Transient Heat Conduction on Quantum Hardware: Replicated Temperature-Field Reconstruction and Shot-Count Sensitivity with a Variational Quantum Linear Solver
Abstract
Implicit discretizations of transient heat conduction repeatedly generate sparse linear systems, making diffusion problems a natural test bed for quantum linear solvers. This study evaluates how accurately a variational quantum linear solver (VQLS) recovers the temperature field produced by one backward-Euler step of one-dimensional conduction on IBM quantum hardware. A pre-specified confirmatory experiment compares N=4 interior temperatures encoded on two qubits with N=8 temperatures encoded on three qubits using ten paired hardware blocks; a separately defined N=16, four-qubit extension is evaluated in five independently scheduled exploratory blocks. Local VQLS optimization reduces relative temperature-field error to approximately 10^{-8} for all three cases, establishing an essentially exact ideal reference at this scale. On hardware, mean relative L2 temperature-field error is 8.07% for N=4 and 20.14% for N=8, a paired increase of 12.06 percentage points (95% CI, 10.71-13.42; exact paired sign-flip p=0.001953). The exploratory N=16 mean is 24.75+/-1.84%. A pre-specified N=16 shot-count follow-up then evaluates the same locked circuit at 2,048, 8,192, and 32,768 shots in six independently scheduled blocks. Mean field error remains 23.64%, 24.57%, and 24.37%, respectively; the 32,768-minus-2,048-shot contrast is +0.74 percentage points (95% CI, -0.65 to 2.12; p=0.281). In contrast, ideal multinomial sampling predicts a reduction from 2.68% to 0.75%. Higher shot counts reduce block-to-block variability but do not recover the mean cold-side temperature field. The results show that full-field readout of a strongly nonuniform thermal state is limited by more than ordinary finite-shot sampling under the tested hardware conditions, motivating observable-focused quantum formulations for heat flux, stored energy, and selected temperatures.
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BibTeX
@misc{mao2026transient,
title = {{Transient Heat Conduction on Quantum Hardware: Replicated Temperature-Field Reconstruction and Shot-Count Sensitivity with a Variational Quantum Linear Solver}},
author = {Yijin Mao and Yuwen Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24004},
eprint = {2610.24004},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24004},
}
RIS
TY - UNPB
TI - Transient Heat Conduction on Quantum Hardware: Replicated Temperature-Field Reconstruction and Shot-Count Sensitivity with a Variational Quantum Linear Solver
AU - Yijin Mao
AU - Yuwen Zhang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.24004
UR - https://arxiv.org/abs/2610.24004
AB - Implicit discretizations of transient heat conduction repeatedly generate sparse linear systems, making diffusion problems a natural test bed for quantum linear solvers. This study evaluates how accurately a variational quantum linear solver (VQLS) recovers the temperature field produced by one backward-Euler step of one-dimensional conduction on IBM quantum hardware. A pre-specified confirmatory experiment compares N=4 interior temperatures encoded on two qubits with N=8 temperatures encoded on three qubits using ten paired hardware blocks; a separately defined N=16, four-qubit extension is evaluated in five independently scheduled exploratory blocks. Local VQLS optimization reduces relative temperature-field error to approximately 10^{-8} for all three cases, establishing an essentially exact ideal reference at this scale. On hardware, mean relative L2 temperature-field error is 8.07% for N=4 and 20.14% for N=8, a paired increase of 12.06 percentage points (95% CI, 10.71-13.42; exact paired sign-flip p=0.001953). The exploratory N=16 mean is 24.75+/-1.84%. A pre-specified N=16 shot-count follow-up then evaluates the same locked circuit at 2,048, 8,192, and 32,768 shots in six independently scheduled blocks. Mean field error remains 23.64%, 24.57%, and 24.37%, respectively; the 32,768-minus-2,048-shot contrast is +0.74 percentage points (95% CI, -0.65 to 2.12; p=0.281). In contrast, ideal multinomial sampling predicts a reduction from 2.68% to 0.75%. Higher shot counts reduce block-to-block variability but do not recover the mean cold-side temperature field. The results show that full-field readout of a strongly nonuniform thermal state is limited by more than ordinary finite-shot sampling under the tested hardware conditions, motivating observable-focused quantum formulations for heat flux, stored energy, and selected temperatures.
ER -