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A mixed-precision quantum-classical algorithm for solving linear systems

Océane Koska, Marc Baboulin, Arnaud Gazda

Abstract

We address the problem of solving a system of linear equations via the Quantum Singular Value Transformation (QSVT). One drawback of the QSVT algorithm is that it requires huge quantum resources if we want to achieve an acceptable accuracy. To reduce the quantum cost, we propose a hybrid quantum-classical algorithm that improves the accuracy and reduces the cost of the QSVT by adding iterative refinement in mixed-precision A first quantum solution is computed using the QSVT, in low precision, and then refined in higher precision until we get a satisfactory accuracy. For this solver, we present an error and complexity analysis, and first experiments using the quantum software stack myQLM.

A mixed-precision quantum-classical algorithm for solving linear systems

Abstract

We address the problem of solving a system of linear equations via the Quantum Singular Value Transformation (QSVT). One drawback of the QSVT algorithm is that it requires huge quantum resources if we want to achieve an acceptable accuracy. To reduce the quantum cost, we propose a hybrid quantum-classical algorithm that improves the accuracy and reduces the cost of the QSVT by adding iterative refinement in mixed-precision A first quantum solution is computed using the QSVT, in low precision, and then refined in higher precision until we get a satisfactory accuracy. For this solver, we present an error and complexity analysis, and first experiments using the quantum software stack myQLM.

Paper Structure

This paper contains 22 sections, 1 theorem, 23 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Quantum Singular Value Transformation (QSVT)
  4. Block-encoding of matrices
  5. QSVT definition
  6. QSVT implementation
  7. Solving linear systems with QSVT
  8. Mixed-precision iterative refinement for linear systems
  9. Iterative refinement for QSVT-based linear system solution
  10. Algorithm
  11. Convergence and accuracy
  12. Complexity analysis
  13. Quantum cost
  14. Classical cost
  15. Remarks on data communication
  16. ...and 7 more sections

Key Result

Theorem 3.1

Suppose the QSVT can solve $Ax=b$ with low accuracy $\epsilon_l$ with $\epsilon_l \kappa < 1$ and that we apply mixed-precision iterative refinement as given in Algorithm alg:iter_ref_QSVT with high precision $u$ when computing the residual and solution update. Then after $i$ iterations we have $\Ve

Figures (5)

  • Figure 1: CPU-QPU communication scheme for Algorithm \ref{['alg:iter_ref_QSVT']} (BE=block-encoding, SP=state preparation).
  • Figure 2: Circuit for the block-encoding of the tridiagonal matrix in Equation (\ref{['eq:tridiag']}).
  • Figure 3: Scaled residual until convergence for $\kappa = 10$, targeted accuracy $\epsilon=10^{-11}$, and various values of $\epsilon_l$.
  • Figure 4: Scaled residual until convergence, $\kappa = 100,200,300$.
  • Figure 5: Complexity in calls to block-encoding for QSVT with and without iterative refinement, $\kappa=2$.

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 3.1
  • proof