用于求解期权价格动力学的高效哈密顿量模拟
Efficient Hamiltonian Simulation for Solving Option Price Dynamics
AI总结：
本文提出将Black-Scholes方程映射为Schrödinger方程，通过单辅助量子比特嵌入非厄米传播子并结合周期边界条件复制初始条件，利用Quantum Signal Processing在容错量子计算机上高效求解期权价格动力学，以9量子比特实现与经典算法相当的精度。
摘要：
金融衍生品定价，尤其是在不同到期时间和执行价格下的欧式期权定价，是金融领域中的一个重要问题。在假设波动率和利率为常数的情况下，描述普通期权价格动态的模型是Black-Scholes模型，这是一个线性抛物型偏微分方程，其终值由期权合约的收益函数给出，且没有额外的边界条件。本文提出了一种数字量子算法，通过将Black-Scholes方程映射到Schrödinger方程，在量子计算机上求解该方程。所得哈密顿量的非厄米性质通过仅使用一个额外的辅助量子比特，将其传播子嵌入到扩大的Hilbert空间中来解决。此外，由于选择由离散化动量算符定义所给出的周期性边界条件，我们复制了初始条件，这显著提高了协议的稳定性和性能。该算法展示了一种可行的方法，即使用高效的哈密顿量模拟技术（如Quantum Signal Processing）在数字量子计算机上求解金融衍生品的价格动力学。我们的方法不同于基于Monte Carlo积分的方法，后者专注于在假设动力学已知的情况下对解进行采样。我们报告了在容错量子计算机上使用9个量子比特模拟其动力学时，预期精度水平可与经典数值算法相媲美，并且由于上述嵌入协议，后选择过程的预期成功概率超过60%。
英文摘要：
Pricing financial derivatives, in particular European-style options at different time-maturities and strikes, means a relevant problem in finance. 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 by mapping it to the Schrödinger equation. The non-Hermitian nature of the resulting Hamiltonian is solved by embedding its propagator into an enlarged Hilbert space by using only one additional ancillary qubit. Moreover, due to the choice of periodic boundary conditions, given by the definition of the discretized momentum operator, we duplicate the initial condition, which substantially improves the stability and performance of the protocol. The algorithm shows a feasible approach for using efficient Hamiltonian simulation techniques as Quantum Signal Processing to solve the price dynamics of financial derivatives on a digital quantum computer. Our approach differs from those based on Monte Carlo integration, exclusively focused on sampling the solution assuming the dynamics is known. We report expected accuracy levels comparable to classical numerical algorithms by using 9 qubits 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%.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
