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.

We present a bound on the length of the path defined by the ground states of a continuous family of Hamiltonians in terms of the spectral gap . We use this bound to obtain a significant improvement over the cost of recently proposed methods for quantum adiabatic state transformations and eigenpath traversal. In particular, we prove that a method based on evolution randomization, which is a simple extension of adiabatic quantum computation, has an average cost of order 1/ 2 , and a method based on fixed-point search has a maximum cost of order 1/ 3/2 . Additionally, if the Hamiltonians satisfy a frustration-free property, such costs can be further improved to order 1/ 3/2 and 1/ , respectively. Our methods offer an important advantage over adiabatic quantum computation when the gap is small, where the cost is of order 1/ 3 .

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The rapid developments that have occurred in quantum computing platforms over the past few years raise important questions about the potential for applications of this new type of technology to fluid dynamics and combustion problems, and the timescales on which such applications might be possible. As a concrete example, here a quantum algorithm is developed and employed for predicting the rate of reactant conversion in the binary reaction of F + rO → (1 + r) P roduct in non-premixed homogeneous turbulence. These relations are obtained by means of a transported probability density function equation. The quantum algorithm is developed to solve this equation and is shown to yield the rate of the reactants' conversion much more efficiently than current classical methods, achieving a quadratic quantum speedup, in line with expectations for speedups arising from quantum metrology techniques more broadly. This provides an important example of a quantum algorithm with a real engineering application, which can build a connection to present work in turbulent combustion modelling and form the basis for further development of quantum computing platforms and their applications to fluid dynamics.

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