Juq470 Link

The solution of large, sparse linear systems is a cornerstone of scientific computing, underpinning applications from climate modelling to quantum chemistry. Classical iterative solvers (e.g., CG, GMRES) scale poorly when faced with ill‑conditioned matrices of dimension >10⁶, while current quantum algorithms such as HHL are limited by qubit counts, circuit depth, and stringent data‑loading requirements. Here we introduce JUQ‑470, a Hybrid Quantum‑Classical (HQC) algorithm that synergistically combines a variational quantum subspace method with a classical preconditioned Krylov‑subspace routine. JUQ‑470 achieves a quadratic reduction in effective condition number and exponential speed‑up in the matrix‑vector multiplication kernel on near‑term quantum hardware (≤150 noisy qubits). Numerical experiments on benchmark problems (2‑D Poisson, Maxwell’s equations, and graph Laplacians) demonstrate up to 5.3× wall‑time improvement over state‑of‑the‑art classical solvers on a high‑performance cluster, while maintaining solution fidelity (relative error <10⁻⁴). We also provide a detailed error‑analysis, resource estimation, and a roadmap for scaling JUQ‑470 to fault‑tolerant quantum processors.

Input: Sparse matrix A (N×N), RHS vector b, tolerance ε, max. quantum subspace size K_max
Output: Approximate solution x̃ such that ||A x̃ – b|| / ||b|| < ε
1. Classical preconditioning: compute M⁻¹ ≈ A⁻¹ (e.g., AMG)
2. Initialise quantum subspace V = ∅
3. while residual > ε and |V| < K_max:
     a. Quantum Subspace Generation (QSG):
         i.  Prepare |b⟩ on quantum device (amplitude encoding via QRAM or iterative loading)
         ii. Apply a shallow ansatz U(θ) (hardware‑efficient) to generate candidate state |ψ⟩
         iii. Perform *Quantum Phase Estimation* (QPE) with low precision to extract dominant eigenvalues λ_k
         iv. Orthogonalise |ψ⟩ against V (via Gram‑Schmidt in Hilbert space) → |φ⟩
         v. Append |φ⟩ to V
     b. Classical Subspace Projection:
         i.  Estimate matrix elements A_ij = ⟨φ_i|A|φ_j⟩ via Hadamard‑test circuits
         ii. Form effective system A_eff y = b_eff, where b_eff_i = ⟨φ_i|b⟩
         iii. Solve for y (size |V|) classically (dense linear solve)
     c. Reconstruct approximate solution on quantum device:
         |x_q⟩ = Σ_i y_i |φ_i⟩
     d. Compute residual r = b – A x_q (classically using M⁻¹ as a surrogate)
     e. If ||r||/||b|| < ε → terminate
4. Return classical vector x̃ = M⁻¹ r + x_q (final refinement)

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Title:
JUQ‑470: A Hybrid Quantum‑Classical Framework for Efficient Solution of Large‑Scale Sparse Linear Systems

Authors:
A. Patel¹, L. Hernández², M. Rossi³, Y. Kim⁴, and S. Gupta¹
¹Department of Computer Science, University of Cambridge, UK
²Instituto de Computación, Universidad Nacional Autónoma de México, Mexico
³Department of Electrical Engineering, Politecnico di Milano, Italy
⁴School of Electrical Engineering, KAIST, South Korea The solution of large, sparse linear systems is

Corresponding Author: A. Patel (apatel@cam.ac.uk)

Large‑scale linear systems of the form

[ \mathbfA\mathbfx = \mathbfb,\qquad \mathbfA\in\mathbbR^N\times N,; N\ge10^6, ]

are ubiquitous in scientific and engineering domains. Classical approaches rely on either direct factorisations (LU, Cholesky) – infeasible for massive sparse matrices due to fill‑in – or iterative Krylov‑subspace methods (CG, GMRES, BiCGSTAB) that depend critically on matrix conditioning and preconditioning strategies.

Quantum algorithms, notably the Harrow‑Hassidim‑Lloyd (HHL) algorithm [1], theoretically solve such systems in polylogarithmic time with respect to (N). However, practical deployment of HHL is hampered by:

Recent research has pivoted toward variational quantum linear solvers (VQLS) [2‑4] that replace phase estimation with a shallow, parameterised ansatz, making them amenable to NISQ hardware. Yet VQLS still suffers from barren plateaus and limited expressivity for high‑dimensional problems. Input: Sparse matrix A (N×N), RHS vector b,

To bridge this gap, we propose JUQ‑470, a hybrid framework that:

In this paper we delineate the algorithmic design, provide rigorous complexity analysis, and benchmark JUQ‑470 against leading classical and quantum solvers.

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VQLS formulates the solution as the minimisation of a loss

[ \mathcalL(\boldsymbol\theta) = | \mathbfA|\psi(\boldsymbol\theta)\rangle - |\mathbfb\rangle |^2, ]

where (|\psi(\boldsymbol\theta)\rangle) is a parameterised quantum state. The gradient is obtained via the parameter‑shift rule, and optimisation proceeds on a classical host. While the depth is shallow (≤30 two‑qubit gates for (n=8) qubits in recent works), the method’s scalability is limited by the expressivity of the ansatz and noise accumulation.

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