Quantum Speedup of Monte Carlo Integration with respect to the Number of Dimensions and its Application to Finance
arXiv:2011.02165
Abstract
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.
arXiv comment13 pages, no figure
References in corpus (14)
- Quantum Networks for Elementary Arithmetic Operations
- Quantum computing for finance: overview and prospects
- Methodology for quantum logic gate constructions
- Quantum speedup of Monte Carlo methods
- Architectures for a quantum random access memory
- Quantum Risk Analysis
- Polynomial-time T-depth Optimization of Clifford+T circuits via Matroid Partitioning
- Quantum computational finance: Monte Carlo pricing of financial derivatives
- Amplitude estimation without phase estimation
- Quantum circuits of T-depth one
- On the advantages of using relative phase Toffolis with an application to multiple control Toffoli optimization
- Quantum unary approach to option pricing
- Prospects and challenges of quantum finance
- Reduction of Qubits in Quantum Algorithm for Monte Carlo Simulation by Pseudo-random Number Generator