Abstract Discrete stochastic processes (DSP) are instrumental for modeling the dynamics of probabilistic systems and have a wide spectrum of applications in science and engineering. DSPs are usually analyzed via Monte-Carlo methods since the number of realizations increases exponentially with the number of time steps, and importance sampling is often required to reduce the variance. We propose a quantum algorithm for calculating the characteristic function of a DSP, which completely defines its probability distribution, using the number of quantum circuit elements that grows only linearly with the number of time steps. The quantum algorithm reduces the Monte-Carlo sampling to a Bernoulli trial while taking all stochastic trajectories into account. This approach guarantees the optimal variance without the need for importance sampling. The algorithm can be further furnished with the quantum amplitude estimation algorithm to provide quadratic speed-up in sampling. The Fourier approximation can be used to estimate an expectation value of any integrable function of the random variable. Applications in finance and correlated random walks are presented. Proof-of-principle experiments are performed using the IBM quantum cloud platform.
Abstract Discrete stochastic processes (DSP) are instrumental for modeling the dynamics of probabilistic systems and have a wide spectrum of applications in science and engineering. DSPs are usually analyzed via Monte-Carlo methods since the number of realizations increases exponentially with the number of time steps, and importance sampling is often required to reduce the variance. We propose a quantum algorithm for calculating the characteristic function of a DSP, which completely defines its probability distribution, using the number of quantum circuit elements that grows only linearly with the number of time steps. The quantum algorithm reduces the Monte-Carlo sampling to a Bernoulli trial while taking all stochastic trajectories into account. This approach guarantees the optimal variance without the need for importance sampling. The algorithm can be further furnished with the quantum amplitude estimation algorithm to provide quadratic speed-up in sampling. The Fourier approximation can be used to estimate an expectation value of any integrable function of the random variable. Applications in finance and correlated random walks are presented. Proof-of-principle experiments are performed using the IBM quantum cloud platform.
Stochastic processes play a fundamental role in physics, mathematics, engineering and finance. One potential application of quantum computation is to better approximate properties of stochastic processes. For example, quantum algorithms for Monte Carlo estimation combine a quantum simulation of a stochastic process with amplitude estimation to improve mean estimation. In this work we study quantum algorithms for simulating stochastic processes which are compatible with Monte Carlo methods. We introduce a new ``analog'' quantum representation of stochastic processes, in which the value of the process at time t is stored in the amplitude of the quantum state, enabling an exponentially efficient encoding of process trajectories. We show that this representation allows for highly efficient quantum algorithms for simulating certain stochastic processes, using spectral properties of these processes combined with the quantum Fourier transform. In particular, we show that we can simulate T timesteps of fractional Brownian motion using a quantum circuit with gate complexity polylog(T), which coherently prepares the superposition over Brownian paths. We then show this can be combined with quantum mean estimation to create end to end algorithms for estimating certain time averages over processes in time O(polylog(T)ε−c) where $3/2
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 methods use random sampling to estimate numerical quantities which are hard to compute deterministically. One important example is the use in statistical physics of rapidly mixing Markov chains to approximately compute partition functions. In this work, we describe a quantum algorithm which can accelerate Monte Carlo methods in a very general setting. The algorithm estimates the expected output value of an arbitrary randomized or quantum subroutine with bounded variance, achieving a near-quadratic speedup over the best possible classical algorithm. Combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions, which use multiple-stage Markov chain Monte Carlo techniques. The quantum algorithm can also be used to estimate the total variation distance between probability distributions efficiently.
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.
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.
We propose an approximate classical algorithm for estimating quantum circuits' transition amplitudes, ⟨Ψf∣U^∣Ψ0⟩, based on Monte Carlo sums over bidirectionally sampled Feynman paths. The algorithm generates Feynman paths, i.e., sequences of intermediate computational basis states, by combining information from random walks initiated at both the input and output states, Ψ0 and Ψf, respectively. It further combines this bidirectional approach with Multiple Importance Sampling, a powerful variance reduction technique that avoids variance peaks caused by divisions by the very low sampling probabilities associated with hard to reach paths. Empirical data demonstrate the superior convergence rate exhibited by the bidirectional algorithm over a unidirectional, forward-only sampling approach. This superiority is due to the effectiveness of the proposed algorithm in finding and weighting paths with non-zero amplitude in an extremely sparse path space. Sum of paths algorithms exhibit very favourable space complexity. Monte Carlo replaces exhaustive enumeration of the exponentially large path space by stochastic sampling, the computational cost being determined by the variance of the estimator. This work demonstrates these two features by presenting results on very large quantum circuits, with up to 4096 qubits.