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

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 $sim10^-14$ once the depth reaches Lapprox5, an 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-fapprox 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 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, $16times16$ problem evolved to t=1, where the state-vector VQE holds a sim10^-7 algorithmic-ceiling infidelity against the sampling-limited sim10^-5 of the corresponding shot-based simulation, a difference of measurement regime rather than algorithmic superiority.