Quantum computing of nonlinear reacting flows via the probability density function method

TL;DR

This work addresses the challenge of simulating nonlinear reacting flows with stiff chemical source terms on quantum hardware by transforming the problem into a high-dimensional linear PDF transport equation. The entire space-time evolution is encoded in a single linear system via the history-state quantum linear solver, bypassing iterative time-stepping and intermediate measurements. A dedicated measurement scheme is developed to extract PDF moments efficiently, using a diagonal unitary and low-order polynomial approximations to achieve polynomial in $ obreak ext{log}(N)$ cost. Validation on a perfectly stirred reactor demonstrates the framework's ability to capture PDF evolution and statistics, highlighting a promising pathway for quantum-enabled nonlinear PDE simulations, albeit with hardware resource constraints and future work on optimization and diffusion integration.

Abstract

Quantum computing offers the promise of speedups for scientific computations, but its application to reacting flows is hindered by nonlinear source terms and the challenges of time-dependent simulations. We present a quantum framework to address these issues. We employ a probability density function (PDF) formulation to transform the nonlinear reacting-flow governing equations into high-dimensional linear ones. The entire temporal evolution is then solved as a single large linear system using the history state method, which avoids the measurement bottleneck of conventional time-marching schemes and fully leverages the advantages of quantum linear system algorithms. To extract the quantity of interest from the resulting quantum state, we develop an efficient algorithm to measure the statistical moments of the PDF, bypassing the need for costly full-state tomography. A computational complexity analysis indicates the potential for a near-exponential speedup over classical algorithms. We validate the framework by simulating a perfectly stirred reactor, demonstrating its capability to capture the PDF evolution and statistics of a nonlinear reactive system. This work establishes a pathway for applying quantum computing to nonlinear reacting flows.

Figures (5)

• Figure 1: Comparison of our history state method (left) with a conventional time-stepping approach (right) for solving the PDF transport equation. Our history state method solves the entire space-time evolution with a single QLSA call, avoiding the measurement bottleneck in the iterative conventional approach.
• Figure 2: Quantum circuit to measure the statistics from $\ket{\psi}$ encoding PDF on $n$ qubits.
• Figure 3: Operational cost for the exact operator $U$ and its low-order approximations, against the number of qubits $n$.
• Figure 4: PDF evolution obtained via the history state approach, using $n=9$ qubits for temporal and compositional spaces.
• Figure 5: Mean $\overline{\phi}$ and variance $\overline{\phi\prime^2}$ obtained with exact measurement operator $U$ and its low-order approximations.