_Introduction

Quantum computing is fast emerging as a technology heralding profound implications for the financial sector by presenting the opportunity of revolutionary advancements in computational power...

For some capital markets and financial services businesses, especially those with a high dependency on very computationally intensive tasks, failure to adopt future quantum computing technology may even outright jeopardise a firm’s ability to compete. In this article we will explore the current state-of-the-art and the future potential application of quantum computing within financial services, highlighting both the areas of promise and its limitations. Our goal is to provide senior executives with actionable insights on preparing for a quantum computing future and motivating them to anticipate drivers for change which are emerging from a complex but commercially rewarding field.

In an attempt to put the opportunities presented by quantum computing into context, we aim to explain the significance of the quantum computational advantage. We will also seek to demystify a little understood fundamental aspect of quantum computing which is key to how quantum calculations are actually performed. This is the role of the “oracle”. What it is and does and exactly how this enables the power of this technology, determining where quantum computing can dramatically outperform today’s computing technologies as a complete game-changer, and where it cannot.

Our goal is to enable executives to formulate clear, actionable roadmaps to help prepare and position their organisations for this future technology by understanding where the real business opportunities exist, and to prepare to take advantage of quantum computing when it becomes a practical and deployable business solution.

Given the pace of change and development advances in quantum computing, that future may not be very far away. In just the past week, Google has announced a milestone achievement in overcoming one of the obstacles to practical quantum computing. This is a limitation of quantum computing which has been grappled with for decades but is now on the cusp of being solved at scale.

Understanding Quantum Computing’s Potential and Limitations in Financial Services

Anyone who has read anything about quantum computing is probably aware that quantum computing leverages principles of quantum mechanics to solve problems at speeds that today’s “classical” computers cannot achieve. Few actually understand however how the nature of that ability to perform faster calculations or scale those calculations comes about from the technology, or indeed the degrees of scaling that are possible. The complexities of the science may be baffling, but a basic understanding of the nature, scope, capabilities and applicability of quantum computing to activities in financial services is what we seek in this article. We suggest approaches to enable executives to guide future projects and identify clear opportunities for future competitive advantage.

Putting the quantum computing calculation advantage into perspective

Invariably articles about quantum computing will say that quantum computers can do more, faster and bigger calculations than classical computers. But what exactly do we mean by “more”? Let’s try to put this into perspective, and get a sense of scale.

Very simply, the difference between classical computing and quantum computing is that the bits in a classical computer can only hold and interrogate one permutation of those bits at a time. A quantum computer can consider every possible permutation of those bits all at once. At the top level it is that simple - that is the difference between classical computing and quantum computing. But what a difference that makes. As ever that isn’t quite the entire story because this is tempered by certain limitations in the quantum approach to performing those calculations. The slight fly in the ointment for quantum computing is that performing the calculation just once does give an answer, but there are probabilities associated with that particular result being correct. Quantum computing does have an approach to dealing with this issue, but this is a point we will come back to later.

When you are sitting at your desk, and assuming you are using a fairly average desktop computer of today, the chances are it has a 64-bit processor. That fairly average CPU may well contain 14 cores. What that ultimately means is that the CPU is capable of holding 64 x 14 bits, a total of 896 bits, at any single point in time in a single permutation of all of those bits.

That relatively small number of 896 means that the total number of permutations of the settings of that set of bits is 2⁸⁹⁶. Given that the number of atoms in the observable universe is something around 10⁸⁰, having simply considered the possible permutations of an average CPU we have suddenly arrived at a very big number describing all of the unique ways in which those bits can be set. Let’s contextualise that in terms of computing power and the ability to process this also. If we wanted our computer to consider every possible permutation of that set of bits, and say the computer can check each one of them at the rate of 1 billion per second, we’re going to have to wait a while as this would take an amount of time many multiples the current age of the universe.

But as mentioned above, a quantum computer holds, and can consider all of these permutations at once. So then why don’t we just run the calculation once through the quantum computer and produce the answer?

Well, that is where the nature of quantum mechanics comes into play and the inherent advantages of the science also come with a limiting factor. A quantum computer assesses all of those many possible answers at once through probabilities. When the computer gives a result, it comes with a probability as to whether it is correct or not. If you run one single quantum calculation and it produces a result that only has a 20% probability of being the correct answer, that isn’t an answer you will want to depend upon.

Fortunately there is a way of addressing this, and that is to repeat the calculation multiple times, because those probabilities also have a frequency distribution and the “right” answer will be the one that emerges most often, if you repeat the calculation a sufficient number of times. The result with the highest frequency which also reflects the right result, in effect, rises to the top - it is “amplified”. Once enough repetitions have been performed for the calculation in question, the answer has been found. The number of repetitions is driven by a desired statistical confidence level and varies depending on the nature of the calculation. In the quantum computing world 100% certainty is not achievable, but extremely high probabilities are. For example in the case of cryptography where high levels of confidence are essential a confidence level of 99.999%, referred to as “five nines” is considered to be adequate.

The quantum computing advantage is the trade-off between the difference in the number of times a classical computer has to perform each permutation of a calculation versus how many times a quantum

computer has to repeat one calculation to achieve sufficient confidence in the result. The advantage being that the number of repetitions in the quantum computer is significantly less than the total number of calculations required by a classical computer. This quantum advantage is described in various ways and expressions such as “speedup” and “scaling” are used in the context of this computational advantage.

But what then determines the number of times we would need to repeat the quantum calculation? There are various factors influencing this, one of which is the actual engineering of quantum computers themselves. As of today, quantum computers are regarded as “noisy” and it is that noise which introduces errors when the calculations are performed, which increases the number of times a calculation needs to be repeated. The noise arises from multiple sources, not least of which are physical engineering challenges.

But Google’s latest announcement regarding their Willow quantum computing chip is that the challenge of reducing errors is being overcome. Not only have Google found a way to reduce errors, but the reduction of those errors shows exponential improvement.

A key factor that is critically important in determining how many times a quantum calculation needs to be performed is the nature of the quantum algorithm required for a particular type of calculation. Some quantum algorithms require more repetitions to amplify the likelihood of identifying the correct result than others. This in turn affects the degree to which computational speedup can be achieved. The nature of the algorithm’s method itself determines the form of speedup that can be achieved.

Amongst the various quantum algorithm speedup possibilities some are of direct relevance to the computational activities of financial services firms. For the moment let’s consider just two forms of speedup, quadratic and exponential.

Quadratic speedup is of particular relevance when calculating Monte Carlo simulations. Whereas exponential speedup plays a role in calculations for financial modelling such as forecasting and value at risk calculations as well as risk factor modelling.

So, given that we anticipate quantum computational advantage and speedup benefits from quantum computers, in order to understand the degree of benefit let’s illustrate this graphically:

Considering quadratic speedup, the chart shows that there is a dramatic reduction in the number of calculations required to be performed by a quantum algorithm versus a classical algorithm. The slight curve in the line for the classical algorithm reflects the fact that certain calculations such as optimisation require more, and increasingly intensive computation with classical computers.

For calculation problems especially relevant to financial services firms such as Monte Carlo simulations, we can already see the potential enormous benefit if we are able to deploy quantum computers for these types of calculation.

But this is only one form of speedup so let’s give some consideration to our other example which offers an ability to perform large calculations even faster, the clue being the word “exponential”:

If an important aspect of your firm’s activities is performing value at risk or forecasting calculations or various other forms of financial modelling then you could certainly make use of quantum algorithms. You would benefit from exponential speed up, meaning that the deployment of quantum computing would offer an utter game changer in terms of computing power available to you.

Given that the nature of the calculation gives rise to the selection of the underlying quantum algorithm, the business computational need therefore determines the degree of quantum computational advantage that can be achieved.

Let’s reflect back on our desktop PC with 896 bits. If we are trying to perform a calculation that can benefit from exponential speed up, without going into the details of the underlying algorithm (but it could be Shor’s algorithm if you are really interested) a quantum calculation considering 896 bits, due to these quantum computing scaling advantages, requires approximately 719 million calculation operations. A quantum computer can process around a million calculations per second, so in this simple example we can estimate that this calculation with a mind-bending number of permutations of bits (2⁸⁹⁶) would take around 719 seconds, or about 12 minutes.

The story however, is somewhat different when it comes to quadratic speedup, and unfortunately not as advantageous for this example. In the case of quadratic speedup the number of iterations required is the square root of 2⁸⁹⁶or 2⁴⁴⁸, and once again if we assume a million operations per second, the amount of time required to perform this is 2.3×10¹²¹ years - a trillion trillion trillion trillion trillion trillion trillion trillion trillion trillion years.

If you have ever looked for an example of a vast chasm in performance outcome based upon a seemingly trivial difference in calculation approach, this one is surely hard to beat. That doesn’t mean that quadratic speedup offers no benefit. But it does mean that the type of calculation in the context of quantum computing is very important. As we move into a world where quantum computing becomes a reality in financial services it is incredibly important for executives to understand this. Quantum computing is definitely not a one size fits all solution.

Quantum computers are exceptionally powerful when solving problems with exponential complexity, such as factoring large numbers or optimising complex systems. However, they do not outperform classical computers for tasks with low computational complexity, deterministic outcomes, sorting data or calculations with simple arithmetic. Quantum computers are extremely good at taking a large number of possibilities and identifying the best result and there is no obvious underlying rule to determine what that result might be. However, if we were to use quantum computer to calculate the result of a + b = c, no matter how big a or b are, the quantum computer isn’t going to offer you any dramatic improvement for that calculation.

Quantum computing is best seen as a complementary tool, not a universal replacement for classical systems.

A reader might consider that for the purposes of this article deliberately large and provocative numbers such as our CPU example of 2⁸⁹⁶ bits have been chosen, but this is not the case. Here and now, today, IBM’s Condor processor utilises 1,121 qubits, comfortably exceeding our example of 896 classical computing bits.

Hopefully we are conveying that while the potential of quantum computing is, in certain areas truly staggering, it is not a panacea for all computational problems, and that the nature of the calculation you need to perform also needs to be taken into account. Executives should be equipped to identify opportunities where quantum computing can have enormous impact and separate these from circumstances where expectations should be tempered. Investing ahead of time to understand this subject matter will enable executives to plan their business models in anticipation of the availability of this exceptionally powerful upcoming technology.

Quantum magic: Understanding the Quantum “Oracle”

Knowing that a quantum computer holds a collection of qubits all in a state of superposition, in other words collectively holding every permutation of those bits at once, how does this get converted into an answer?

For anyone who has seen any of the numerous explanations of quantum computing there is usually much discussion of quantum entanglement, superposition and other aspects of the science but little explanation of how this actually creates a result from a quantum calculation.

Let us try to cast some light on this by explaining a key aspect of the quantum calculation process, the oracle - an essential yet often barely understood aspect of how a quantum computer works. In simple terms the oracle is a powerful technical shortcut in a problem-solving process.

Imagine being in a maze where you don’t know the right path. The oracle is like a guide that tells you whether you’re closer to the exit after each step, without revealing the whole map. The oracle doesn’t know the answer, but it can determine whether a possible answer is more or less likely to be correct, and so it enables you to narrow down the correct answer, and in this example find the exit to the maze. This echoes the underlying difference between classical and quantum computing, in that a classical computer would need to check every possible path in that maze to find the exit, but the capability of the oracle gives quantum computing a powerful shortcut by ruling out incorrect possibilities.

The oracle is tailored to understand information about what makes an answer correct, without actually knowing what the answer is. It is coded to understand a particular mathematical property, specific pattern or perhaps a target value to identify whether a particular potential solution is more or less likely to be correct, and this is expressed in the form of a mathematical function.

The oracle itself is a subroutine that is contained within a larger algorithm, such as one of the algorithms that enable the speedup possibilities we discussed earlier. The oracle is implemented in a quantum circuit by arranging a set of quantum gates corresponding to the mathematical function identifying the correct answer.

When the oracle encounters a state of the qubits that the underlying mathematical function identifies is likely to be correct, the oracle has a technical way of marking that particular result. We mentioned earlier the concept of amplifying the correct answer, and this is exactly where that amplification takes place.

Then depending on the type of algorithm required and therefore the degree of speedup, the calculation needs to be repeated a certain number of times. After enough calculation repetitions with the oracle marking and amplifying individual answers, the result with the highest frequency is the right answer.