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IEEE International Conference on Quantum Computing and Engineering (QCE)

Quantum algorithms for Monte Carlo integration using pseudo-random numbers

Koichi Miyamoto

Published Oct 1, 2021• 3 citations
Publisher

Abstract

Quantum algorithms for Monte Carlo integration (QMCI), which are based on quantum amplitude estimation (QAE), provide quadratic speed-up compared with classical counterparts, and are therefore widely investigated, along with applications to industries such as finance. One typical feature of problems in finance is high dimensionality, or, in other words, the necessity to create numerous random numbers (RNs) to calculate the integrand. In such a situation, the original implementation of QMCI, in which states encoding the probability distributions of the RNs are generated on different quantum registers, requires a large number of qubits. In this poster, we propose an implementation of QMCI, in which pseudo-random numbers (PRNs) are sequentially generated on one register, and therefore the qubit number is tremendously reduced. Moreover, we show that, when the integrand has the form such that contributions from different RNs are separable into different terms, we can also achieve time complexity reduction with respect to the number of dimensions by combining the nested QAE and use of PRNs.

DOI
10.1109/qce52317.2021.00075

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Quantum algorithms for Monte Carlo integration (QMCI), which are based on quantum amplitude estimation (QAE), provide quadratic speed-up compared with classical counterparts, and are therefore widely investigated, along with applications to industries such as finance. One typical feature of problems in finance is high dimensionality, or, in other words, the necessity to create numerous random numbers (RNs) to calculate the integrand. In such a situation, the original implementation of QMCI, in which states encoding the probability distributions of the RNs are generated on different quantum registers, requires a large number of qubits. In this poster, we propose an implementation of QMCI, in which pseudo-random numbers (PRNs) are sequentially generated on one register, and therefore the qubit number is tremendously reduced. Moreover, we show that, when the integrand has the form such that contributions from different RNs are separable into different terms, we can also achieve time complexity reduction with respect to the number of dimensions by combining the nested QAE and use of PRNs.
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.
It is known that quantum computers can speed up Monte Carlo simulation compared to classical counterparts. There are already some proposals of application of the quantum algorithm to practical problems, including quantitative finance. In many problems in finance to which Monte Carlo simulation is applied, many random numbers are required to obtain one sample value of the integrand, since those problems are extremely high-dimensional integrations, for example, risk measurement of credit portfolio. This leads to the situation that the required qubit number is too large in the naive implementation where a quantum register is allocated per random number. In this paper, we point out that we can reduce qubits keeping quantum speed up if we perform calculation similar to the classical one, that is, estimate the average of integrand values sampled by a pseudo-random-number generator (PRNG) implemented on a quantum circuit. We present not only the overview of the idea but also concrete implementation of PRNG and application to credit risk measurement. Actually, reduction of qubits is a trade-off against increase of circuit depth. Therefore, full reduction might be impractical, but such a trade-off between speed and memory space will be important in adjustment of calculation setting considering machine specs, if large-scale Monte Carlo simulation by quantum computer is in operation in the future.
Numerical integration with Monte Carlo methods is a central computational task in many scientific and industrial applications, including financial derivative pricing and risk management. Classical Monte Carlo algorithms are computationally demanding: achieving an accuracy εεε typically requires a number of function evaluations scaling as O(1/ε2)O(1/ε^2)O(1/ε2). Quantum-accelerated Monte Carlo methods based on quantum amplitude estimation can in principle quadratically improve this dependence. However, \textit{quasi}-Monte Carlo methods have not been explored in the quantum context. In this work, we introduce a quantum quasi-Monte Carlo algorithm that combines low-discrepancy nets with quantum amplitude estimation. The proposed method prepares the quasi-random point set coherently in superposition. The method does not yield an asymptotic improvement over classical quasi-Monte Carlo, since the total error separates into a discretization error, determined by the finite net, and a quantum estimation error. Instead, we explore a pre-asymptotic advantage window: for a target accuracy that would classically require 2q2^q2q low discrepancy points, one can prepare a higher-resolution net of size 2Q2^Q2Q, with Q>qQ>qQ>q, in superposition and reach the same accuracy using significantly fewer function queries. This window can be controlled by tuning the circuit resolution and amplitude-estimation parameters, making the approach relevant for practical regimes where the number of queries is finite rather than asymptotically large.
Quantum Monte Carlo integration (QMCI), a quantum algorithm for calculating expectations that provides a quadratic speed-up compared to its classical counterpart, is now attracting increasing interest in the context of its industrial and scientific applications. In this paper, we propose the first application of QMCI to solving McKean-Vlasov stochastic differential equations (MVSDEs), a nonlinear class of SDEs whose drift and diffusion coefficients depend on the law μt\mu _{t}μt​ of the solution XtX_{t}Xt​ —appearing in fields such as finance and fluid mechanics. We focus on the problem setting where the coefficients depend on μt\mu _{t}μt​ through expectations of some functions E[φk(Xt)]\mathbb{E}[\varphi _{k}(X_{t})]E[φk​(Xt​)] , and the goal is to compute the expectation of a function E[ϕ(XT)]\mathbb{E}[\phi (X_{T})]E[ϕ(XT​)] at a terminal time T . We devise a quantum algorithm that leverages QMCI to compute these expectations, combined with a high-order time discretization method for SDEs and extrapolation of the expectations in time. The proposed algorithm estimates E[ϕ(XT)]\mathbb{E}[\phi (X_{T})]E[ϕ(XT​)] with accuracy ϵ , making O(1/ϵ1+2/p)O(1/\epsilon ^{1+2/p})O(1/ϵ1+2/p) queries to the quantum circuit for time evolution over one step, where p∈(1,2]p\in (1,2]p∈(1,2] is the weak order of the SDE discretization method. This demonstrates the speed-up over the well-known classical algorithm called the particle method with complexity of O(1/ϵ3)O(1/\epsilon ^{3})O(1/ϵ3) . We conduct numerical demonstrations of our quantum algorithm applied to examples of MVSDEs, with some parts emulated classically, and observe that the accuracy and complexity behave as expected.
Quantum computing is a very active and diverse area, with immense research and development occurring in quantum hardware design and algorithm creation for applications in various scientific fields. The key motivation behind such efforts is the potential for significant speed-up relative to simulations on classical computers. Quantum Monte Carlo Integration (QMCI) is a quantum algorithm that promises a quadratic advantage in terms of the required number of samples for a required accuracy over classical Monte Carlo Integration (MCI), which is a key statistical resource for integrating multi-dimensional space. In this study, we explore the reality of this potential, specifically for applications in Computational Fluid Dynamics (CFD). We perform an eigenvalue stability analysis for a 1D convection-diffusion equation, in which the coefficients depend on a random number. The quantum algorithm utilized uses quantum amplitude estimation (QAE) to accelerate the integral estimation process. Numerical values obtained from QMCI results are compared against numerical integration techniques, and acceptably low errors are observed. The performance of QMCI is compared against MCI by equating the number of oracle calls in QMCI with the number of samples required in MCI. No significant difference in this metric is observed, with QMCI requiring more oracle calls. This suggests that the expected quantum supremacy is not attainable for the considered numerical integration problem and its quantum-circuit implementation. Extension to available Noisy-Intermediate-Scale-Quantum (NISQ) hardware is expected to reveal more about the algorithm implementation and the simulator emulation. Potential future directions that can be explored in algorithmic development based on this preliminary research are outlined.
Quantum algorithms present a quadratically improved complexity over classical ones for certain sampling tasks. For instance, the Quantum Amplitude Estimation (QAE) algorithm promises to speed up the estimation of the mean of certain functions, given access to the quantum state corresponding to the probability distribution to be sampled from. Classically, samples are often obtained by simulating a Markov chain. In this work, we experimentally use encodings of Markov chains to prepare quantum states and run a quantum Markov Chain Monte Carlo (qMCMC) algorithm on Quantinuum System Model H2 and Quantinuum Helios quantum computers. We first demonstrate that it is possible to obtain accurate results by operating directly on the physical qubits. Crucially, to pave the way toward fault-tolerant execution, we also implement these algorithms using logical qubits protected by the Iceberg Quantum Error Detection (QED) code. By applying multiple rounds of mid-circuit QED during a random walk, we successfully catch and filter out hardware errors. These experiments demonstrate the feasibility of qMCMC on quantum hardware, bringing us closer to realizing quadratic quantum speedups with broad applications ranging from molecular modeling to machine learning and finance.