Sample-based quantum linear solvers measure quantum Krylov states $e^{-iAT}|b\rangle$ and reconstruct the solution of $Ax=b$ classically on the observed basis states. We develop this approach for positive-definite systems that arise from elliptic partial differential equations. We show that the sampling guarantees hold in any fixed basis with an efficient transform. We then replace the variational reconstruction by a Galerkin solve on the sampled set. For this reconstruction we derive expected and high-probability energy-norm error bounds, a sample complexity in terms of the smallest sampling probability on an essential set, a computable residual certificate, and a support-completeness theorem for the dephased outcome law. We formulate the method for elliptic problems on Riemannian cross sections. In two dimensions the stiffness matrix is metric-independent in conformal coordinates, and curved vessels lead to the same matrix class. Killing symmetries of the cross section give selection rules for the measurement outcomes, and the Lie derivative of the metric gives first-order bounds for geometric perturbations. We separate operator approximations, whose effect we bound, from sampler approximations, which affect coverage but not the target. As examples we treat the implicit step of pulsatile flow in elliptic, lens-shaped, and curved ducts. In simulations of up to 12 qubits, the number of Fourier modes required for 1% error grows from 14 to 60 while the number of unknowns grows from 64 to 4096. A classical residual-greedy selection is nevertheless more efficient than quantum sampling in all cases studied, consistent with the low measured sampling efficiency. On a six-qubit trapped-ion processor, the measured outcome laws follow a replacement-noise model whose strength grows by about 0.6% per two-qubit gate, and a parity selection rule rejects shots at the predicted rate.
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The paper studies sample-based quantum Krylov solvers for positive-definite linear systems from elliptic PDEs: measuring $e^{-iAT}|b\rangle$ identifies basis vectors, and a classical reduced solve reconstructs the solution. It allows measurement in any fixed efficiently transformable basis and replaces variational reconstruction with Galerkin projection, which minimizes energy-norm error on the sampled subspace.
For $x^*=A^{-1}b$, sampling law $q$, and $M$ shots, it proves $\mathbb E\|x_{\mathcal S}-x^*\|_A^2\leq\lambda_{\max}\sum_z|\hat x_z|^2e^{-Mq(z)}$. An essential set $\mathcal E$ is completely discovered with probability at least $1-\delta$ when $M\geq\log(|\mathcal E|/\delta)/q_{\min}(\mathcal E)$; a suitable coefficient-tail condition then guarantees energy accuracy. Spectral decomposition establishes that the fully dephased sampling law contains the solution support. Residual certificates and perturbation bounds are also derived.
The PDE formulation uses Riemannian cross sections, two-dimensional conformal invariance, and isometry-based measurement filters. Exact-evolution simulations of duct-flow systems up to 12 qubits show Fourier compression: one lens example requires 14–60 modes for 1% error as the system grows from 64 to 4096 unknowns. However, classical residual-greedy selection is more accurate at matched subspace size throughout the reported tests.
Six- and eight-qubit hardware experiments test outcome laws and parity filtering rather than dimension reduction: samples cover 89–100% of admissible modes. Noise approximately follows a replacement model, with fitted survival decay around 0.6% per logical two-qubit gate. No quantum advantage is demonstrated; efficient time scheduling remains heuristic, scalable circuit preparation is unresolved, and curved-vessel and clinical applicability require separate validation.
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Sample-based quantum linear solvers measure quantum Krylov states e−iAT∣b⟩ and reconstruct the solution of Ax=b classically on the observed basis states. We develop this approach for positive-definite systems that arise from elliptic partial differential equations. We show that the sampling guarantees hold in any fixed basis with an efficient transform. We then replace the variational reconstruction by a Galerkin solve on the sampled set. For this reconstruction we derive expected and high-probability energy-norm error bounds, a sample complexity in terms of the smallest sampling probability on an essential set, a computable residual certificate, and a support-completeness theorem for the dephased outcome law. We formulate the method for elliptic problems on Riemannian cross sections. In two dimensions the stiffness matrix is metric-independent in conformal coordinates, and curved vessels lead to the same matrix class. Killing symmetries of the cross section give selection rules for the measurement outcomes, and the Lie derivative of the metric gives first-order bounds for geometric perturbations. We separate operator approximations, whose effect we bound, from sampler approximations, which affect coverage but not the target. As examples we treat the implicit step of pulsatile flow in elliptic, lens-shaped, and curved ducts. In simulations of up to 12 qubits, the number of Fourier modes required for 1% error grows from 14 to 60 while the number of unknowns grows from 64 to 4096. A classical residual-greedy selection is nevertheless more efficient than quantum sampling in all cases studied, consistent with the low measured sampling efficiency. On a six-qubit trapped-ion processor, the measured outcome laws follow a replacement-noise model whose strength grows by about 0.6% per two-qubit gate, and a parity selection rule rejects shots at the predicted rate.
The paper studies sample-based quantum Krylov solvers for positive-definite linear systems from elliptic PDEs: measuring e−iAT∣b⟩ identifies basis vectors, and a classical reduced solve reconstructs the solution. It allows measurement in any fixed efficiently transformable basis and replaces variational reconstruction with Galerkin projection, which minimizes energy-norm error on the sampled subspace.
For x∗=A−1b, sampling law q, and M shots, it proves E∥xS−x∗∥A2≤λmax∑z∣x^z∣2e−Mq(z). An essential set E is completely discovered with probability at least 1−δ when M≥log(∣E∣/δ)/qmin(E); a suitable coefficient-tail condition then guarantees energy accuracy. Spectral decomposition establishes that the fully dephased sampling law contains the solution support. Residual certificates and perturbation bounds are also derived.
The PDE formulation uses Riemannian cross sections, two-dimensional conformal invariance, and isometry-based measurement filters. Exact-evolution simulations of duct-flow systems up to 12 qubits show Fourier compression: one lens example requires 14–60 modes for 1% error as the system grows from 64 to 4096 unknowns. However, classical residual-greedy selection is more accurate at matched subspace size throughout the reported tests.
Six- and eight-qubit hardware experiments test outcome laws and parity filtering rather than dimension reduction: samples cover 89–100% of admissible modes. Noise approximately follows a replacement model, with fitted survival decay around 0.6% per logical two-qubit gate. No quantum advantage is demonstrated; efficient time scheduling remains heuristic, scalable circuit preparation is unresolved, and curved-vessel and clinical applicability require separate validation.
To solve linear systems Ax=b, we propose a sample-based quantum linear solver combining quantum Krylov-state sampling with classical variational diagonalization. Rather than simulating the cost Hamiltonian Hb(A)=A†(I−∣b⟩⟨b∣)A, we measure quantum Krylov states e−iAT∣b⟩ generated by the original Hermitian operator A in the computational basis. We then classically diagonalize the projected cost Hamiltonian to reconstruct an approximate solution. Using a representation of A−1∣b⟩ as a linear combination of quantum Krylov states, we derive lower bounds on computational-basis sampling probabilities from solution amplitudes. We establish support-discovery guarantees for finite nonnegative-time grid sampling and signed-time sampling, which permits both positive and negative evolution times. The former bounds the maximum evolution time, whereas the latter improves ideal shot complexity at the cost of occasional long evolutions. For a condition-number bound κ on A, a lower bound βL on the solution probability weight of each of L target basis states, and failure probability δ, signed-time sampling achieves O(κ2βL−1log(L/δ)) shots, matching a worst-case lower bound up to constants within an evolution-only observation model in a specified parameter regime. Finally, we analyze the effects of finite-length qDRIFT compilation and circuit reuse on sampling and derive an infidelity bound for the reconstructed solution.
Quantum linear-system algorithms (QLSAs) have rigorous worst-case complexity guarantees, but their runtimes are often chosen from spectral information assumed in advance. What is largely lacking is an a posteriori progress flag: most QLSA workflows, unlike the classical counterparts, do not provide a built-in mechanism to signal whether a particular instance has already converged.
For discretizations of elliptic PDEs −∇⋅(a(x)∇u(x))=f(x), with divergence--gradient structure −∇⋅(a(x)∇)≈Ah=Gh†Gh, we formulate a stable first-order ODE whose limiting solution block is the desired Galerkin solution. The PDE-dependent scale is then \(\norm{G_h}=\bigO(h^{-1})\), comparable to factorized QLSA constructions with square-root condition-number scaling. We design an augmented dynamics with residual variables, in which measuring a residual register gives an on-the-fly convergence indicator without reconstructing the solution vector. For smooth right-hand sides, dynamic stopping can reduce the evolution time and gate count relative to a fixed worst-case schedule, and may also reduce exposure to accumulated hardware errors. Numerical experiments for a two-dimensional finite element Poisson problem show that the residual-register probability follows the actual error decay and, for some right-hand sides, can stop the quantum circuit well before a conservative worst-case runtime estimate is reached.
Solving systems of linear equations, Ax=b, is a ubiquitous task in computational science. The Harrow-Hassidim-Lloyd algorithm requires phase estimation, controlled rotations, and post-selection, impractical on noisy intermediate-scale quantum hardware. We extend sample-based quantum diagonalization (SQD), developed for quantum chemistry, to linear systems. Following the sample-based Krylov variant, the Hermitian matrix A is decomposed in the Pauli basis, the Trotterized time evolution of ∣b⟩ is sampled at several times to select computational-basis configurations, and the system is solved classically by a Galerkin projection onto the selected subspace. We prove that the subspace solution is exact when the sampled configurations contain the Krylov support of b and optimal in the energy norm for symmetric positive-definite matrices, so that noise-induced configurations cannot degrade it as long as they do not displace the ideal ones. On IBM Quantum hardware, the sampled distribution of a 4×4 example deviates from the exact one by eleven times the shot-noise level, yet the solution is recovered exactly. For a localized problem of dimension N=1024, SQD selects 33 configurations, reaches a relative error of 10−5, and its truncated subspaces coincide with the optimal choice; readout errors enlarge the subspace without degrading the accuracy. The tests also show the limitations: the selected subspace coincides with that of inexpensive classical selection, banded matrices require exponentially many Pauli terms in the binary encoding, and Trotter errors populate spurious configurations unless the circuits are deep. An MIT-licensed Python package and notebooks that reproduce the numerical results are provided.
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≈0.4, and final time T=1. Statevector, ideal-shot (105 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 ℓ2 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.
The Hamiltonian of a natural quantum system is usually determined indirectly, by proposing microscopic models and comparing their predicted observables with experiment. A more direct approach is to extract information about the Hamiltonian from measurements on the system itself. In many settings, however, direct access to the full system is unavailable and measurements can be performed only through controllable probes, as in quantum sensing and boundary spectroscopy. Existing Hamiltonian learning protocols under such restricted access typically require a special symmetry, a known parametric form, or controlled operations beyond free evolution. We introduce Quantum Krylov Learning (QKL), a restricted-access framework that avoids these requirements. QKL uses single temperature probe autocorrelators and their operator Krylov structure to extract model-independent spectral information about the unknown system. Rather than fitting a parametrized Hamiltonian, it reconstructs the operator Krylov Jacobi matrix representing the dynamical modes accessible to the probe. The method assumes that the unobserved system is at, or near, infinite temperature. Under this condition, the Jacobi matrix can be reconstructed directly from the autocorrelation of a probe observable, without oracle access to the Hamiltonian. The required operations are free evolution under the unknown H, together with probe state preparation and measurement. The learning core therefore requires no symmetry, no parametric model, and no controlled-unitary gates. We show a Lieb--Robinson bound for the operator Krylov chain and derive a data-driven criterion for the required observation window. We also establish an end-to-end sample complexity guarantee: under generic conditions on the spectral gaps and weights, the number of measurement shots required to achieve spectral precision ε is polynomial in 1/ε.
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.