Turbulent Mixing Simulation via a Quantum Algorithm

Abstract

Probability density function (PDF) methods have been very useful in describing many physical aspects of turbulent mixing. In applications of these methods, modeled PDF transport equations are commonly simulated via classical Monte Carlo techniques, which provide estimates of moments of the PDF at arbitrary accuracy. In this work, we use recently developed techniques in quantum computing and quantum enhanced measurements (quantum metrology) to construct a quantum algorithm that accelerates the computation of such estimates. Our quantum algorithm provides a quadratic speedup over classical Monte Carlo methods in terms of the number of repetitions needed to achieve the desired precision. We illustrate the power of our algorithm by considering a binary scalar mixing process modeled by means of the coalescence/dispersion (C/D) closure. The equation is first simulated using classical Monte Carlo methods, where we provide error estimates for the computation of central moments. We then simulate the quantum algorithm for this problem by sampling from the same probability distribution as that of the output of a quantum computer, and show that significantly less resources are required to achieve the same precision.

Quantum computational finance: Monte Carlo pricing of financial derivatives

“…To boost this success probability, multiple runs of phase estimation can be performed. The median of these multiple runs will have a higher success probability [15,22,48], as will be shown now. Letθ 1 , .…”

“…The simulations are performed such that quantum and classical estimates have a similar confidence (> 99.5%), which implies a number of independent single-qubit phase estimation runs of D ≈ 24 [15], see also Appendix F. Fig. 3 shows a comparison between the error scalings for our quantum algorithm (blue solid curve with markers) and classical MC (orange dashed curve).…”

“…This quadratic dependency in is analogous to the classical Monte Carlo dependency Eq. (15). Obtaining a quadratic speedup for the number of repetitions is the core task of amplitude estimation.…”

“…2 (b) illustrates how Q imprints a phase of 2θ via the reflections just discussed. The eigenvalues of Q are e ±iθ with corresponding eigenstates |ψ ± [15]. The task is to resolve these eigenvalues via phase estimation, as shown in Fig.…”

“…It is evident that the scaling exponent of quantum phase estimation is almost twice the classical one. Indeed, the green solid curve shows the upper bound on the errors in quantum estimation defined as [15], see also Appendix F:…”