A Modular Engine for Quantum Monte Carlo Integration
Ismail Yunus Akhalwaya, Connolly, Adam, Roland Guichard, Steven Herbert, Cahit Kargı, Alexandre Krajenbrink, Michael Lubasch, Conor Mc Keever, Julien Sorci, Michael Spranger, Ifan Williams
We present the Quantum Monte Carlo Integration (QMCI) engine developed by Quantinuum. It is a quantum computational tool for evaluating multi-dimensional integrals that arise in various fields of science and engineering such as finance. This white paper presents a detailed description of the architecture of the QMCI engine, including a variety of distribution-loading methods, a novel quantum amplitude estimation method that improves the statistical robustness of QMCI calculations, and a library of statistical quantities that can be estimated. The QMCI engine is designed with modularity in mind, allowing for the continuous development of new quantum algorithms tailored in particular to financial applications. Additionally, the engine features a resource mode, which provides a precise resource quantification for the quantum circuits generated. The paper also includes extensive benchmarks that showcase the engine's performance, with a focus on the evaluation of various financial instruments.
We present the Quantum Monte Carlo Integration (QMCI) engine developed by Quantinuum. It is a quantum computational tool for evaluating multi-dimensional integrals that arise in various fields of science and engineering such as finance. This white paper presents a detailed description of the architecture of the QMCI engine, including a variety of distribution-loading methods, a novel quantum amplitude estimation method that improves the statistical robustness of QMCI calculations, and a library of statistical quantities that can be estimated. The QMCI engine is designed with modularity in mind, allowing for the continuous development of new quantum algorithms tailored in particular to financial applications. Additionally, the engine features a resource mode, which provides a precise resource quantification for the quantum circuits generated. The paper also includes extensive benchmarks that showcase the engine's performance, with a focus on the evaluation of various financial instruments.
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 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.
We investigate the feasibility of integrating quantum algorithms as subroutines of simulation-based optimisation problems with relevance to and potential applications in mathematical finance. To this end, we conduct a thorough analysis of all systematic errors arising in the formulation of quantum Monte Carlo integration in order to better understand the resources required to encode various distributions such as a Gaussian, and to evaluate statistical quantities such as the Value-at-Risk (VaR) and Conditional-Value-at-Risk (CVaR) of an asset. Finally, we study the applicability of quantum Monte Carlo integration for fundamental financial use cases in terms of simulation-based optimisations, notably Mean-Conditional-Value-at-Risk (Mean-CVaR) and (risky) Mean-Variance (Mean-Var) optimisation problems. In particular, we study the Mean-Var optimisation problem in the presence of noise on a quantum device, and benchmark a quantum error mitigation method that applies to quantum amplitude estimation -- a key subroutine of quantum Monte Carlo integration -- showcasing the utility of such an approach.
Quantum Monte Carlo (QMC) methods deliver highly accurate electronic structure calculations but are computationally intensive. The quantum Monte Carlo kernel library (QMCkl) provides a modular, portable collection of high-performance kernels implementing the core building blocks of QMC calculations. It offers a C-compatible application programming interface, supports the TREXIO standard for input, and covers essential QMC kernels including atomic and molecular orbitals, cusp corrections, the Jastrow factor, and the necessary derivatives also to perform variational and structural optimization. QMCkl separates algorithmic development from hardware-specific tuning by combining human-readable reference implementations with performance-optimized kernels that produce identical numerical results. The library enables consistent, efficient, and reproducible simulations across different QMC codes and architectures and achieves substantial speedups in the evaluation of the energy and its derivatives. Beyond QMC, QMCkl can accelerate deterministic quantum chemistry workflows and visualization tools, promoting cross-code interoperability and simplifying high-performance scientific software development.
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 of the solution Xt —appearing in fields such as finance and fluid mechanics. We focus on the problem setting where the coefficients depend on μt through expectations of some functions E[φk(Xt)] , and the goal is to compute the expectation of a function 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)] with accuracy ϵ , making O(1/ϵ1+2/p) queries to the quantum circuit for time evolution over one step, where 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) . 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.
Efficient and robust uncertainty quantification (UQ) is essential in aerospace systems, as emphasized by both NASA’s CFD Vision 2030 and Certification by Analysis (CbA) 2040 initiatives. However, the extensive computational resources required by traditional UQ methods, especially when coupled with high-fidelity computational fluid dynamics (CFD) simulations, pose significant barriers. Quantum computing (QC) has been identified by the CFD Vision 2030 study as a promising technology to address these challenges, but has not yet been extensively explored in the context of UQ. In this study, a Quantum Computing-based Monte Carlo (QCMC) algorithm is implemented using a Qiskit simulator. This method utilizes Quantum Amplitude Estimation (QAE), specifically tailored for near-term Noisy Intermediate-Scale Quantum (NISQ) computers. The QCMC method demonstrates comparable accuracy to classical Monte Carlo (MC) methods on a two-dimensional heat diffusion problem while reducing the required number of samples (oracle calls) by ∼8X. By substantially lowering the computational overhead of UQ, this quantum computing-based algorithm could be a promising tool required to enable robust uncertainty-informed aerospace design.