Quantum algorithms for numerical differentiation of expected values with respect to parameters

The quantum algorithms for Monte Carlo integration (QMCI), which are based on quantum amplitude estimation (QAE), speed up expected value calculation compared with classical counterparts, and have been widely investigated along with their applications to industrial problems such as financial derivative pricing. In this paper, we consider an expected value of a function of a stochastic variable and a real-valued parameter, and how to calculate derivatives of the expectation with respect to the parameter. This problem is related to calculating sensitivities of financial derivatives, and so of industrial importance. Based on QMCI and the general-order central difference formula for numerical differentiation, we propose two quantum methods for this problem, and evaluate their complexities. The first one, which we call the naive iteration method, simply calculates the formula by iterative computations and additions of the terms in it, and then estimates its expected value by QAE. The second one, which we name the sum-in-QAE method, performs the summation of the terms at the same time as the sum over the possible values of the stochastic variable in a single QAE. We see that, depending on the smoothness of the function and the number of qubits available, either of two methods is better than the other. In particular, when the function is nonsmooth or we want to save the qubit number, the sum-in-QAE method can be advantageous.