Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distributed Variational Quantum Linear Solver

Tong Shen, Zeru Zhu, Ji Liu

Abstract

This paper develops a distributed variational quantum algorithm for solving large-scale linear equations. For a linear system of the form $Ax=b$, the large square matrix $A$ is partitioned into smaller square block submatrices, each of which is known only to a single noisy intermediate-scale quantum (NISQ) computer. Each NISQ computer communicates with certain other quantum computers in the same row and column of the block partition, where the communication patterns are described by the row- and column-neighbor graphs, both of which are connected. The proposed algorithm integrates a variant of the variational quantum linear solver at each computer with distributed classical optimization techniques. The derivation of the quantum cost function provides insight into the design of the distributed algorithm. Numerical quantum simulations demonstrate that the proposed distributed quantum algorithm can solve linear systems whose size scales with the number of computers and is therefore not limited by the capacity of a single quantum computer.

Distributed Variational Quantum Linear Solver

Abstract

This paper develops a distributed variational quantum algorithm for solving large-scale linear equations. For a linear system of the form , the large square matrix is partitioned into smaller square block submatrices, each of which is known only to a single noisy intermediate-scale quantum (NISQ) computer. Each NISQ computer communicates with certain other quantum computers in the same row and column of the block partition, where the communication patterns are described by the row- and column-neighbor graphs, both of which are connected. The proposed algorithm integrates a variant of the variational quantum linear solver at each computer with distributed classical optimization techniques. The derivation of the quantum cost function provides insight into the design of the distributed algorithm. Numerical quantum simulations demonstrate that the proposed distributed quantum algorithm can solve linear systems whose size scales with the number of computers and is therefore not limited by the capacity of a single quantum computer.

Paper Structure

This paper contains 16 sections, 1 theorem, 24 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Quantum Linear System Problem
  3. Variational Quantum Linear Solver
  4. Multi-Agent Quantum Computing
  5. Distributed Multi-Agent VQLS
  6. Problem Formulation
  7. Distributed Optimization Reformulation
  8. The Distributed VQLS Algorithm
  9. Numerical Experiments
  10. Large-Scale Demonstration
  11. Effect of Block Partitioning
  12. Failure of Distributed Gradient Descent
  13. Conclusion
  14. Proof of Lemma \ref{['lemma:goal_fixed']}
  15. Hadamard Test Circuits
  16. ...and 1 more sections

Key Result

Lemma 1

If $\mathbb G^{{\rm row}}$ and $\mathbb G^{{\rm col}}$ are connected, then $x^*$ is a least squares solution to $Ax=b$ if, and only if, there exist $\{z_i^*\}_{i=1}^m$ such that $(x^*,\{z_i^*\}_{i=1}^m)$ is a minimizer of $\sum_{i=1}^m \|\bar{A}_i x-\bar{b}_i-\bar{L}z_i\|^2$. $\blacktriangleleft$$\b

Figures (8)

  • Figure C1: Global residual norm and parameter consensus error trajectories for the proposed distributed VQLS algorithm applied to the 51-qubit Ising-inspired linear system in \ref{['eq:Ising_A']} and \ref{['eq:Ising_b']} with a $2 \times 2$ block partition and a 4-agent network
  • Figure C2: Global residual norm and consensus error performance comparison of the distributed VQLS algorithm under different block partitions for the $13$-qubit perturbed cluster-state stabilizer system
  • Figure C3: Global residual norm trajectories of the proposed distributed VQLS algorithm and its three variants applied to the 7-qubit Ising-inspired linear system in \ref{['eq:Ising_A']} and \ref{['eq:Ising_b']} with a $4 \times 4$ block partition and a 16-agent network
  • Figure D1: Hadamard test circuit for \ref{['eq:overlap_test']}
  • Figure D2: Hadamard test circuit for \ref{['eq:hadamard1']}
  • ...and 3 more figures

Theorems & Definitions (1)

  • Lemma 1