A Residual-Based Quantum Linear System Algorithm with Dynamic Stopping and Applications to Elliptic PDEs

Abstract: 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 $-\nabla\cdot(a(x)\nabla u(x))=f(x),$ with divergence--gradient structure [ -\nabla\cdot \big(a(x)\nabla) \approx A_h=G_h^\dagger G_h, ] 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.

• The paper introduces a novel quantum linear system algorithm that integrates residual evaluation and dynamic stopping for adaptive convergence.
• It reformulates elliptic PDEs as a first-order ODE using divergence-gradient factorization, allowing on-the-fly error tracking through measurable residuals.
• Numerical experiments show that the adaptive stopping rule can reduce runtime significantly compared to conservative, worst-case quantum schedules.

Residual-Based QLSA with Dynamic Stopping for Elliptic PDEs: Technical Summary

This paper develops a novel quantum linear system algorithm (QLSA) framework that embeds residual evaluation and dynamic stopping directly into quantum dynamics, with a focus on elliptic PDE discretizations. The approach brings a key feature of classical iterative linear algebra — adaptive, residual-based stopping criteria — into the quantum setting where typical QLSAs run for a predetermined, worst-case schedule independent of instance-specific convergence.

Motivation and Context

Existing QLSAs such as HHL and QSVT-based solvers achieve near-optimal black-box complexity in terms of sparsity, condition number, and error. However, their run-time is chosen based on a priori spectral information — for example, bounds on the matrix's smallest and largest singular values — and does not adaptively exploit favorable spectral properties of specific right-hand sides or instance structure. In classical numerical methods, residual-based stopping is standard, allowing for early termination once sufficient accuracy is detected.

For large-scale problems such as elliptic PDEs, after finite-element discretization, the system matrix $A_h$ typically exhibits condition number $\kappa(A_h) = \mathcal{O}(h^{-2})$, with $h$ the meshwidth. The actual computational difficulty, in practice, depends not only on $\kappa(A_h)$, but also on the spectral decomposition of the right-hand side $b_h$. Smooth right-hand sides may have negligible overlap with high-frequency, ill-conditioned modes, enabling faster practical convergence in the classical Krylov and conjugate gradient methods due to adaptive residual monitoring.

ODE-Based Quantum Linear System Solvers and Residual Embedding

The paper reformulates the quantum solution of $A_h x = b_h$ as an initial value problem of a stable first-order ODE characterized by the divergence-gradient factorization $A_h = G_h^\top G_h$ inherent in elliptic PDE discretization. Specifically:

• The first-order system augments the solution dynamics with residual variables, so that the limiting state encodes the Galerkin solution, while the evolution's transient naturally tracks residuals.
• The scaling of the ODE generator is $\mathcal{O}(h^{-1})$, matching the cost of QLSAs employing square-root condition-number scaling via block-encoding or factorized constructions [33].
• The residual block can be measured at intermediate times, providing an on-the-fly convergence indicator without a full solution reconstruction.

This structure enables the realization of a quantum residual-based stopping rule: the simulation dynamically measures the probability amplitude in the residual register and decides to terminate the evolution once convergence into a small-residual regime is detected, as formalized in their Theorem 1.2 and 4.2.

Dynamic Stopping and Complexity Bounds

The stopping criterion is organized as follows:

• For a sequence of checkpoint times $t_k$, the algorithm prepares, evolves the ODE, and measures the residual register. If the outcome's probability estimate crosses a fixed (constant, not $\epsilon$-dependent) threshold, the algorithm enters a deterministic coasting phase to ensure the target error $\kappa(A_h) = \mathcal{O}(h^{-2})$0 is finally reached.
• This method's sampling overhead is logarithmic (not polynomial) in the number of checkpoints and the inverse failure probability, avoiding expensive high-precision amplitude estimation.
• The final cost for attaining an $\kappa(A_h) = \mathcal{O}(h^{-2})$1-accurate normalized solution is

$\kappa(A_h) = \mathcal{O}(h^{-2})$2

(ignoring output-normalization and input state preparation factors), aligning with the optimal mesh-dependent scaling for quantum elliptic PDE solvers but now adaptive to instance difficulty.

Numerical Validation and Practical Implications

Numerical experiments on benchmark two-dimensional Poisson problems discretized by finite elements corroborate that the probability in the residual register closely tracks the true solution error. For multiple right-hand sides — especially those with more regularity and less high-frequency content — the quantum residual-based stopping rule allows the algorithm to halt significantly earlier than a conservative, worst-case schedule would dictate.

Key implications:

• Adaptivity: Unlike conventional QLSAs, this method adapts quantum runtime to instance difficulty, reducing exposure to hardware noise and error by potentially running shorter circuits.
• Separation of Concerns: The framework distinguishes between the quantum resources needed for the base ODE evolution and those for the residual measurement and dynamic stopping, enabling fine-tuned resource estimation and error tracking.
• Compatibility: The approach is agnostic to the underlying quantum ODE solver and could be combined both with state-of-the-art nonunitary simulation algorithms and block-encoding techniques.

Theoretical and Practical Impact

On the theoretical front, this work bridges a gap between classical and quantum numerical linear algebra, embedding residual tracking and adaptivity into the quantum algorithmic workflow — an aspect that had been missing from QLSA literature, which usually focuses on worst-case guarantees.

From a practical standpoint, the technique provides:

• Mechanisms for early stopping, reducing computation time in favorable cases.
• A rigorous, a posteriori error estimator based on directly measurable quantum observables, facilitating instance-optimal quantum scientific computing for PDEs.
• Potential extensibility to other stable linear systems, preconditioning schemes, and adaptive algorithms beyond the elliptic PDE setting — provided compatible factorization or stable first-order ODE formulations exist.

Speculation on Future Developments

Future work could generalize these mechanisms to:

• Nonsymmetric or non-elliptic steady-state problems (e.g., advection-diffusion), where similar first-order dynamics with measurable residuals can be constructed.
• Development of observable-specific or adjoint-based stopping rules, enabling early termination once quantities of interest (not just the full solution) have converged.
• Improved integration with multilevel and preconditioned quantum solvers, harnessing both mesh hierarchy and residual adaptivity for further efficiency gains.

Conclusion

This paper introduces a robust residual-based dynamic stopping framework for QLSAs applied to elliptic PDE discretizations, making quantum solvers more adaptive and resource-efficient by directly embedding convergence monitoring into the quantum dynamics. This innovation closes an important gap with classical methods, enabling quantum algorithms to more intelligently leverage problem instance properties and monitor computational progress efficiently. The methodology is expected to be readily generalizable to other stable linear systems and stimulating further advances in quantum numerical algorithms for scientific computing.