Monte Carlo integration using quantum computers has been widely investigated, including applications to concrete problems. It is known that quantum algorithms based on quantum amplitude estimation (QAE) can compute an integral with a smaller number of iterative calls of the quantum circuit which calculates the integrand, than classical methods call the integrand subroutine. However, the issues about the iterative operations in the integrand circuit have not been discussed so much. That is, in the high-dimensional integration, many random numbers are used for calculation of the integrand and in some cases similar calculations are repeated to obtain one sample value of the integrand. In this paper, we point out that we can reduce the number of such repeated operations by a combination of the nested QAE and the use of pseudorandom numbers (PRNs), if the integrand has the separable form with respect to contributions from distinct random numbers. The use of PRNs, which the authors originally proposed in the context of the quantum algorithm for Monte Carlo, is the key factor also in this paper, since it enables parallel computation of the separable terms in the integrand. Furthermore, we pick up one use case of this method in finance, the credit portfolio risk measurement and estimate to what extent the complexity is reduced.

使用量子计算机的蒙特卡洛积分已得到广泛研究，包括对具体问题的应用。众所周知，与经典方法称为被整数子程序相比，基于量子幅度估计（QAE）的量子算法可以计算具有较少迭代次数的量子电路来计算积分，该迭代电路计算被整数。然而，关于迭代运算的问题在集成电路讨论不多。也就是说，在高维积分中，许多随机数用于计算被积物，并且在某些情况下，重复类似的计算以获得一个被积物的样本值。在本文中，我们指出，如果被积体对于来自不同随机数的贡献具有可分离的形式，则可以通过嵌套QAE和使用伪随机数（PRN）的组合来减少此类重复操作的次数。作者最初在蒙特卡洛量子算法的上下文中提出的PRN的使用也是本文的关键因素，因为它可以在被积物中并行计算可分离项。此外，我们在财务中选择了这种方法的一个用例，