Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Efficient Quantum Algorithm for Linear System Problem in Tensor Format

Zeguan Wu, Sidhant Misra, Tamás Terlaky, Xiu Yang, Marc Vuffray

TL;DR

The paper tackles solving linear systems where the coefficient matrix and right-hand side have low-rank tensor structure (LSP-TF) by presenting a circuit-implementable quantum linear system algorithm (QLSA) based on adiabatic-inspired evolution. It introduces a Hamiltonian decomposition into Type-1 and Type-2 components and develops explicit quantum circuits, including a binary-spectral encoding, QFT-based adder, and quantum multiplier, to implement the necessary time evolutions. A detailed cost analysis shows the total runtime scales polylogarithmically with the problem dimension under reasonable assumptions, potentially offering exponential speedups over general classical solvers for large tensor-format problems. The work also discusses trotterization error control, approximate data handling, and practical extensions to higher-dimensional tensor factors, highlighting the practical relevance for discretized PDEs and structured linear systems in quantum computing.

Abstract

Solving linear systems is at the foundation of many algorithms. Recently, quantum linear system algorithms (QLSAs) have attracted great attention since they converge to a solution exponentially faster than classical algorithms in terms of the problem dimension. However, low-complexity circuit implementations of the oracles assumed in these QLSAs constitute the major bottleneck for practical quantum speed-up in solving linear systems. In this work, we focus on the application of QLSAs for linear systems that are expressed as a low rank tensor sums, which arise in solving discretized PDEs. Previous works uses modified Krylov subspace methods to solve such linear systems with a per-iteration complexity being polylogarithmic of the dimension but with no guarantees on the total convergence cost. We propose a quantum algorithm based on the recent advances on adiabatic-inspired QLSA and perform a detailed analysis of the circuit depth of its implementation. We rigorously show that the total complexity of our implementation is polylogarithmic in the dimension, which is comparable to the per-iteration complexity of the classical heuristic methods.

An Efficient Quantum Algorithm for Linear System Problem in Tensor Format

TL;DR

The paper tackles solving linear systems where the coefficient matrix and right-hand side have low-rank tensor structure (LSP-TF) by presenting a circuit-implementable quantum linear system algorithm (QLSA) based on adiabatic-inspired evolution. It introduces a Hamiltonian decomposition into Type-1 and Type-2 components and develops explicit quantum circuits, including a binary-spectral encoding, QFT-based adder, and quantum multiplier, to implement the necessary time evolutions. A detailed cost analysis shows the total runtime scales polylogarithmically with the problem dimension under reasonable assumptions, potentially offering exponential speedups over general classical solvers for large tensor-format problems. The work also discusses trotterization error control, approximate data handling, and practical extensions to higher-dimensional tensor factors, highlighting the practical relevance for discretized PDEs and structured linear systems in quantum computing.

Abstract

Solving linear systems is at the foundation of many algorithms. Recently, quantum linear system algorithms (QLSAs) have attracted great attention since they converge to a solution exponentially faster than classical algorithms in terms of the problem dimension. However, low-complexity circuit implementations of the oracles assumed in these QLSAs constitute the major bottleneck for practical quantum speed-up in solving linear systems. In this work, we focus on the application of QLSAs for linear systems that are expressed as a low rank tensor sums, which arise in solving discretized PDEs. Previous works uses modified Krylov subspace methods to solve such linear systems with a per-iteration complexity being polylogarithmic of the dimension but with no guarantees on the total convergence cost. We propose a quantum algorithm based on the recent advances on adiabatic-inspired QLSA and perform a detailed analysis of the circuit depth of its implementation. We rigorously show that the total complexity of our implementation is polylogarithmic in the dimension, which is comparable to the per-iteration complexity of the classical heuristic methods.
Paper Structure (17 sections, 21 theorems, 86 equations, 2 algorithms)

This paper contains 17 sections, 21 theorems, 86 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Notation
  4. LSP-TF
  5. Contributions
  6. QLSA
  7. Trotterization Method
  8. QLSA Circuit Design for LSP-TF
  9. Circuit for Type-1 Hamiltonian
  10. Algorithm \ref{['algo: generic implement diagonal tensor']} Step 2 & 3
  11. Algorithm \ref{['algo: generic implement diagonal tensor']} Step 4 & 5
  12. Circuit for Type-2 Hamiltonian
  13. Circuit Cost Analysis
  14. Cost Analysis for Approximate Data
  15. Trotterization Cost Analysis
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $0<\epsilon\leq 1/(3n)$, $\epsilon_0 = \epsilon^2/(\kappa\log^2\kappa)$ and $p=\lceil 1- \log_2\epsilon_0 \rceil$. Our implementation of Algorithm algo: qlsa prepares an $\mathcal{O}(\epsilon)$-approximate solution using $\mathcal{T}$ classical arithmetic operations, $\mathcal{T}$ single qubit u

Theorems & Definitions (29)

  • Theorem 2.1
  • Remark 2.1
  • Definition 2.1: QLSP
  • Lemma 2.1: Restated Theorem 6 of childs2021theory
  • Remark 2.2
  • Lemma 2.2: Restated Corollary 7 in childs2021theory
  • Lemma 3.1
  • proof
  • Remark 3.1
  • Lemma 3.2
  • ...and 19 more