As Orrell states:
Perhaps the most useful contribution of quantum finance will be to change the way we think about the financial system. Instead of seeing stock prices as particles that are randomly jostled from their stable resting place by interactions with many independent investors, we begin to see them as fundamentally indeterminate quantities. In quantum physics, particles can never be perfectly still because that would violate the uncertainty principle.[cxxiv]
This is an ambitious longterm vision for changing the very foundations of current economic models. However, this paper is primarily concerned with the nearterm utility of quantum computing for the finance industry. Therefore, the discussion that follows will be focused on QC’s promise to solve particular problems in finance, in areas where the performance of QCs can already, or will shortly, exceed those of classical computers.
Computational Problems in Finance That Can Be Solved
In a recent report on quantum computing in the financial industry,[cxxv] BCG estimated that quantum computing could add $70 billion in operating income for financial services companies after the technology has sufficiently matured. While the technology is still in its infancy, a rapid rise in financialservices applications is plausible. Quantum computing has the potential to be a game changer for the competitiveness of banks and other financial institutions. The three capabilities it is projected to revolutionize are:
· Optimization. Current optimizations have to use unrealistic assumptions to simplify scenarios so that they are solvable problems for classical computers. But QCs promise to solve problems in which the full complexity of the real world is captured.
· Simulation and pricing. Monte Carlo simulations can take days or weeks to run, but QCs promise to run them in real time.
· Machine learning. Machine learning is currently limited by the inability of classical computers to handle complex and computationally intensive algorithms. Quantum computing will overcome this constraint and promise to make large complex systems understandable to machines in a short period of time.
IBM Quantum confirms that the types of financialindustry problems that quantum computing can solve may be divided into these three capability categories. For example, portfolio optimization and diversification fall into the optimization category, option pricing and portfolio risks in the simulation and pricing category, credit scoring and fraud detection in the machinelearning category. IBM has already developed quantum algorithms in these areas.[cxxvi]
According to McKinsey, the rationale for using QCs is based on the need for financial institutions to crunch large and unstructured datasets. It sees powerful use cases in capital markets, corporate finance, portfolio management, and encryptionrelated activities. Quantum computing can offer real competitive advantage in an increasingly commoditized world. McKinsey sees four capital markets industry archetypes: sellers, buyers, matchmakers (including trading platforms and brokers), and rule setters. Buyers and rule setters require more complex models. For example, quantdriven hedge funds aim to profit through analytical complexity could be a natural constituency for ultrapowerful processing. Large banks, which take on multiple roles in financial markets, are also significant early experimenters. Areas where artificial intelligence techniques such as machine learning have already improved traditional classification and forecasting are ripe for quick wins.[cxxvii]
The specific finance applications of quantum computing tools may also be grouped by finance vertical, according to Multiverse (a quantumalgorithm provider for the financial industry): [cxxviii]
· Capital markets
o Portfolio optimization
o Optimal trajectory detection for investment/divestment
o Training of traders on best trajectories
o Trend/Anomaly detection for trading and investing
o Market crash predictions
· Credit and risk
o Credit scoring on automated lending
o Loan portfolio supervision and alert
o ALCO/ALM matching optimization
o Capital allocation optimization
o Customer buying propensity
o Insurance feature selection (automated)
o Insurance individual pricing
· Fraud detection
o Creditcard fraud
o Instant money transfer fraud
o Antimoneylaundering
o Tax fraud detection
As previously mentioned, a recent analysis of 100 hundred use cases for potential nearterm value creation was done by McKinsey. There were 28 use cases in Finance – the most of any industry analyzed – and the value at stake in both the medium and long term was high. Finance is clearly what McKinsey calls a “firstwave industry” for quantum computing, and the authors of the report made a call to action to executives in the firstwave industries:
We believe that industries such as finance, travel, logistics, global energy and materials, and advanced industries will start reaping significant value from the hybrid classical/quantum approach in the early 2020s. Business leaders in these firstwave sectors need to develop a quantum strategy quickly or they will be left behind by innovative companies such as Barclays, BASF, BMW, Dow, ExxonMobil, and others that already have taken strategic steps into quantum computing.[cxxix]
This conclusion is shared by a recent Fitch Solutions analysis of cloud computing technology megatrends to 2050. The report finds that quantum computing has the greatest disruption potential; that quantum computing is closer to reality than it has ever been, with expectations it might happen within the next ten years; and that the winners will be the companies and governments investing in the technology.[cxxx]
In a recent report on the impact of emerging technologies in financial services – the result of a collaboration between the World Economic Forum (WEF) and Deloitte – the potential of quantum computing to “solve a narrow, but critical range of problems significantly more efficiently than classical computers”[cxxxi] is also recognized. Portfolio optimization, credit scoring, risk analysis, and cryptography are the major areas of impact seen by the WEF.
In only the past couple of years, several major companies in the financial industry have announced quantum computing experiments. For example, BBVA has been exploring portfolio optimization, CaixaBank risk management, and JPMorgan asset pricing. Many financial institutions are also actively partnering with quantum players or networks, or even directly investing in startups. The list of financial firms involved in quantum computing is getting longer. It includes global FIs such as Allianz, Barclays, Citigroup, Goldman Sachs, HSBC, JPMorgan, and Mizuho, but also major regional or national players such as ABN Amro Bank, Anthem, Bank of Canada, BBVA, BMO, BNP Paribas, CaixaBank, Commonwealth Bank of Australia, NatWest Group, Nomura, RBS, Scotiabank, Standard Chartered, UBS, and Wells Fargo.
Financial services companies that may find quantum computing too exotic could enter the field by investing in quantuminspired computing. In a survey and study commissioned by Fujitsu,[cxxxii] 70 percent of survey respondents made aware of the abilities of the Fujitsu Digital Annealer to solve combinatorial optimization problems without the need for a QC stated that it would accelerate their journey to a quantum future.
Trading, Portfolio and Other Optimization Applications
Quantitative investors are hoping that quantum computing will solve many of the current computational problems in portfolio optimization, arbitrage strategy, and trading cost minimization. Classical computers encounter problems with the complex computing load imposed by adding more realistic assumptions and constraints to models:
Adding noncontinuous, nonconvex functions such as interest rate yield curves, trading lots, buyin thresholds, and transaction costs to investment models makes the optimization surface so complex that classical optimizers often crash, simply take too long to compute, or, worse yet, mistake a local optimum for the global optimum. To get around this problem, analysts often simplify or exclude such constraints, sacrificing the fidelity of the calculation for reliability and speed. Such tradeoffs, many experts believe, would be unnecessary with quantum combinatorial optimization.[cxxxiii]
Researchers have developed a number of techniques that aim to use various quantum approaches to solve portfolio problems. The main approaches put forward are:
· Algorithms based on quantum annealing
· Gatebased quantum algorithms
· Quantuminspired models based on tensor networks
Whichever approach is used, there is a need to translate the realworld problem into a polynomial unconstrained binary optimization (PUBO) expression. This is not a trivial problem in itself. In a recent preprint article, researchers from Zapata propose quantum enhanced optimizers (QEOs), a type of blackbox solver which is independent of the details of the objective function, which can scale to large problems when combinatorial problems are intractable due to realworld complexity.[cxxxiv]
The quantum computing technology that is getting the most current and nearterm attention in the finance industry is quantum annealing, due to annealing’s natural strengths in modeling and solving optimization problems. DWave Systems is a leader in the field. They focus on NPhard optimization problems, and have been pursuing a highly empirical approach to these problems. In the case of an NP problem, it is possible to verify the solution in polynomial time. In the case of NPhard problems, it is possible to find a solution but without knowing whether is optimal or not. (The distinction between P and NP problems is a theoretical one. DWave’s Catherine McKeough points out that in the 30 years since P vs NP was posed, no one has proposed an experiment to settle it, likely because such an experiment cannot be designed.[cxxxv]) For an optimization problem, the lowest point is the ground state, but there are also local minimums with higher state neighbors. Problem containing several of the latter are choppy which makes it hard to find the optimal solution.
Adiabatic quantum computing (AQC) is an alternative to the gate model, but AQC is polynomially equivalent to the gate model. Bot approaches are universal (Turing equivalent). Quantum annealing algorithms can be run either on classical computers or on AQC platforms. The quantum version provides probabilistic results, which means that a solution has to be run a hundred or a thousand times to provide an answer distribution. Compared to the polynomial solutions running on a classical computer, the annealing solution running on DWave’s QC converges much faster, and there is no gain in running it for a long time.[cxxxvi]
Sam Mugel, the CTO of Multiverse, a quantum software company that runs its algorithms on DWave hardware, sets out a few practical criteria for selecting a good problem that can be better solved on current QCs than on classical computers. First, the input questions should be quite small, so that only a low number of qubits are needed. Second, there should be many possible solutions or states to explore. Third, it should be a high value problem. And fourth, it is a good choice when the current best classical solution is a bruteforce solution. This is typically the case when the problem is known to be NPhard with no classical method of finding the optimal solution. By combining classical and QCs in socalled hybrid solvers, big gains can be made. A powerful classical computer with ample resources can manage the problem and data management, shooting off to the QC a small, extremely difficult problem that quantum computing is best at solving. Annealingbased systems are ready for use today.[cxxxvii]
Notable Case Studies and POCs
A number of recent case studies, experiments, and proofs of concept (POCs) conducted between financial institutions and QC companies have been made public. These cases reveal not only what is being done and by whom, but sometimes also the rationale of the financial institutions involved:
Barclays created an internal QC working group in 2017, with modelers running programs on IBM’s quantum cloud.[cxxxviii] In one example, they worked with IBM on a POC for a quantum algorithm that can be used in securities transaction settlement.[cxxxix] Transaction settlement (which trades to settle when) is computationally complex and difficult to optimize because of a combination of various legal constraints and optionality in collateralizing assets and utilizing credit facilities. While it takes a long time for a classical computer to solve, the researchers published a joint paper[cxl] describing how a QC with a small number of qubits could execute the most complex parts of the algorithm.
BBVA is following six lines of research, working hand in hand with Spain’s Senior Council for Scientific Research (CSIC), Accenture, Fujitsu, Zapata Computing, and Multiverse. While the project is still in an exploratory phase, the early results suggest that the technology can solve certain complex problems — such as investmentportfolio optimization — “quickly, accurately, and efficiently.” According to Carlos Kuchkovky, BBVA global head of research and patents: “Although this technology is still in an early stage of development, its potential to impact the sector is already a reality. Our research is helping us identify the areas where quantum computing could represent a greater competitive advantage, once the tools have sufficiently matured. We believe this will be, for certain concrete tasks, in the next two to five years,” explains.”[cxli]
BMO Financial Group and Scotiabank collaborated with Xanadu to benchmark a quantum Monte Carlo algorithm for a variety of trading products. According to Xanadu, the algorithm shows the potential disruptive potential of quantum computing for derivatives pricing over the coming years, leading to near realtime pricing and significantly lower power overhead.[cxlii]
Commonwealth Bank of Australia (CBA) and Rigetti Computing conducted a joint experiment applying the quantum approximate optimization algorithm (QAOA) to portfolio rebalancing. CBA and Rigetti used a simulated gatemodel QC.[cxliii] The experiment was a success, identifying portfolios within 5 percent of the optimal adjusted returns and with optimal risk for a small portfolio, and demonstrating the potential tractability of this application on more advanced quantum hardware.[cxliv]
Commerzbank teamed up with Fujitsu for a quantuminspired POC that optimized the selection process for a securitized loan portfolio. Using the Fujitsu Digital Annealer, Commerzbank was able to handle multiple loan selection factors simultaneously (e.g. regulatory requirements, absolute volume limits, percentage limits for specific asset characteristics etc.) in order to achieve greater risk diversification in the portfolio.[cxlv]
GloFund leveraged cloudbased quantum computing to rebalance portfolios in optimal ways with significantly more speed. GloFund claims that quantum computing allows it to take into account more constraints (e.g. regulatory requirements, volume limits, percentage limits etc.) and examine a larger set of inputs (e.g. different security classes) when rebalancing their portfolios. Historically, this computationallyintensive calculation took several hours or days to complete. That forced them to sacrifice accuracy of the calculation (e.g. simulated annealing, threshold accepting etc.) to make faster, more efficient portfolio decisions. However, by running quantum algorithms, GloFund is able to fully solve the combinatorial optimization of the portfolio and conduct advanced market simulations. Portfolio managers at GloFund are able to process significantly more portfolio combinations simultaneously to find rapid and more accurate results.[cxlvi]
Goldman Sachs is working with QC Ware to explore the acceleration of Monte Carlo algorithms with quantum computing.[cxlvii]
JPMorgan Chase and IBM tested a methodology to price options and portfolios of options on a gatebased QC using amplitude estimation, an algorithm which provides a quadratic speedup compared to classical Monte Carlo methods. A simple errormitigation scheme significantly reduced errors from noisy twoqubit gates.[cxlviii]
JPMorgan Chase also tried out Honeywell’s trappedion QC to produce what is called a quantum oracle, a blackbox operation used as an input to another algorithm.[cxlix] The oracle’s purpose was to ease the computation of Fibonacci[15] numbers, which has application in investing and information security.[cl]
NatWest Bank is using Fujitsu’s quantuminspired Digital Annealer to optimize its composition of the bank’s £120bn HQLAs[16] (highquality liquid assets) portfolio, including bonds, cash, and government securities. It has completed a highly complex calculation that needs to be undertaken regularly by the bank, at 300 times the speed of a traditional computer, with an even higher degree of accuracy. Natwest also believes this process reduces the risk of human error, and that it can complete a comprehensive risk assessment for its portfolio much faster, as well as, gaining access to a far wider range of results and permutations, therefore helping to ensure an optimized spread and reduced risk.[cli]
Nomura Asset Management (NAM) is exploring quantumcomputing applications with Tohoku University, focusing on portfolio optimization and stockreturn prediction.[clii]
RBS is using quantuminspired computing to help portfolio managers optimize the composition of the bank’s $150bn high quality liquid assets portfolio. The bank is investigating which other portfolios could be calculated by the same technology.[cliii] RBS has also evaluated algorithms from 1Qbit (in which it is an investor) to determine amounts to set aside for bad loans. John Stewart, RBS’s head of innovation, believes that RBS is “18 to 24 months” ahead of its rivals in using quantum computing. He justifies RBS’s investment as an insurance policy against getting caught offguard: “Maybe a milliondollar investment in order to understand something that could jeopardise your multibilliondollar business is a great tradeoff at this stage.”[cliv]
Standard Chartered has worked with Nasa and the Universities Space Research Association to investigate the benefits that quantum computers can bring to optimizing investment portfolios.[clv]
In January 2021, Multiverse confirmed that its clients BMO and Scotiabank were studying trading problems, Goldman Sachs was working on option pricing, Bankia was interested in minimum holding periods, and VW in making financial market predictions.[clvi] Earlier it was disclosed by DWave that for Bankia, Multiverse and DWave partnered to solve the NPhard problem of dynamic portfolio optimization – determining the optimal trading trajectory for an investment portfolio of assets over a period of time, taking into account transaction costs and other possible constraints.[clvii]
A closer glimpse into the nature of Multiverse’s work (sans client names) can be found in posted papers in which it shares progress made in running four years of market data through its tensor network algorithms in order to determine optimal[17] portfolios. The algorithms ran very fast on DWave hardware, suggesting that it was ready for “commerciallyvaluable applications.” The portfoliooptimization results were also very encouraging: The algorithm found a set of holdings that would give a 60 percent return at 15 percent volatility.[clviii] (The details can be found in a preprint paper by Mugel et al.[clix]) Multiverse has also demonstrated the optimization of an investment portfolio spanning 4 years and 7 assets using DWave and Tensor Networks, and have demonstrated a 50 percent return on investment at 14 percent volatility. According to DWave, timeseries clustering and postprocessing algorithms were key to this successful demonstration.
Similar work has been done by Multiverse competitor, Chicago Quantum, who has published a number of results over the last year from its efforts to prove that quantum techniques can select an efficient portfolio, run using the DWave annealer. Their latest demonstrations involved running the full list of U.S. common stocks from all major U.S. equity exchanges through their algorithm to prove that it could have picked a portfolio with superior returns over a certain past period.[clx] Two papers lay out the results from portfolio optimization exercises with respectively 40[clxi] and 60[clxii] U.S. stocks, contrasting QC with classical results.
Accenture has also used DWave’s hybrid solver for its banking clients to pilot quantum applications currency arbitrage, credit scoring, and trading optimization.[clxiii]
What to Expect Next
Milestones Reached and What They Portend
Some significant quantumcomputing milestones have been reached over the last two years:
Google made news when it declared socalled quantum supremacy in 2019, solving a contrived mathematical problem not realistically possible with classical computers on its 54 qubit Sycamore QC in 200 seconds vs. 10,000 years estimated for a supercomputer. In 2020 there were several significant developments:[clxiv]
· IBM, Google and others demonstrated chemical molecularbonding simulations with practical application potential such as novel material design, understanding chemical processes such as nitrogen fixation which potentially improves food production
· IBM also published its ambitious 1 million qubit roadmap out to 2030, with envisaged quantum advantage from a 1,121qubit system by 2023) 2020 Honeywell Quantum Solutions—10x Quantum Volume annually (2025: QV of 650,000) (model H0 QV 128; model H1 with 32 atomic ions released in Oct 2020)
· USTC (China) demonstrated that its QC could do Gaussian boson sampling (detected 76 photons approx. 200 seconds) that would take a supercomputer 2.5 billion years
· IonQ launched its new computer with 32 QB and QV 4 million, equaling 22 Algorithmic Qubits
· Many ongoing experiments and POCs (see previous section) also continued
BCG expects the first gains of quantum computing to accrue to firms in industries with complex simulation and optimization requirements. The financial industry has several such challenges in portfolio optimization, arbitrage strategy, and trading costs. A “slow build” is forecast until about 2024, but value is then expected to increase rapidly as the technologies matures and becomes more commercially viable. A distinct earlymover advantage is predicted:
Since quantum computing is a stepchange technology with substantial barriers to adoption, early movers will seize a large share of the total value, as laggards struggle with integration, talent, and IP...Quantum computing is a candidate for a precipitous breakthrough that may come at any time. Companies that have invested to integrate quantum computing into the workflow are far more likely to be in a position to capitalize—and the leads they open will be difficult for others to close. This will confer substantial advantage in industries in which classically intractable computational problems lead to bottlenecks and missed revenue opportunities.[clxv]
Quantum Computing was placed at the peak of the Gartner Hype Cycle for Compute Infrastructure 2020.[clxvi] But Gartner also points out that while quantum computing may be overhyped, it could offer companies real competitive advantage, and that there are real risks in ignoring it. One major risk of being a late mover is jeopardizing intellectual property (IP) and patent portfolios, as early movers move quickly to patent innovations, for example, a rival bank patenting an innovation in Monte Carlo simulations.[clxvii] Another major risk is being locked out of the QC talent pool as first movers rapidly absorb scarce QC talent.
But maybe the biggest risk of being a late mover is the loss of competitive advantage in trading activities: Whichever financial institution makes a huge breakthrough in quantum computing will choose not to announce it, but to rather reap the rewards in obscurity for as long as possible, akin to the start of highfrequency trading.[clxviii]
Plausible Timeframe for Partial vs. Full Scale Solutions
The first official Quantum Computing Roadmap was published by ARDA’s Los Alamos National Laboratory in 2002.[clxix] The roadmap was originally intended to be updated annually. However, it was last refreshed in 2004, which makes it very dated. Still the 268page document has a lot of depth on the different quantum computing technologies, and it provides some historical context. For example, it was originally predicted that a 50qubit QC would be attained by 2012. Subsequent roadmaps have recent come from major private organizations pursuing the technology, such as IBM, Honeywell, and IonQ, as well as from some consultants.
IBM Roadmap
In September 2020, IBM released an ambitious quantum hardware roadmap, which showed a pathway to a QC (codenamed “Condor”) with over 1,000 qubits by the end of 2023, and eventually on to a million qubits and beyond.[clxx] In 2021, IBM will release its 127qubit IBM Quantum processor codenamed “Eagle”. The Condor milestone will include error correction and enough scale to explore what IBM calls quantum advantage, where its QC can solve problem more efficiently than the world’s best supercomputers.
Then, in February 2021, IBM followed up by releasing a roadmap for building an open quantumsoftware ecosystem.[clxxi] IBM identified three key segments in which they expect opensource developers to create a base for those working higher up the software stack:
· Quantumkernel developers create the highperforming quantum circuits at the lowest level (i.e. closest to the hardware). IBM will release the Qiskit runtime environment later in 2021. It will increase the capacity to run more circuits faster, and to store quantum programs so that others may run them as a service. A wider variety of circuits will also be made available.
· Quantumalgorithm developers leverage those circuits to develop quantum algorithms that surpass classical computing solutions. IBM’s Circuit Composer[clxxii] has a customizable set of tools that allow developers to build, visualize, and run quantum circuits on quantum hardware or simulators.
· Quantummodel developers apply these algorithms to realworld use cases to develop quantum models for optimization, chemistry, physics, machine learning, and so forth.
By 2023, IBM plans to offer families of prebuilt runtimes in the domains of natural science, optimization, machine learning, and finance. These will be callable from a cloudbased API. Beyond 2025, IBM has a vision of frictionless quantum computing, where developers and users will no longer need to concern themselves with the hardware. The usage and creation of opensource tool and converting these to run native in the cloud are key components of IBM’s quantum roadmap.
Honeywell Roadmap
When it released the Model H1 QC in 2020, Honeywell also provided a highlevel roadmap for successor QCs, up the “large scale” Model H5 in 2030.[clxxiii] The roadmap mentions the technologies that Honeywell expect to add with every generation, but it is not specific on the number of qubits that will be achieved.
IonQ Roadmap
This quantum startup released a roadmap in December 2020.
In the roadmap IonQ introduced a new metric, Algorithmic Qubits (AQ), while rejecting IBM’s Quantum Volume (QV) metric for the reason that it will become unusable when QCs become large. AQ takes the log 2 base of QV. According to IonQ, AQ represents the number of "useful" encoded qubits in a particular QC and therefore is a good proxy for the ability to execute real quantum algorithms for a given input size. AQ is smaller than QV, for instance, IBM's quantum computing roadmap has a 1,121qubit system in 2023, which would be equal to an AQ of 65.
IonQ aims to have a 64 AQ machine by 2025, a 384 AQ machine by 2027, and a 1024 machine by 2028.[clxxiv]
The Highlevel View
In a recent report,[clxxv] BCG estimated the total benefit of quantum computing across all industries to reach $450 billion to $850 billion by 2050 (roughly evenly split between revenue and cost improvements.) BCG also estimated the impact and timeline for the financial industry: The first applications expected will be quantum annealers for optimization, expected over the next 3 – 5 years. In the 5 – 10year horizon, shallow quantum approximate optimization algorithms (QADA) are expected, with convex optimizers and fullscale quantum heuristics only expected at least a decade or two out. But even during the noisy intermediatescale quantum (NISQ) era, when high error rates will limit what QCs can do, quantuminspired algorithms and hybrid approaches (of quantum and classical computing) may create significant value for institutions and help them and prepare for the big quantum advances yet to come.
It is therefore advisable not to overgeneralize from the assessment that fullscale faulttolerant QCs are decades away to assert that no realworld applications are possible in the First Wave, defined as the next 35 years. Indeed, particular areas of finance, such as portfolio optimization, lend themselves well to what early quantum hardware and software are already able to do. The rapid progress made on early working quantum technologies, in particular annealing, suggests that optimization solutions are feasible in the short run. In the second wave, improved risk analytics have the potential to move risk management from a defensive to an offensive trading strategy.
First and Second Wave Applications in Finance
A summary of the expected impact of quantum computing on the finance industry compiled from various sources is presented in Table 2 below.
Table SEQ Table \* ARABIC2. Expected Impact of Quantum Computing on Finance over Time
Stages: 
First Wave (<5 yrs.)

Second Wave (5 – 10 yrs.) 
Third Wave (a decade +) 
Technology Status 
NISQ era of noisy machines, most still <100 qubits and low quantum volume. Quantuminspired algorithms can run on classical computers. Hybrid solutions that are part classical, part quantum. Long coherence times on quantum annealers. 
Qubits in the thousands allow for partial error correction. Most cloud services providers offer access to quantum computing, “quantumasaservice.” 
Largescale machines with tens of thousands of qubits. Fullscale fault tolerance (as decoherence is controlled via quantum error correction) gives broad quantum advantage. But only 2,000 – 5,000 operational machines exist worldwide. 
New Finance Applications 
Annealingbased portfolio optimization algorithms run on either classical or QCs. 
Improved forecasting and risk assessments that better predict black swan events. New standards for postquantum cryptography to replace current encryption techniques (RSA, AES 256) 
Nearrealtime risk assessments e.g. for quant hedge funds. Tailevent trading. Security based on postquantum cryptography. 
Financeindustry Impact 
Quantum computing becomes a competitive differentiator. Income gains from portfolio optimization up to $0.5bn. 
Income gains from portfolio optimization and risk analytics beyond $5bn. 
$40 billion to $70 billion in operating income to banks and other financial services from all quantum applications. 
Applications in Other Industries 
Molecule simulations for pharma and material science. Quantum neural networks. 
Transport and supply chain network optimization. Auto and aircraft computational fluid dynamics. 
De novo drug discovery and design with large biologics. New materials, extension of market life of patents. 
(Sources: BCG, McKinsey, IBM, Honeywell, IonQ, NEC, WEF)
Conclusions and Recommendations
Fullscale quantum computers that fulfil all the breathless media predictions for the technology are still at least a decade or more away. It will take that long to increase the number of useable qubits into the thousands while carrying a full errorcorrection overhead, which will consume most of the qubits absent any other breakthroughs. However, there are several niche QC applications available at present that can deliver nearterm business value despite the limits of the current hardware. The financial industry is bestplaced of all industries to be a first mover because the types of probabilistic optimizations possible with alreadyproven quantum technology are directly applicable to valuable finance problems such as portfolio optimization. This explains the level of activity and investment in the technology by many of the largest financial institutions in the world.
A range of portfoliooptimization applications from quantuminspired algorithms that run on classical digital hardware to hybrid and full quantum algorithms are available from a growing ecosystem of hardware and software vendors. In contrast to previous revolutionary technologies, access to quantum computing is widely available over the cloud, often at no or low cost. That both broadens and accelerates the adoption of the technology.
Given the extent of public information on activity and progress with portfolio optimization, it is likely, even plausible, that there are already financialservices companies using quantum technology to guide their trading strategies. It would be easy for a company to do this quietly since no hardware purchase is necessary. It only requires vendor relationships, cloud access, and internal modeling expertise.
A number of pathways for financial institutions to get involved in the technology can be discerned from publiclyavailable cases and from the way the industry ecosystem is taking shape. For example, as a first move an institution could try out quantuminspired portfoliooptimization algorithms running on classical hardware, and then graduate to running portfoliooptimization algorithms on quantum hardware over the cloud. That could build internal expertise in the use of the technology while proving the concept. Afterwards, there are opportunities to make larger and more aggressive investments, including taking a stake in quantum startups, or joining a consortium. The extent of the commitment will be determined by whether the organization only wants to try out the technology, shape the development of the technology, or own the technology.
The risk of not exploring quantum computing technology at
all seems to exceed the relatively modest cost of exploration at this point.
Quantum engineers and physicists are in high demand and in short supply, a fact
that is driving many of the current participants to build up teams now. Given
the scarcity of human resources with quantum computing expertise, those institutions
who are first to build up quantum computing teams may be able to lock in their
advantage for years. The internal capabilitybuilding dimension should not be
an afterthought but be matched to the corporate strategy, and aligned with the
organization’s technology strategy.
[1] All processing is done via logic gates (AND, OR, NOT, XOR i.e. exclusive OR), which compare two or more bits at a time. For example, addition relies on XOR, where the sum digit is 1 when either of the two input digits are 1, but not both. If both are 1, then the sum digit is 0 with a carry to the nextlevel digit. If both are 0, then the sum digit is also zero, but with no carry. Thus addition, or any other mathematical function, is at its core a logic operation.
[2] The Antikythera mechanism, an ancient Greek handpowered orrery, dated as far back as 100200 BC is considered to be the earliest analog computer.
[3] However, the physics of modern semiconductors intrinsically rely on quantum effects, and it is the understanding of those quantum effects that enabled the inventors of the transistor to create the device in the first place. That is why some use the term, The Second Quantum Revolution, to describe the age of quantum computing. The First Quantum Revolution involved devices based on quantum mechanics, such as the transistor and laser.
[4] Einstein famously called this phenomenon “spooky action at a distance.”
[5] For more details, see Table 1 in the subsequent section.
[6] Complex algebra uses an expanded number system where there are both real and imaginary numbers. A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit number defined as QUOTE <?msoapplication progid="Word.Document"?> 161"> <?msoapplication progid="Word.Document"?> 161">
[7] A vector that points to a specific point in space which corresponds to a particular quantum state.
[8] In computer science, the order of sophistication and problem difficulty are as follows: P (easy), NP (medium hard), NPcomplete (hard), and NPhard (hardest). A hotly debated question is whether there are ways that NP and NPcomplete problems can be solved in polynomial time, effectively making them all P. The is called the P = NP problem, and is currently the subject of a $1m Millennium Prize for a correct solution.
[9] The term “annealing” in computer science is borrowed from metallurgy, where it refers to thermal annealing, i.e. the heat treatment that increases metal ductility while reducing hardness. This heat treatment accelerates the thermodynamically spontaneous processes that return a metal to its stressfree equilibrium state. The stressfree equilibrium is the lowest energy state of the metal structure. Thus, annealing is generally applied as a term to any process whereby the lowest energy of a system is found.
[10] The Traveling Salesman Problem (TSP) is an NPhard problem in combinatorial optimization: “Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?” TSP is wellstudied and often used as a benchmark for optimization methods.
[11] The QC equivalent of an SDK.
[12] Ions are electricallycharged atoms meaning they have more or less electrons than protons.
[13] Customers include Lockheed Martin, Google, NASA, the University of Southern California, and Los Alamos National Laboratory.
[14] The Sharpe ratio is a widelyused method for calculating the riskadjusted return of a portfolio. It equals the portfolio return less the riskfree rate, divided by the standard deviation of the portfolio’s expected return.
[15] A series in which each number is the sum of the prior two numbers.
[16] HQLAs are assets such as cash and bonds that every UK bank must hold as a buffer in case it runs into financial trouble.
[17] In each realworld case it is up to the asset manager to frame the problem in terms of what is optimal.
[i] Dietz, Miklos, Nico Henke, Jared Moon, Jens Backes, Lorenzo Pautasso, and Zaheen Sadeque, “How quantum computing could change financial services,” McKinsey & Company, Dec. 2020.
[ii] Levy, Max G., “New Quantum Algorithms Finally Crack Nonlinear Equations,” Quanta Magazine, Jan. 5, 2021, https://www.quantamagazine.org/newquantumalgorithmsfinallycracknonlinearequations20210105/ .
[iii] Deutsch, David. "Quantum theory, the Church–Turing principle and the universal quantum computer." Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 400, no. 1818, pp. 97117, 1985.
[iv] Shor, Peter W, "Algorithms for quantum computation: discrete logarithms and factoring." In Proceedings 35th annual symposium on foundations of computer science, pp. 124134. IEEE, 1994.
[v] For a more complete discussion of the mathematics, see “Shor’s algorithm,” IBM Quantum, https://quantumcomputing.ibm.com/docs/iqx/guide/shorsalgorithm .
[vi] Grover, L.K., A fast quantum mechanical algorithm for database search, Proceedings of the TwentyEighth Annual ACM Symposium on Theory of Computing – STOC ’96, 1996, pp. 212–219, https://doi.org/10.1145/237814.237866.
[vii] Y. Li, M. Tian, G. Liu, C. Peng and L. Jiao, "Quantum Optimization and Quantum Learning: A Survey," in IEEE Access, vol. 8, pp. 2356823593, 2020, doi: 10.1109/ACCESS.2020.2970105.
[ix] See for example, “What is Quantum Annealing?,” DWave System Documentation, https://docs.dwavesys.com/docs/latest/c_gs_2.html /
[x] McMahon, Peter, “To Crack the Toughest Optimization Problems, Just Add Lasers,” IEEE Spectrum, Nov. 27, 2018, https://spectrum.ieee.org/computing/hardware/tocrackthetoughestoptimizationproblemsjustaddlasers .
[xi] Burkacky, Ondrej, Niko Mohr, and Lorenzo Pautasso, “Will quantum computing drive the automotive future?, ” McKinsey & Company, Sep. 2, 2020, https://www.mckinsey.com/industries/automotiveandassembly/ourinsights/willquantumcomputingdrivetheautomotivefuture .
[xii] Pakin, Scott, “The Problem with Quantum Computers,” Scientific American, Jun. 10, 2019, https://blogs.scientificamerican.com/observations/theproblemwithquantumcomputers/
[xiii] Dyakonov, Mikhail, “The Case Against Quantum Computing,” IEEE Spectrum, Nov. 15, 1998, https://spectrum.ieee.org/computing/hardware/thecaseagainstquantumcomputing .
[xiv] See, for example, the Comments section below the original article at https://spectrum.ieee.org/computing/hardware/thecaseagainstquantumcomputing .
[xv] Versluis, Richard, “Here’s a Blueprint for a Practical Quantum Computer,” IEEE Spectrum, Mar. 24, 2020, https://spectrum.ieee.org/computing/hardware/heresablueprintforapracticalquantumcomputer.
[xvi] Ibaraki, Stephen, “What You Need for Your Quantum Computing Pilots In 2021,” Forbes, Jan. 29, 2021, https://www.forbes.com/sites/stephenibaraki/2021/01/29/whatyouneedforyourquantumcomputingpilotsin2021/?sh=24c3955d3d1f .
[xviii] Wolchover, Natalie, “Classical computing embraces quantum ideas,” Quanta Magazine, Dec. 18, 2012, www.quantamagazine.org/20121218classicalcomputingembracesquantumideas/.
[xix] Martinis, John, “Quantum Supremacy Using a Programmable Superconducting Processor,” Google AI Blog, Oct. 23, 2019, https://ai.googleblog.com/2019/10/quantumsupremacyusingprogrammable.html .
[xx] Arute, Frank, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Rupak Biswas et al. "Quantum supremacy using a programmable superconducting processor." Nature 574, no. 7779 (2019): 505510.
[xxi] Preskill, John. "Quantum computing and the entanglement frontier." arXiv preprint arXiv:1203.5813 (2012). Available at https://arxiv.org/abs/1203.5813v3 .
[xxii] Haughton, Richard, “Race for quantum supremacy hits theoretical quagmire,” Nature, Nov. 14, 2017, https://www.nature.com/news/raceforquantumsupremacyhitstheoreticalquagmire1.22993#/b1.
[xxiii] SmithGoodson, Paul, “IonQ Releases A New 32Qubit TrappedIon Quantum Computer With Massive Quantum Volume Claims,” Forbes, Oct. 7, 2020, https://www.forbes.com/sites/moorinsights/2020/10/07/ionqreleasesanew32qubittrappedionquantumcomputerwithmassivequantumvolumeclaims/?sh=6665ed1e3b39
[xxiv] Swayne, Matt, “Honeywell Sets New Record Ffor Quantum Computing Performance,” The Quantum Daily, Mar. 2021, https://thequantumdaily.com/2021/03/09/honeywellsetsnewrecordforquantumcomputingperformance/ .
[xxv] Haughton, Richard, “Race for quantum supremacy hits theoretical quagmire,” Nature, Nov. 13, 2017, https://www.nature.com/news/raceforquantumsupremacyhitstheoreticalquagmire1.22993
[xxviii] SmithGoodson, Paul, “Quantum USA Vs. Quantum China: The World's Most Important Technology Race,” Forbes, Oct. 10, 2019, https://www.forbes.com/sites/moorinsights/2019/10/10/quantumusavsquantumchinatheworldsmostimportanttechnologyrace/?sh=b2cf82772de9
[xxxii] Srinivasan, Prianka , “Could quantum computing be the technology that drives your quantum leap forward?,” EY, Oct 3., 2019, https://www.ey.com/en_gl/disruption/couldquantumcomputingbethetechnologythatdrivesyourquantumleapforward
[xxxiii] Decker, Susan, and Christopher Yasiejko, “Forget the Trade War. China Wants to Win Computing Arms Race, ” Bloomberg, Apr. 8, 2018, https://www.bloomberg.com/news/articles/20180408/forgetthetradewarchinawantstowinthecomputingarmsrace .
[xxxiv] Chen, Shiyin, “Chinese Scientists Claim Breakthrough in Quantum Computing Race,” Bloomberg, Dec. 3, 2020, https://www.bloomberg.com/news/articles/20201204/chinesescientistsclaimbreakthroughinquantumcomputingrace .
[xxxv] SmithGoodson, Paul, “Quantum USA Vs. Quantum China: The World's Most Important Technology Race,” Forbes, Oct. 10, 2019, https://www.forbes.com/sites/moorinsights/2019/10/10/quantumusavsquantumchinatheworldsmostimportanttechnologyrace/?sh=b2cf82772de9 .
[xxxvi] See “TQD Exclusive: A Detailed Review of Qubit Implementations for Quantum Computing,” The Quantum Daily, May 21, 2020, https://thequantumdaily.com/2020/05/21/tqdexclusiveadetailedreviewofqubitimplementationsforquantumcomputing/
[xxxvii] Ruess, Frankl, “The Next Decade in Quantum Computing—and How to Play,” BCG, Nov. 15, 2018, https://www.bcg.com/publications/2018/nextdecadequantumcomputinghowplay
[xxxviii] Ibid.
[xl] Kahn, Jeremy, “DWave unveils its most powerful quantum computer to date,” Fortune, Sep. 29, 2020, https://fortune.com/2020/09/29/dwave5000qubitquantumcomputer/ .
[xli] Silcoff, Sean, “Valuation of quantum computer maker DWave slashed by more than half after company struggles to raise financing,” The Globe and Mail, Oct., 21, 2020, https://www.theglobeandmail.com/business/articleinvestorssharplycutvalueofholdingsinquantumcomputermakerd/ .
[xliv] “Honeywell Releases Next Generation of Quantum Computer,” Honeywell, Oct. 29, 2020, https://www.honeywell.com/us/en/press/2020/10/honeywellreleasesnextgenerationofquantumcomputer .
[xlv] “The World’s Highest Performing Quantum Computer is Here, “ Honeywell, Jun., 2020, https://www.honeywell.com/us/en/news/2020/06/theworldshighestperformingquantumcomputerishere
[xlvi] “Scannell, Ed, Honeywell's quantum computer makes a quantum volume leap,” TechTarget, Nov. 2, 2020, https://searchdatacenter.techtarget.com/news/252491505/Honeywellsquantumcomputermakesaquantumvolumeleap .
[xlviii] Chow, Jerry, and Gambetta, Jay, “Quantum Takes Flight: Moving from Laboratory Demonstrations to Building Systems,” IBM Research Blog, Jan. 8, 2020, https://www.ibm.com/blogs/research/2020/01/quantumvolume32/ .
[xlix] Saran, Cliff, “IBM celebrates quantum cloud fourth anniversary with coding challenge,” Apr. 28, 2020, Computer Weekly, https://www.computerweekly.com/news/252482251/IBMcelebratesquantumcloudfourthanniversarywithcodingchallenge .
[l] Crosman, Penny, “JPMorgan Chase, Barclays join IBM quantum computing network,” American Banker Vol. 182 Issue 240, Dec. 15, 2017.
[li] Swayne, Matt, “Intel’s New Chip May Make Quantum Computers Commercially Viable,” The Quantum Daily, Dec 10, 2019, https://thequantumdaily.com/2019/12/10/intelsnewchipmaymakequantumcomputerscommerciallyviable/
[lii] Dignen, Larry, “Intel's 'Horse Ridge' control chip may make quantum computing more viable, scalable,” ZDNet, Dec. 9, 2019, https://www.zdnet.com/article/intelshorseridgecontrolchipmaymakequantumcomputingmoreviablescalable/ .
[liv] SmithGoodson, Paul, “IonQ Releases A New 32Qubit TrappedIon Quantum Computer With Massive Quantum Volume Claims,” Forbes, Oct. 7, 2020, https://www.forbes.com/sites/moorinsights/2020/10/07/ionqreleasesanew32qubittrappedionquantumcomputerwithmassivequantumvolumeclaims/?sh=6665ed1e3b39
[lv] Castellanos, Sara, “QuantumComputing Startup IonQ Plans Public Debut in $2 Billion SPAC Merger,” The Wall Street Journal, Mar. 8, 2021, https://www.wsj.com/articles/quantumcomputingstartupionqplanspublicdebutin2billionspacmerger11615201201 .
[lx] “Silicon Quantum Computing: What Are They Up To in the Land Down Under?,” Quantum Computing Report, Oct. 2, 2020, https://quantumcomputingreport.com/siliconquantumcomputingwhataretheyuptointhelanddownunder/ .
[lxi] “CBA Backs Australia's First Quantum Computing Company,” CBA Story, Aug. 23, 2017, https://www.commbank.com.au/guidance/newsroom/quantumcompanylaunch201708.html .
[lxiii] SmithGoodson, Paul, “Xanadu Brings Photonic Quantum Computing To The Cloud,” Forbes, Sep. 9, 2020, https://www.forbes.com/sites/moorinsights/2020/09/09/xanadubringsphotonicquantumcomputingtothecloud/?sh=537b422064e6
[lxiv] Silcoff, Sean, “Toronto quantum computer startup Xanadu eyes $100million funding after launching online service,” The Globe and Mail, Sep. 2, 2020, https://www.theglobeandmail.com/business/articletorontoquantumcomputerstartupxanadueyes100millionfunding/
[lxv] Benmeleh, Yaacov , Giles Turner, and Matt Day, “Amazon Is Laying the Groundwork for Its Own Quantum Computer,” Bloomberg, Dec 1., 2020, https://www.bloomberg.com/news/articles/20201201/amazonislayingthegroundworkforitsownquantumcomputer .
[lxvi] Swayne, Matt, “Cisco Showing Signs That It’s Gearing Up For a Longterm Quantum Future,“ The Quantum Daily, Feb. 11, 2021, https://thequantumdaily.com/2021/02/11/ciscoshowingsignsthatitsgearingupforalongtermquantumfuture/ .
[lxvii] Simonite, Tom, “Microsoft’s Big Win in Quantum Computing Was an ‘Error’ After All,” WIRED, Feb. 12, 2021, https://wwwwiredcom.cdn.ampproject.org/c/s/www.wired.com/story/microsoftwinquantumcomputingerror/amp .
[lxviii] Zhang, Hao, Michiel WA de Moor, Jouri DS Bommer, Di Xu, Guanzhong Wang, Nick van Loo, ChunXiao Liu et al. "Large zerobias peaks in InSbAl hybrid semiconductorsuperconductor nanowire devices." arXiv preprint arXiv:2101.11456, 2021. Available at https://arxiv.org/abs/2101.11456 .
[lxix] Swayne, Matt, “Paper Backing Microsoft’s Quantum Technology to be Retracted, Quantum Hardware Field Narrows,” The Quantum Daily, Feb. 13, 2021, https://thequantumdaily.com/2021/02/13/paperbackingmicrosoftsquantumtechnologytoberetractedquantumhardwarefieldnarrows/
[lxxi] Fortson, Danny, “How RBS made a quantum leap into the future,” The Sunday Times, Jul. 5, 2020, https://www.thetimes.co.uk/article/howrbsmadeaquantumleapintothefuturev8ptxv7qj
[lxxiii] “Alibaba Cloud and CAS Launch One of the World’s Most Powerful Public Quantum Computing Services,” Alibaba Cloud, March 1, 2018, https://www.alibabacloud.com/pressroom/alibabacloudandcaslaunchoneoftheworldsmost.
[lxxvii] Swayne, Matt, “Paddling Back: Baidu Makes Good on Its Quantum Promise with Machine Learning Toolkit,” The Quantum Daily, May 28, 2020, https://thequantumdaily.com/2020/05/28/paddlingbackbaidumakesgoodonitsquantumpromisewithmachinelearningtoolkit/ .
[lxxxi] “CQNS Efficient Stock Portfolio Picks (dividend paid & medium risk),” Chicago Quantum, Feb, 7, 2021, https://chicagoquantum.medium.com/feb72021cqnsefficientstockportfoliopicksdividendpaidmediumrisk597e542a6211
[lxxxiii] Svore, Krysta, “Azure Quantum is now in Public Preview,” Microsoft Quantum, Feb. 1, 2021, https://cloudblogs.microsoft.com/quantum/2021/02/01/azurequantumpreview/.
[lxxxiv] LeprinceRinquet, Daphne, “Microsoft's quantum cloud computing plans take another big step forward,” ZDNet, Feb 1., 2021, https://www.zdnet.com/article/microsoftsquantumcloudcomputingplanstakeanotherbigstepforward/.
[xci] Microsoft Quantum Team, “Toshiba joins Azure Quantum, providing Ising machine to solve large combinatorial optimization problems,” Microsoft Quantum, https://cloudblogs.microsoft.com/quantum/2020/09/22/toshibajoinsazurequantumnetworkmachinesolvinglargecombinatorialoptimizationproblems/ . https://cloudblogs.microsoft.com/quantum/2020/09/22/toshibajoinsazurequantumnetworkmachinesolvinglargecombinatorialoptimizationproblems/
[xcii] https://aws.amazon.com/marketplace/pp/ToshibaDigitalSolutionsCorporationSimulatedBi/B07T8STKZR
[xciii] “Toshiba to launch quantum cryptography services by fiscal 2025,” The Japan Times, Oct. 19, 2020, https://www.japantimes.co.jp/news/2020/10/19/business/corporatebusiness/toshibaquantumcryptography/ .
[xcix] Matt Swayne, “The World’s Top 12 Quantum Computing Research Universities,” The Quantum Daily, Nov. 28, 2019, https://thequantumdaily.com/2019/11/18/theworldstop12quantumcomputingresearchuniversities/.
[cix] https://www.sydney.edu.au/science/ourresearch/researchareas/physics/quantumsciencegroup.html
[cxiii] “HSBC joins European quantum consortium, Finextra, Oct. 6, 2020, https://www.finextra.com/newsarticle/36686/hsbcjoinseuropeanquantumconsortium .
[cxiv] “What is This QuantumInspired Stuff All About?,” Quantum Computing Report, https://quantumcomputingreport.com/whatisthisquantuminspiredstuffallabout/
[cxv] Langston, Jennifer, “How the quest for a scalable quantum computer is helping fight cancer,” Microsoft Innovation Stories, Jul. 15, 2019, https://news.microsoft.com/innovationstories/quantumcomputingmricancertreatment/
[cxvi] Microsoft Quantum Team, “Microsoft Quantum collaborates with Willis Towers Watson to transform riskmanagement solutions,” Microsoft Quantum, May 22, 2019, https://cloudblogs.microsoft.com/quantum/2019/05/22/microsoftquantumcollaborateswithwillistowerswatsontotransformriskmanagementsolutions/.
[cxix] Kahane, Thierry, “QuantumInspired Computing – What is it and why does it matter?,” Fujitsu Blog, 21 Feb. 2020, https://blog.global.fujitsu.com/fgb/20200221/quantuminspiredcomputingwhatisitandwhydoesitmatter/.
[cxx] Ruess, Frankl, “The Next Decade in Quantum Computing—and How to Play,” BCG, Nov. 15, 2018, https://www.bcg.com/publications/2018/nextdecadequantumcomputinghowplay .
[cxxi] Ménard, Alexandre, Ivan Ostojic, Mark Patel, and Daniel Volz, “A game plan for
quantum computing,” McKinsey & Company, Feb. 6, 2020, https://www.mckinsey.com/businessfunctions/mckinseydigital/ourinsights/agameplanforquantumcomputing .
[cxxii] Ibaraki, Stephen, “What You Need for Your Quantum Computing Pilots In 2021,” Forbes, Jan. 29, 2021. https://www.forbes.com/sites/stephenibaraki/2021/01/29/whatyouneedforyourquantumcomputingpilotsin2021/?sh=24c3955d3d1f
[cxxiii] Schatsky, David and Ramya Kunnath Puliyakodil, “From fantasy to reality – Quantum computing is coming to the marketplace,” Signals for Strategists, Deloitte University Press, 2017, https://www2.deloitte.com/us/en/insights/focus/signalsforstrategists/quantumcomputingenterpriseapplications.html .
[cxxiv] Orrell, David. Quantum Economics (p. 172). Icon Books Ltd. Kindle Edition.
[cxxv] Bobier, JeanFrançois, JeanMichel Binefa, Matt Langione, and Amit Kumar, “It’s Time for Financial Institutions to Place Their Quantum Bets,” BCG, Oct. 16, 2020, https://www.bcg.com/publications/2020/howfinancialinstitutionscanutilizequantumcomputing .
[cxxvi] Egger, Daniel J., Claudio Gambella, Jakub Marecek, Scott McFaddin, Martin Mevissen, Rudy Raymond, Andrea Simonetto, Stefan Woerner, and Elena Yndurain, "Quantum computing for Finance: state of the art and future prospects," IEEE Transactions on Quantum Engineering, 2020.
[cxxvii] Dietz, Miklos, Nico Henke, Jared Moon, Jens Backes, Lorenzo Pautasso, and Zaheen Sadeque, “How quantum computing could change financial services,” McKinsey & Company, Dec. 2020, https://www.mckinsey.com/industries/financialservices/ourinsights/howquantumcomputingcouldchangefinancialservices .
[cxxviii] Mugel, Samuel, Quantum Computing for Finance, Jan 29, 2021, https://youtu.be/yJk4eMGOkRE .
[cxxix] Ménard, Alexandre, Ivan Ostojic, Mark Patel, and Daniel Volz, “A game plan for
quantum computing,” McKinsey & Company, Feb. 6, 2020, https://www.mckinsey.com/businessfunctions/mckinseydigital/ourinsights/agameplanforquantumcomputing .
[cxxx] Cloud Computing Industry Trends & Regional Outlooks, Fitch Solutions, 2020. Available at https://store.fitchsolutions.com/cloudcomputingindustrytrendsregionaloutlooks .
[cxxxi] Forging New Pathways: The next evolution of innovation in Financial Services, World Economic Forum, Sep. 20, 2020, https://www.weforum.org/reports/forgingnewpathwaysthenextevolutionofinnovationinfinancialservices .
[cxxxii] Mayes, Nick, “Is Business Ready to Make the Quantum Leap?,” teknowlogy Group, May 2019. (Commissioned by Fujitsu)
[cxxxiii] Langione, Matt , Corban TillemannDick, Amit Kumar, and Vikas Taneja, “Where Will Quantum Computers Create Value—and When?,” BCG, May 13, 2019, https://www.bcg.com/publications/2019/quantumcomputerscreatevaluewhen .
[cxxxiv] Alcazar, Javier, and Alejandro PerdomoOrtiz. "Enhancing Combinatorial Optimization with Quantum Generative Models." arXiv preprint arXiv:2101.06250, 2021. Available at https://arxiv.org/pdf/2101.06250 .
[cxxxv] McKeough, Catherine, “Quantum Annealing: Theory and Practice,” Guest lecture at UBC Computer Science, Sep. 30, 2019, https://youtu.be/y8gPz8sc7tY .
[cxxxvi] Ibid.
[cxxxvii] Mugel , Samuel, Quantum Computing for Finance, Jan. 29, 2021 , https://youtu.be/yJk4eMGOkRE .
[cxxxviii] Crosman, Penny, “Why Banks like Barclays Are Testing Quantum Computing.” American Banker 183 (136):1, 2018.
[cxxxix] Saran, Cliff, “Barclays demonstrates proofofconcept quantum clearing algorithm,” Computer Weekly, Oct. 17, 2019, https://www.computerweekly.com/news/252472462/Barclaysdemonstratesproofofconceptquantumclearingalgorithm .
[cxl] Braine, Lee, Daniel J. Egger, Jennifer Glick, and Stefan Woerner. "Quantum algorithms for mixed binary optimization applied to transaction settlement." arXiv preprint arXiv:1910.05788, 2019, https://arxiv.org/pdf/1910.05788v1.pdf .
[cxlii] “BMO Financial Group and Scotiabank Partner with Xanadu on Quantum Computing Speedups for Trading Products,” Xanadu press release, Newswire Canada, Aug. 20, 2019, https://www.newswire.ca/newsreleases/bmofinancialgroupandscotiabankpartnerwithxanaduonquantumcomputingspeedupsfortradingproducts866739711.html .
[cxliii] Pearce, Rohan, “Rigetti, CBA test quantum computing’s financial potential,” Computerworld, Nov. 15, 2019, https://www.computerworld.com/article/3474204/rigetticbatestquantumcomputingsfinancialpotential.html.
[cxliv] Forging New Pathways: The next evolution of innovation in Financial Services, World Economic Forum, Sep. 20, 2020, https://www.weforum.org/reports/forgingnewpathwaysthenextevolutionofinnovationinfinancialservices .
[cxlv] Forging New Pathways: The next evolution of innovation in Financial Services, World Economic Forum, Sep. 20, 2020, https://www.weforum.org/reports/forgingnewpathwaysthenextevolutionofinnovationinfinancialservices .
[cxlvi] Forging New Pathways: The next evolution of innovation in Financial Services, World Economic Forum, Sep. 20, 2020, https://www.weforum.org/reports/forgingnewpathwaysthenextevolutionofinnovationinfinancialservices .
[cxlvii] Castellanos, Sara, “Goldman Taps Startup to Explore Quantum Computing,” The Wall Street Journal, Dec. 10, 2019, https://www.wsj.com/articles/goldmantapsstartuptoexplorequantumcomputing11575986520
[cxlviii] Stamatopoulos, Nikitas, Daniel J. Egger, Yue Sun, Christa Zoufal, Raban Iten, Ning Shen, and Stefan Woerner. "Option pricing using quantum computers," Quantum 4, 2020, p. 291, https://quantumjournal.org/papers/q20200706291/pdf/ .
[cxlix] Swayne, Matt, “JP Morgan Chase Unleashes Honeywell’s Quantum Computer on Tough Fintech Problems,“ The Quantum Daily, Jul. 2, 2020, https://thequantumdaily.com/2020/07/02/jpmorganchaseunleasheshoneywellsquantumcomputerontoughfintechproblems/ .
[cl] Gilliam, Austin, Marco Pistoia, and Constantin Gonciulea. "Canonical construction of quantum oracles." arXiv preprint arXiv:2006.10656, 2020. Available at https://arxiv.org/abs/2006.10656 .
[cli] White Paper – The Case for Quantum and QuantumInspired Computing in Financial Services, Fujitsu, https://sp.ts.fujitsu.com/dmsp/Publications/public/wpdafinancialsectorwwen.pdf
[clii] “Quantum Computer Research of Nomura Asset Management,” Nov. 2019, Nomura, https://www.nomuraconnects.com/focusedthinkingposts/quantumcomputerresearchofnomuraassetmanagement/ .
[cliii] Mayes, Nick, “Is Business Ready to Make the Quantum Leap?,” teknowlogy Group, May 2019. (Commissioned by Fujitsu)
[cliv] Fortson, Danny, “How RBS made a quantum leap into the future,” The Sunday Times, Jul. 5, 2020, https://www.thetimes.co.uk/article/howrbsmadeaquantumleapintothefuturev8ptxv7qj .
[clv] Walker, Owen, “Nasa says quantum computing is the future for funds,” The Financial Times, Oct. 20, 2019, https://www.ft.com/content/195ce103241930e4a19d96e8cb8a3c9b .
[clvi] Mugel, Samuel, Quantum Computing for Finance, Jan. 29, 2021, https://youtu.be/yJk4eMGOkRE.
[clviii] Mugel, Samuel, Quantum Computing for Finance, Jan. 29, 2021, https://youtu.be/yJk4eMGOkRE.
[clix] Mugel, Samuel, Carlos Kuchkovsky, Escolastico Sanchez, Samuel FernandezLorenzo, Jorge LuisHita, Enrique Lizaso, and Roman Orus. "Dynamic Portfolio Optimization with Real Datasets Using Quantum Processors and QuantumInspired Tensor Networks." arXiv preprint arXiv:2007.00017 (2020).
[clxi] Cohen, Jeffrey, Alex Khan, and Clark Alexander. "Portfolio optimization of 40 stocks using the DWAVE quantum annealer, " arXiv preprint arXiv:2007.01430, 2020. Available at https://arxiv.org/pdf/2007.01430 .
[clxii] Cohen, Jeffrey, Alex Khan, and Clark Alexander. "Portfolio Optimization of 60 Stocks Using Classical and Quantum Algorithms," arXiv preprint arXiv:2008.08669 (2020). Available at https://arxiv.org/pdf/2008.08669 .
[clxiii] Kahn, Jeremy, “DWave unveils its most powerful quantum computer to date,” Fortune, Sep. 29, 2020, https://fortune.com/2020/09/29/dwave5000qubitquantumcomputer/ .
[clxiv] Ibaraki, Stephen, “What You Need for Your Quantum Computing Pilots In 2021,” Forbes, Jan. 29, 2021, https://www.forbes.com/sites/stephenibaraki/2021/01/29/whatyouneedforyourquantumcomputingpilotsin2021/?sh=24c3955d3d1f .
[clxv] Langione, Matt , Corban TillemannDick, Amit Kumar, and Vikas Taneja, “Where Will Quantum Computers Create Value—and When?,” BCG, May 13, 2019, https://www.bcg.com/publications/2019/quantumcomputerscreatevaluewhen .
[clxvi] Harvey, Tony , Daniel Bowers , Chirag Dekate, “ Hype Cycle for Compute Infrastructure, 2020,” Gartner, Jul. 8, 2020, https://www.gartner.com/en/documents/3987268/hypecycleforcomputeinfrastructure2020 .
[clxvii] Panetta, Kasey, “The CIO’s Guide to Quantum Computing,” Apr. 18, 2019, Gartner, https://www.gartner.com/smarterwithgartner/theciosguidetoquantumcomputing/ .
[clxviii] “Wall Street’s latest shiny new thing: quantum computing,” The Economist, Dec. 16, 2020, https://www.economist.com/financeandeconomics/2020/12/19/wallstreetslatestshinynewthingquantumcomputing .
[clxix] “A Quantum Information Science and Technology Roadmap,” Report of the Quantum Information Science and Technology Experts Panel, ARDA, Apr. 2, 2004, https://qist.lanl.gov/pdfs/qc_roadmap.pdf .
[clxx] Gambetta, Jay, “IBM’s Roadmap For Scaling Quantum Technology,” IBM Research Blog, Sept. 15, 2020, https://www.ibm.com/blogs/research/2020/09/ibmquantumroadmap/ .
[clxxi] Wehden, Karl, Ismael Faro, and Jay Gambetta, “IBM’s roadmap for building an open quantum software ecosystem,” IBM Research Blog, Feb. 4, 2021, https://www.ibm.com/blogs/research/2021/02/quantumdevelopmentroadmap/ .
[clxxiii] SmithGoodson, Paul, “Honeywell Quantum Solutions Releases Its New System Model H1 Quantum Computer,” Forbes, Oct. 30, 2020, https://www.forbes.com/sites/moorinsights/2020/10/30/honeywellquantumsolutionsreleasesitsnewsystemmodelh1quantumcomputer/?sh=40d85eee250f .
[clxxiv] Dignan, Larry, “IonQ introduces Algorithmic Qubits to counter Quantum Volume in quantum computing,” ZDNet, Dec. 9, 2020, https://www.zdnet.com/article/ionqintroducesalgorithmicqubitstocounterquantumvolumeinquantumcomputing/ .
[clxxv] Bobier, JeanFrançois, JeanMichel Binefa, Matt Langione, and Amit Kumar, “It’s Time for Financial Institutions to Place Their Quantum Bets,” BCG, Oct. 16, 2020, https://www.bcg.com/publications/2020/howfinancialinstitutionscanutilizequantumcomputing .
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