Pricing financial derivatives with exponential quantum speedup Javier Gonzalez-Conde, 1,2, ∗ ´ Angel Rodr´ ıguez-Rozas, 3 Enrique Solano, 1, 4, 5, 6 and Mikel Sanz 1,4,7, † 1 Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain 2 Quantum Mads, Uribitarte Kalea 6, 48001 Bilbao, Spain 3 Risk Division, Banco Santander, Avenida de Cantabria S/N, 28660 Boadilla del Monte, Madrid, Spain 4 IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, 48009, Bilbao, Spain 5 International Center of Quantum Artificial Intelligence for Science and Technology (QuArtist) and Department of Physics, Shanghai University, 200444 Shanghai, China 6 Kipu Quantum, Kurwenalstrasse 1, 80804 Munich, Germany 7 IQM, Nymphenburgerstr. 86, 80636 Munich, Germany (Dated: June 28, 2021) Pricing financial derivatives, in particular European-style options at different time-maturities and strikes, is a relevant financial problem. The dynamics describing the price of vanilla options when constant volatilities and interest rates are assumed, is governed by the Black-Scholes model, a linear parabolic partial differential equation with terminal value given by the pay-off of the option contract and no additional boundary conditions. Here, we present a digital quantum algorithm to solve Black-Scholes equation on a quantum computer for a wide range of relevant financial parameters by mapping it to the Schr¨ odinger equation. The non-Hermitian nature of the resulting Hamiltonian is solved by embedding the dynamics into an enlarged Hilbert space, which makes use of only one additional ancillary qubit. Moreover, we employ a second ancillary qubit to transform initial condition into periodic boundary conditions, which substantially improves the stability and performance of the protocol. This algorithm shows a feasible approach for pricing financial derivatives on a digital quantum com- puter based on Hamiltonian simulation, technique which differs from those based on Monte Carlo simulations to solve the stochastic counterpart of the Black Scholes equation. Our algorithm remarkably provides an exponen- tial speedup since the terms in the Hamiltonian can be truncated by a polynomial number of interactions while keeping the error bounded. We report expected accuracy levels comparable to classical numerical algorithms by using 10 qubits and 94 entangling gates to simulate its dynamics on a fault-tolerant quantum computer, and an expected success probability of the post-selection procedure due to the embedding protocol above 60%. I. INTRODUCTION In finance, European-style vanilla options are financial derivative contracts written on an underlying asset, which give the holder the right to buy or sell such asset on a specified fu- ture date at a predetermined strike price. One of the funda- mental tasks of quantitative finance is calculating a fair price of such option contract before their expiration time. This task is far from being straightforward due to the randomness asso- ciated to the time evolution of both the underlying stock and the interest rates, whose dynamics can be modelled via either a stochastic processes or a partial differential equation, and connected by the celebrated Feynman-Kac formula. One of the first successful approaches to this problem was achieved by F. Black and M. Scholes in 1972, who proposed the cele- brated Black-Scholes model [1], in which a lognormal distri- bution of the underlying stock price is assumed. Even though a closed-form solution exists for this dynamics, numerical solutions are still considered relevant since they serve as a benchmark for more sophisticated models. Numerical solu- tions also turn out to be fundamental when hedging a portfo- lio with a great number of coupled options. Several classical methods proposed in the literature includes finite differences, finite elements, Monte Carlo methods, and Fourier (spectral) methods [2–6]. ∗ Corresponding author: javier.gonzalezc@ehu.eus † Corresponding author: mikel.sanz@ehu.eus Quantum technologies have experienced a rapid develop- ment in the last decade. Recently, Google has achieved quan- tum advantage, meaning that they have performed a calcula- tion employing a superconducting processor faster than the most powerful supercomputers available today [7]. One of the fields which will expectably experience a deep impact due to this upcoming technology is finance. Indeed, the emer- gence of scalable quantum technologies will affect forecast- ing, pricing and data science, and will undoubtedly have an economic impact in the following years [8, 9]. Certainly, there already exist several efforts in this direction, for instance, an attempt to predict financial crashes [10, 11], the application of the principal component analysis to interest-rate correla- tion matrices [12], quantum methods for portfolio optimiza- tion [13–17], quantum generative models for finance [18], a quantum model for pricing collateral debt obligations [19], a protocol to optimize the exchange of securities and cash be- tween parties [20], an application to improve Monte Carlo methods in risk analysis [21, 22], among many others. Re- garding the option pricing problem, it has been studied the problem of solving Black-Scholes model employing Monte Carlo methods to solve the associated stochastic differential equation (SDE). In Ref. [23], the authors proposed a theoreti- cal approach based on solving the SDE using quantum Monte Carlo with quadratic speedup. Afterwards, an experimental implementation in the IBM Tokyo quantum processor was at- tained in Refs. [24–26], employing a gate-based methodol- ogy to price options and portfolios of options. More recently, another approach to solve the SDE was proposed in [27], in which an unary representation of the asset value is used to arXiv:2101.04023v2 [quant-ph] 25 Jun 2021