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**Quantum finance**

Quantum computers are famous for being able to solve particular tasks such as factorizing numbers (Shor’s algorithm) and searching an unstructured database (Grover’s algorithm). The fact that quantum computers can solve these faster than on a classical computer is a theoretically proven result: for Shor’s algorithm the speedup is exponential, whereas for Grover’s algorithm it is quadratic. With the development of quantum computers in both universities and industry in rapid development, there is much excitement that the day that it is possible to run these algorithms on an real, physical quantum computer, and demonstrate their power. On the other hand, it is well-known that a quantum computer does not necessarily speed up the solution to an arbitrary problem. The Shor and Grover algorithms can be used in certain instances to help solve problems, but it is often not a trivial question of how to do this. For example, one can apply Grover’s algorithm to many types of combinatorial problems, such as the travelling salesman problem. However, there are also many classical algorithms (heuristic or otherwise) which are also very good at giving near-optimal results. Often just applying Grover’s algorithm is not faster than these classical algorithms, which makes the application of quantum computing not quite as simple as it appears. In this way, there is potentially a great market for quantum computers in industries where heavy numerical work that needs to be done. Examples of these include pharmaceutical, biophysics, engineering, and financial industries. Here we discuss how quantum computers can possibly help in financial problems.

The first application is in tasks such as portfolio optimization. Consider the following prototypical problem. Investor A has $1000 to invest, and has a choice between N stocks to invest in. Estimates of the growth, volatility of each stock are known. What is the optimal way to distribute the money between the stocks? Various strategies for how to distribute the capital exist, such as simply maximizing profits, minimizing risk, using various models. For example one simple approach is to use a Kelly criterion to maximize the profits [1]. In this approach, the main consideration is to maximize profits, but being aware of the possibility that if the price of a particular stock drops to zero, then there is henceforth no possibility of recovery from that point on. Hence considering long-term growth, it is better to diversify across several stocks. The optimal portfolio for this case can be solved with complexity polynomial in N, but more generally it is more complex than this. In general, the number of different types of portfolios are exponential with the number of different stocks, hence is a complex optimization problem. Quantum algorithms can be used to help in these situations, by using adiabatic quantum computing [2], and quantum amplitude estimation [3].

Another application is in the modelling of the time dynamics of stock prices. The Black-Scholes model is the best known model which simulates the movement of stock prices. In physics terms, it is a model of exponential growth combined with Brownian motion. An interesting observation is that it actually takes the same form as the Schrodinger equation in imaginary time. This equivalence has been exploited in the past to convert the dynamics of the Black-Scholes equation into a path integral formalism [4]. Another direction is actually to have quantum extension of the Black-Scholes model that explicitly takes into quantum mechanics. Instead of having classical fluctuations it is possible to incorporate quantum fluctuations, which produces different dynamics that are potentially more relevant in a realistic model of stock price movements [5]. In our work, we are examining potential methods to perform quantum simulation of they types of models such that they can be efficiently calculated using quantum mechanics.

**References**

[1] T. Byrnes, T. Barnett, International Journal of Theoretical and Applied Finance 21, 1850033 (2018).

[2] G. Rosenberg et al., arxiv 1508:06182.

[3] S. Woerner, npj Quantum Information 5, 15 (2019).

[4] B. Baaquie, J. Phys. I I France 7, 1733 (1997) .

[5] W. Segal, PNAS 95, 072 (1998).

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