Performance evaluation of variational quantum eigensolver and quantum dynamics algorithms on the advection-diffusion equation

Abstract

Near-term quantum algorithms are a promising route to solving partial differential equations, but gauging their true potential requires separating algorithmic performance from sampling and hardware noise. We benchmark a ground-state variational quantum eigensolver (VQE), cast as a variational quantum linear solver, against the Trotterization, variational quantum imaginary time evolution, and adaptive variational quantum dynamics simulation methods applied to the one-dimensional advection-diffusion equation in the recent quantum-dynamics study by Alipanah et al. [Phys. Rev. Res. 7, 043318 (2025)] at matched grid and problem size. On a noiseless state-vector simulator the $N=4$ VQE drives the final-time infidelity to a numerical floor ($\sim 10^{-14}$) once the depth reaches $L\approx 5$, an \emph{algorithmic ceiling} set by exact expectation values. Evaluating the same solver with a finite number $S$ of measurement shots, still without hardware noise, makes the infidelity sampling limited, following $1-f\approx c/S$ (a best-case readout-sampling estimate, with the solution's signs assumed known), providing a regime-matched comparison with the shot-based emulator of Alipanah \emph{et al.}\ and explaining the gap to their noisy hardware runs ($>10^{-1}$). The benchmark thus decomposes the near-term error budget into algorithmic, sampling, and hardware contributions, with a matched-depth resource comparison. The formulation applies without modification across $N=4,5,6$ qubits and to a two-dimensional (eight-qubit, $16\times 16$) problem evolved to $t=1$, where the state-vector VQE holds a $\sim 10^{-7}$ algorithmic-ceiling infidelity against the sampling-limited $\sim 10^{-5}$ of the corresponding shot-based simulation, a difference of measurement regime rather than algorithmic superiority.

13 pages, 4 figures. Published version

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