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arXiv:2607.12688

Quantum PDE Solvers in Practice: Application-Driven Benchmarking of the Heat Equation

Mahmoud Elkarargy, Abdelaziz Rahwan, Abdelrahman Elsayed, Forat Hatem

quant-phcs.SESubmitted Jul 14, 2026• Announced Jul 15, 2026• 0 citations
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Abstract

Quantum PDE solvers are difficult to evaluate in practice because published studies use different discretizations, output models, reconstruction rules, and hardware assumptions. We present a reproducible, application-driven benchmark for the 1-D Dirichlet heat equation that compares eleven kernels under the same problem instances and readout contract. The benchmark covers coherent linear solvers (HHL, QSVT, and QLS-Fourier), VQLS, imaginary-time methods (QITE, var-QITE, and AVQDS), real-time Hamiltonian simulation and unitary dilations (Hamiltonian simulation, Schade-Hamiltonian, and Schr"odingerisation), and the spectral quantum simulation method (QSM). We use three initial conditions, four grid sizes from $n=4$ to $7$ qubits ($N=16$ to $128$), a CFL-like ratio $r\approx0.4$, and final time $T=1$. Statevector, ideal-shot ($10^5$ shots per step), and noisy Aer backends separate algorithmic, sampling, and device-noise errors. On statevector, QSM and Schade-Hamiltonian reproduce the semi-discrete reference to floating-point precision, Schr"odingerisation reaches approximately $10^{-4}$ error, and QITE is the strongest non-transform method for smooth data. Under the fixed-shot setting, HHL degrades to approximately $0.79$ relative $\ell_2$ error, while several low-depth or postselected methods become readout-limited. A norm-mismatch ablation attributes 23--29% of the $n=7$ smooth-initial-condition error of Hamiltonian simulation, AVQDS, and QLS-Fourier to reconstruction normalization. Compact observables, including total thermal energy and individual Fourier-mode weights, require 1--3 orders of magnitude fewer shots than full-field reconstruction. The resulting public benchmark provides a practical guide for selecting quantum PDE solvers.

Comments
11 pages, 5 figures, 5 tables. Accepted for presentation at the 2026 IEEE International Conference on Quantum Computing and Engineering (QCE26, IEEE Quantum Week 2026); to appear in the conference proceedings

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Quantum PDE solvers are difficult to evaluate in practice because published studies use different discretizations, output models, reconstruction rules, and hardware assumptions. We present a reproducible, application-driven benchmark for the 1-D Dirichlet heat equation that compares eleven kernels under the same problem instances and readout contract. The benchmark covers coherent linear solvers (HHL, QSVT, and QLS-Fourier), VQLS, imaginary-time methods (QITE, var-QITE, and AVQDS), real-time Hamiltonian simulation and unitary dilations (Hamiltonian simulation, Schade-Hamiltonian, and Schr"odingerisation), and the spectral quantum simulation method (QSM). We use three initial conditions, four grid sizes from n=4n=4n=4 to 777 qubits (N=16N=16N=16 to 128128128), a CFL-like ratio r≈0.4r\approx0.4r≈0.4, and final time T=1T=1T=1. Statevector, ideal-shot (10510^5105 shots per step), and noisy Aer backends separate algorithmic, sampling, and device-noise errors. On statevector, QSM and Schade-Hamiltonian reproduce the semi-discrete reference to floating-point precision, Schr"odingerisation reaches approximately 10−410^{-4}10−4 error, and QITE is the strongest non-transform method for smooth data. Under the fixed-shot setting, HHL degrades to approximately 0.790.790.79 relative ℓ2\ell_2ℓ2​ error, while several low-depth or postselected methods become readout-limited. A norm-mismatch ablation attributes 23--29% of the n=7n=7n=7 smooth-initial-condition error of Hamiltonian simulation, AVQDS, and QLS-Fourier to reconstruction normalization. Compact observables, including total thermal energy and individual Fourier-mode weights, require 1--3 orders of magnitude fewer shots than full-field reconstruction. The resulting public benchmark provides a practical guide for selecting quantum PDE solvers.
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)2643-156410.1103/ndc3-bdwt] 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 (∼10^{-14}) once the depth reaches L≈5, 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-f≈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, 16×16) problem evolved to t=1, where the state-vector VQE holds a ∼10^{-7} algorithmic-ceiling infidelity against the sampling-limited ∼10^{-5} of the corresponding shot-based simulation, a difference of measurement regime rather than algorithmic superiority.
Quantum methods for solving partial differential equations (PDEs) generally fall into a few general categories, each leveraging different quantum mechanisms to achieve potential speedups. One well-known approach is the quantum linear system solver, epitomized by the Harrow-Hassidim-Lloyd (HHL) algorithm, which uses phase estimation and eigenvalue inversion to solve large, sparse linear systems arising from discretized linear PDEs [1]. This method can yield exponential improvements over classical solvers under certain conditions (mostly when the matrix is well-conditioned and sparse). More recent developments, such as Quantum Singular Value Transformation (QSVT), provide a framework for matrix inversion in a more flexible way, often improving on the cost and error dependence of earlier solvers [2]. Time-dependent PDEs can be tackled via Hamiltonian simulations. The Trotter-Suzuki method decomposes the exponential of a Hamiltonian into products of exponentials of simpler sub-Hamiltonians, allowing a piecewise approximation of continuous dynamics [3]. Continuous-time quantum walks, which treat a PDE operator as a Hamiltonian that governs the walk, similarly simulate processes like diffusion or wave propagation on discretized grids or graphs [4]. Beyond direct simulation, variational quantum algorithms have emerged as a promising route in the noisy intermediate-scale quantum era. These approaches encode the PDE solution as a parameterized quantum state and optimize the parameters via classical feedback, thereby minimizing a cost function tied to the PDE residual or energy functional [5]. Quantum annealing or adiabatic methods, meanwhile, can solve optimization problems derived from PDE constraints by adiabatically evolving a simple initial Hamiltonian to the final problem Hamiltonian and staying in the system's ground state [3]. In our study, we implement a new parameterized quantum model, specifically designed to solve large linear systems on quantum hardware. This model is less deep and computationally faster than the proposed one in [5], reducing the complexity and speeding up convergence times. Such improvements are particularly valuable in cases where classical algorithms may fail due to the problem's size or complexity. Ongoing research is focusing on generalized quantum models that can be adapted to practical PDE problems for solving large electromagnetic problems where classical computers can fail.
Can a quantum PDE algorithm avoid the polynomial cost of resolving a fine spatial mesh? For standard fixed-order discretizations, direct classical methods require work polynomial in h−1h^{-1}h−1, or equivalently in the number of spatial degrees of freedom Nh=Θ(h−d)N_h=Θ(h^{-d})Nh​=Θ(h−d). Direct quantum implementations of a parabolic semigroup still have coherent complexity O~(T/h)\widetilde{\mathcal O}(\sqrt{T}/h)O(T​/h), and gradient-dependent observables such as heat flux and dissipation introduce additional mesh dependence. Decay of the solution norm will further suppress the postselection probability for preparing a normalized final state. We develop a multilevel quantum algorithm that estimates linear and quadratic observables directlydirectlydirectly and places the fine--coarse cancellation inside the circuit before measurement. A contour-based LCU reconstructs each target-time correction from a coherent family of shifted resolvent differences. Rather than block encoding the fine and coarse inverses separately, we encode their difference through a shifted Ritz--Schur factorization, exposing its O(hℓ2)\mathcal O(h_\ell^2)O(hℓ2​) two-grid normalization. For Fourier hierarchies, the corresponding SELECT oracle consists of a quantum Fourier or sine transform, a spectral-band selector, and reversible diagonal arithmetic. We also give a non-Fourier realization based on energy-orthogonal dyadic midpoint details in one dimension, together with structured tensor-product extensions under fixed-rank coefficient and access assumptions. For readouts with derivative order 0≤χ≤20\leχ\le20≤χ≤2, optimized amplitude estimation removes allallall polynomial dependence on the finest mesh size. Under the stated access assumptions, both linear and quadratic observables can be estimated with complexity O~(1+(Tε)−1)\widetilde{\mathcal O}(1+(Tε)^{-1})O(1+(Tε)−1), with only polylogarithmic dependence on h−1h^{-1}h−1.
Solving systems of linear equations is a key subroutine in many quantum algorithms. In the last 15 years, many quantum linear solvers (QLS) have been developed, competing to achieve the best asymptotic worst-case complexity. Most QLS assume fault-tolerant quantum computers, so they cannot yet be benchmarked on real hardware. Because an algorithm with better asymptotic scaling can underperform on instances of practical interest, the question of which of these algorithms is the most promising remains open. In this work, we implement a method to partially address this question. We consider four well-known QLS algorithms which directly implement an approximate matrix inversion function: the Harrow-Hassidim-Lloyd algorithm, two algorithms utilizing a linear combination of unitaries, and one utilizing the quantum singular value transformation (QSVT). These methods, known as functional QLS, share nearly identical assumptions about the problem setup and oracle access. Their computational cost is dominated by query calls to a matrix oracle encoding the problem one wants to solve. We provide formulas to count the number of queries needed to solve specific problem instances; these can be used to benchmark the algorithms on real-life instances, under benevolent, idealized assumptions, without requiring access to quantum hardware to implement the procedure. We select three data sets: random generated instances that obey the assumptions of functional QLS, linear systems from simplex iterations on MIPLIB, and Poisson equations. Our methods can be easily extended to other data sets and provide a high-level guide to evaluate the performance of a QLS algorithm. In particular, our work shows that HHL underperforms in comparison to the other methods across all data sets, often by orders of magnitude, while the Chebyshev-approximation based methods typically have the best performance.
Spatial derivatives pose a basic challenge for quantum algorithms for partial differential equations: their standard discretizations have operator norms that grow as the mesh is refined, leading to large block-encoding normalization factors. Even when preconditioning improves access to the solution, estimating derivative-dependent physical observables can reintroduce polynomial dependence on the inverse mesh width. We develop a multilevel quantum algorithm for estimating linear and quadratic observables of elliptic equations that addresses this readout cost. The algorithm decomposes the output into corrections across nested discretizations and realizes fine--coarse cancellation coherently, before measurement. A Ritz-complement factorization connects the O(h2)O(h^2)O(h2) size of the corrected response with a block encoding at the same scale, allowing this decay to offset growing readout normalizations. Under efficient quantum access to the corrected coordinates, input data, and readouts, observables with readout scale O(h−χ)O(h^{-χ})O(h−χ), 0≤χ≤20\leχ\le20≤χ≤2, can be estimated to additive accuracy εεε at cost O~(ε−1)\widetilde O(ε^{-1})O(ε−1) using amplitude estimation or O~(ε−2)\widetilde O(ε^{-2})O(ε−2) using direct sampling, with only polylogarithmic dependence on the inverse finest mesh width hL−1h_L^{-1}hL−1​. We give explicit constructions of the required operator oracles for one-dimensional piecewise linear finite elements with diffusion piecewise constant on a fixed coarsest partition, and for Fourier spectral hierarchies. Output-specific bias estimates connect the discrete results to continuum accuracy.
We present a comprehensive review of quantum approaches for solving partial differential equations (PDEs) arising in computational fluid dynamics (CFD). We examine fully quantum approaches, including quantum linear system algorithms (QLSAs), ranging from the Harrow--Hassidim--Lloyd (HHL) algorithm to quantum singular value transformation (QSVT), Hamiltonian simulation, and quantum lattice Boltzmann methods (QLBMs), while emphasizing hybrid quantum--classical approaches, including quantum physics-informed neural networks (QPINNs) and amplitude-encoded variational PDE solvers. We focus on hardware-agnostic algorithms compatible with present noisy processors and emerging fault-tolerant architectures. For each framework, we analyze the mathematical formulation, algorithmic structure, and principal limitations. We also examine tensor-network (TN) representations, since CFD fields, differential operators, and geometrical information can often be encoded efficiently in low-rank form. The TN formalism bridges CFD discretizations and quantum states, operators, and circuits, enabling compact representations to be translated into tensor-programmable variational quantum algorithms (TP-VQAs). We further review benchmark problems, including Poisson, reaction, diffusion, and nonlinear model equations, and assess how well quantum algorithms capture key features of fluid dynamics. Our analysis highlights that potential quantum advantage is highly problem dependent and governed by condition number, representational complexity, state preparation, and measurement constraints. We outline capabilities, limitations, and challenges toward scalable quantum algorithms for CFD.