D5.4 Evaluation of quantum algorithms for pricing and computation of VaR

Abstract

NEASQC Use Case 5 (UC5) works on the development and evaluation of quantum algorithms for financial applications. More specifically, the main line of research in UC5 focuses on Pricing and Value at Risk (VaR) problems. These two problems are computationally demanding tasks that are classically solved using Monte Carlo (MC) techniques. As Quantum Accelerated Monte Carlo (QAMC) techniques promise a quadratic speedup over Classical Monte Carlo (CMC) this roughly motivates why and how this field could benefit from the recent advances in Quantum Computing. This report summarises the development of a new pricing algorithm as well as its experimental assessment. In the first part of the report there is a brief summary of the classical and quantum techniques used to tackle both financial challenges. In the second part of the report the new method is explained in detail and evaluated. This new method includes two well defined parts: a new encoding for the quantum oracle and a new Amplitude Estimation (AE) technique. The new proposals are not yet applicable to current NISQ architectures although they represent a clear advance because: The new encoding algorithm allows pricing derivative products with negative payoffs. In particular, as illustrated in Section 3.1, it works in cases where other proposals fail as it is the case for a payoff of the form V (x) = x−K. The new AE algorithm, the so-called Real Quantum Amplitude Estimation (RQAE), allows pricing financial products with negative values (see Figure 9), which is an absolute novelty in the area. Current AE algorithms in the literature are concerned with the efficient estimation of the probability of finding a particular state, but they are not designed to be sensitive to the phases present in the underlying amplitudes. In contrast, RQAE is specifically tailored to be sensible to the sign of the amplitude. RQAE achieves a similar performance when compared with other previous well-known AE algorithms. In fact, in reference (Manzano et al., 2022), tighter bounds on the convergence of RQAE than any of the existing methods in the literature are proved. In this work, the experimental results show that although the performance is not the best one, it is on par with the cutting edge methods. However, there are some less positive results as: The new encoding algorithm gives a worse performance compared to the standard encoding algorithm. It depends on a factor (Npaths) that can overshadow or even kill the speedup depending on its specific setup. Nevertheless, NEASQC researchers who participate in the Use Case are actively working in new encoding methods to solve this issue. Two identified challenges have to be addressed in the near future. On the one hand, how to improve these algorithms to execute on current Quantum Processing Units (QPU), where deep circuits such as those required by the AE routines are not feasible without error correction techniques. On the other hand, previous works such as (Montanaro, 2015; Rebentrost et al., 2018) implicitly assume that the simulation of the Stochastic Differential Equation (SDE) to solve for these financial calculations can be directly done by translating the classical circuits into quantum circuits. However, this direct translation would require in practice a number of qubits too far from the current possibilities of the NISQ era.