Quantum-accelerated conjugate gradient methods via spectral initialization

TL;DR

This work proposes a quantum-accelerated conjugate gradient (QACG) framework that uses a fault-tolerant quantum subroutine to generate a spectrally informed initial guess for a classical CG solver, thereby reducing the effective condition number that governs the CG iterations. By restricting the quantum inversion to a low-energy subspace via a spectral filter, QACG decouples spectral conditioning between quantum and classical components and treats the remainder with scalable HPC CG. The authors develop time-complexity and resource-estimation models on a STAR fault-tolerant quantum device and demonstrate, for the 3D Poisson problem, that the hybrid approach can outperform purely classical HPC methods under favorable quantum clock speeds and spectral-window choices, while using far fewer logical qubits and gates than an end-to-end QLSA. They further validate the concept with a 1D Poisson numerical experiment showing substantial reductions in CG iterations as more eigenvalues are included in the quantum initialization, and discuss practical open problems such as readout interfaces and broader CAE applicability.

Abstract

Solving large-scale linear systems problems is a central task in scientific and industrial computing. Classical iterative solvers face increasing difficulty as the number of unknowns becomes large, while fully quantum linear solvers require fault-tolerant resources that remain far beyond near-term feasibility. Here we propose a quantum-accelerated conjugate gradient (QACG) framework in which a fault-tolerant quantum algorithm is used exclusively to construct a spectrally informed initial guess for a classical conjugate gradient (CG) solver. Rather than replacing classical kernels, the quantum subroutine functions as a cooperative accelerator that selectively suppresses low-energy spectral components responsible for slow classical convergence. We analyze the total runtime and resource requirements of this integrated quantum-HPC platform for the 3D Poisson equation. A central feature of QACG is a controllable decomposition of the condition number between the quantum and the classical solver, enabling flexible allocation of computational effort across quantum and classical resources. Under explicit architectural assumptions, we identify regimes in which this cooperative strategy yields a runtime advantage over purely classical approaches while requiring substantially fewer quantum resources than end-to-end quantum linear solvers. These results illustrate a concrete pathway toward the scientific and industrial use of early-stage fault-tolerant quantum computing and point to a scalable hybrid paradigm in which quantum devices act as accelerators within high-performance computing workflows rather than as standalone replacements.

Figures (11)

• Figure 1: Conceptual figure of the quantum-accelerated conjugate gradient (QACG) method. Left: CG starts a classical search with a zero vector $x_{(0)}$ when there is no prior knowledge. Right: QACG first utilizes a quantum computer to generate an initial guess $\tilde{x}$ in a low-energy subspace to accelerate the subsequent search by CG (blue arrow).
• Figure 2: Quantum circuits used to construct an initial guess for the 3D Poisson equation: (a) amplitude amplification applied to boost the success probability of the solution state and (b) the HHL operator composed of state preparation, quantum phase estimation, and controlled rotation to implement a spectrally filtered matrix inversion.
• Figure 3: Quantum phase estimation circuit used within HHL to coherently encode eigenvalue information of the system matrix. Controlled time-evolution unitaries and an inverse quantum Fourier transform map eigenphases onto a phase register, enabling subsequent controlled rotations that implement spectral inversion.
• Figure 4: Quantum circuit implementing the unitary $U=\exp(iAt)$ for the 3D periodic Laplacian via spectral decomposition. By applying quantum Fourier transforms on each spatial register, performing diagonal phase rotations corresponding to the Laplacian eigenvalues, and uncomputing the Fourier basis, the circuit realizes exact time evolution under the separable 3D Laplacian.
• Figure 5: Quantum circuit for the 1D Fourier-series loader (FSL). The cascaded entangler $U_c$ prepares a truncated Fourier-coefficient state on an $(m+1)$-qubit register, which is then entangled with the remaining qubits via CNOT ladders and mapped to the spatial domain by an inverse QFT, yielding $\ket|\tilde{b}>$.

Theorems & Definitions (37)

• Definition 1: Quantum acceleration problem
• Definition 2: Finite difference Poisson equation
• Lemma 1: Iteration complexity of CG
• proof
• Definition 3: HPC model
• Theorem 1: Time complexity of HPCG
• proof
• Definition 4: STAR architecture model
• Theorem 2: Time complexity of HHL
• proof