Quantum Finance

# Quants in Quantum: Quantum Machine Learning Algorithms with Applications in Finance

In the 21st century, the financial sector has seen the rise of quantitative traders. Most financial decisions are no longer being conducted by people, but orchestrated through sophisticated algorithms and driven by complex machine learning models. These computerized models are the backbone of the financial market in the modern era.

Parallel to the rise of machine learning models, researchers in quantum computing have made significant strides in the progression towards quantum advantage. These recent advancements have introduced new opportunities to capitalize on the interdisciplinary field of quantum machine learning and to improve existing financial quantitative models. Moreover, improvements in quantum machine learning algorithms could result in great impact and possibly revolutionize the modern marketplace.

## Introduction

According to Moore's Law, the number of transistors on an integrated circuit follows a curious pattern, it seems to double every two years. During the infancy of the computer revolution this phenomenon held true to a remarkable degree, this continual improvement of microchip technology allowed for the rapid advancement of computer technology and applications. However, over the past decade, Moore's Law seems to be faltering. The tiny space onto which transistors are to be manufactured on a microchip is now so small that it has become increasingly difficult to produce faster computer chips.

Quantum computing hopes to solve this computing efficiency problem by offering an alternative method of computing instead of the traditional binary bits (1s and 0s), quantum computers utilize quantum bits called qubits which exist in a state between 1 and 0 called superposition. The unique feature of qubits allows for novel ways of tackling classical problems. For example, prime factorization is a famous problem commonly known to be very difficult on classical computers; however, a quantum computer given enough qubits will be able to exponentiate the factorization process of prime numbers.

Among the new methods and algorithms that quantum computers offer includes the ability to improve the current machine learning models. These models, which have a history of being crippled by their inefficiency, could benefit immensely from the advancements that quantum computing would have to offer. Specifically, the field of finance, which heavily utilizes machine learning models to forecast and make market predictions, would see a revolution in methods, analysis, and financial tools as a result of quantum computing.

## 1.1 Quantum Optimization Algorithms

Not all problems can be solved efficiently. Certain mathematical formulations are so computationally expensive that it would be infeasible for a classical computer to calculate. In computer science, these problems are categorized as NP-Hard problems. Nonetheless, other satisfying solutions to NP-Hard problems may exist. In this case, the optimal (feasible but may not be the best) solution is often selected as the next best option.

Even still, due to limitations in modern computational efficiency, it often remains incredibly difficult to find a satisfying solution, much less an optimal solution. The quantum characteristics of qubits offer a novel method of improving computational efficiency and deciphering the optimization problem. Qubits are the underlying foundation of new quantum optimization algorithms, which theoretically have greater precision and efficiency than standard optimization models. Examples of these algorithms include: the Quantum Evolutionary/Genetic Algorithm and the Quantum Particle Swarm Optimization [1].

## Quantum Evolutionary/Genetic Algorithms

Based upon the biological idea of Darwinism, genetic/evolutionary algorithms (GAs) are a stochastic approach to convergence. In GAs, the information of the solution is encoded in the chromosomes of a population [5]. Furthermore, GAs utilize the idea of natural selection to crossover, mutate, and select chromosomes from a population. The resulting population after n generations of offspring can be evaluated to produce the optimal solution for the optimization algorithm [5].

Quantum computers offer a way to accelerate genetic algorithmic models by providing an alternative method of encoding population information using quantum codes instead of traditional binary bits [2]. These encodings include the Bloch spherical coordinate, real number, and hybrid encoding approaches [2]. Each of the encoding approaches allow different improvements to GA models. For example, the Bloch sphere coordinate encoding technique circumvents the traditional pitfalls of randomness brought upon by measurements and can ensure a higher likelihood of an optimal solution [2]. Fundamentally, these encoding methods enable evolutionary algorithms to encapsulate a diverse gene pool with a small population size. Along with this, the parallel nature of quantum genetic algorithms improves upon the computation efficiency of traditional GAs [4] and generates more precise and faster convergence to an optimal solution.

## Quantum Particle Swarm Algorithms

Another biologically inspired approach to optimization is the particle swarm algorithm (PSOs). Particle swarm algorithms encapture the hive mentality of eusocial insects like bees and ants. In PSOs, the solution to the optimization problem is represented as a population of particles. The individual inter and intra particle interactions/experiences impact the next position of the particle. Ultimately, through a series of random interactions, the population of particles will converge on an optimal solution [6]. In contrast to GAs, the main advantages of PSOs are their simplicity and speed of convergence; nonetheless, PSOs severely lack in reliability and are susceptible to local extremes, making them vulnerable to premature convergence [6].

Quantum mechanics bridges the gap between the speed and reliability of PSOs by enhancing the convergent abilities of particle swarms. The entangled properties of qubits enable a truly random system for generating particle positions [2]. In traditional PSO algorithms, a particle's velocity is used to determine the particle's next position. Quantum PSO algorithms completely disregard the previous movements of particles; instead, the particles are represented using the quantum wave function (where according to the Heseinberg Uncertainty Principle the position and speed of the particle is undetermined) [2]. Ultimately, the increased randomness of the swarm particles permits a higher potential for quantum PSOs to avoid local extrema and premature convergence.

## 1.2 Applications of Optimization in Finance

Everyone likes free money. Quantitative investors (quants) like billionaire hedge fund manager Jim Simons imagined a system where he could sit back and relax while his computer filled his bank account [7]. Since the introduction of quantitative trading in the 1970s [8], financial management has shifted from human analysis to automated tools like machine learning models. These models have become standard in the modern financial industry; however, they still experience many of the same drawbacks and limitations of traditional machine learning models. Therefore, the financial sector may be among the first fields to make use of the innovations in quantum machine learning.

## Portfolio Optimization

Improvements in optimization using quantum computing would allow for a more efficient form of portfolio optimization. A traditional portfolio contains a selection of assets like fiat, bonds, and commodities [9]. The complication arises when determining the percentage and amount of those assets in the portfolio. How would a portfolio manager determine the amount of each asset in his/her portfolio to maximize profit and limit risk? This type of optimization falls under the umbrella of constraint satisfaction optimization. Moreover, given a selection of positive and negative constraints (e.g. monetary, legal, temporal, etc.), portfolio optimization hopes to satisfy these constraints and maximize the expected profit [1].

Fundamentally, constraint satisfaction boils down to an NP-Complete problem - a subset of NP-Hard problems - for which traditional optimization algorithms can be applied with varying success. The introduction of new regulation, call for transparency, transaction costs, and additional risks piles even more complexity onto modern portfolio management [3]. Ultimately, portfolio optimization is evolving into a problem that outpaces the current growth in computational resources [3]. For that reason, quantum computing algorithms could prove revolutionary in shaping portfolio optimization by providing managers with faster and better optimal solutions to optimization problems.

## Arbitrage

Occasionally, an imbalance of an asset price (or a combination of assets) on different markets delivers the opportunity for an investor to profit [10]. This concept is called arbitrage. Though a simple idea, arbitrages can be difficult to find and are even more difficult to execute with optimal efficiency. A simple arbitrage can be modeled as a graph of nodes and edges. Each node is labeled with the asset price, and each edge labeled with the conversion rate corresponding to its connected nodes [1]. An arbitrage is detected when a negative cycle within the graph is located.

The difficulty is in finding the optimal path (the best sequence of edges and nodes) to the arbitrage opportunity, any unnecessary detours could result in profit losses or potential risk. In a similar reduction, arbitrage is classified as an NP-Hard problem (traveling salesman) making it especially difficult for classical computers to solve in a reasonable amount of time. However, the unique property of quantum computers exponentially accelerates the solution finding process through optimization algorithms like the Quantum Evolutionary/Genetic Algorithm and the Quantum Particle Swarm Algorithm. Lucrative would underplay the potential applications for arbitrage using quantum algorithms. According to IBM, the revenue projected for arbitrage using quantum computing is among a 1 to 5 billion dollar marketplace [3]. Essentially, quantum optimization algorithms would outperform any existing models attempting to orchestrate an arbitrage opportunity.

## 1.3 Quantum "Hype"

Despite all the media attention and “quantum hype”, the era of quantum computing is still in its infancy. Although it promises many revolutionary advantages, it remains a challenging task to build and program a useful quantum computer. Ultimately, a functional quantum computer is required before quantum algorithms and quantum machine learning can be applied to financial markets.

There are many skeptics of quantum computing, including Mikhail Dyakonov, a physics professor at Laboratoire Charles Coulomb (L2C) Université Montpellier. He argues that theoretical quantum computers rely on extremely precise measurements that can never be achieved in the real physical world [11]. His concerns are met with equal fervor at the opposite end. Vladan Vuletic, professor of physics at MIT, is among many who remain optimistic. Dr. Vuletic is a co-founder of neutral atom computing company QuEra whose mission is to build the most scalable quantum computers [12]. His company envisions the potential benefits quantum computing has on businesses through techniques like optimization and hopes to bridge the gap to accessible quantum computing.

## Conclusion

There is no way to predict what the future holds for quantum computers in the financial sector. Nonetheless, recent advancements have indicated very promising results. Researchers have already performed instances of portfolio optimization on D-Wave's quantum annealer [1]. These experiments demonstrate the potential that algorithms like GA and PSO have to revolutionize exploit portfolio optimization and arbitrage opportunities. They demonstrate the possibility of exploiting the untapped riches of industry yet to be uncovered.

## References

[1]

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Y. Li, M. Tian, G. Liu, C. Peng and L. Jiao, "Quantum Optimization and Quantum Learning: A Survey," in IEEE Access, vol. 8, pp. 23568-23593, 2020, doi: 10.1109/ACCESS.2020.2970105.[3]

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Katoch, S., Chauhan, S.S. & Kumar, V. A review on genetic algorithm: past, present, and future. Multimed Tools Appl 80, 8091–8126 (2021). https://doi.org/10.1007/s11042-020-10139-6[6]

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“The Best Way to Quantum.” QuEra, https://www.quera.com/.