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Quantum preconditioning method for linear systems problems via Schrödingerization

Shi Jin, Nana Liu, Chuwen Ma, Yue Yu

TL;DR

The paper presents a quantum algorithm for solving linear systems by transforming convergent stationary iterations into Schrödinger-type dynamics through Schrödingerization, enabling direct quantum-state preparation of the solution. By incorporating a BPX multilevel preconditioner, it achieves near-polylogarithmic dependence on the target accuracy $\varepsilon$ and reduces sensitivity to discretization size in high dimensions. The approach hinges on block-encoding techniques for the BPX substructures and a carefully crafted augmented ODE-to-Hamiltonian mapping, yielding an input model and quantum-simulation procedure that can realize $|\bm{x}\rangle$ with high probability. This work thus provides a path to quantum acceleration for high-dimensional Poisson-like problems and other multilevel elliptic systems, with potential impact on numerical PDE workflows and quantum linear algebra. $

Abstract

We present a quantum computational framework that systematically converts classical linear iterative algorithms with fixed iteration operators into their quantum counterparts using the Schrödingerization technique [Shi Jin, Nana Liu and Yue Yu, Phys. Rev. Lett., vol. 133 No. 230602,2024]. This is achieved by capturing the steady state of the associated differential equations. The Schrödingerization technique transforms linear partial and ordinary differential equations into Schrödinger-type systems, making them suitable for quantum computing. This is accomplished through the so-called warped phase transformation, which maps the equation into a higher-dimensional space. Building on this framework, we develop a quantum preconditioning algorithm that leverages the well-known BPX multilevel preconditioner for the finite element discretization of the Poisson equation. The algorithm achieves a near-optimal dependence on the number of queries to our established input models, with a complexity of $\mathscr{O}(\text{polylog} \frac{1}{\varepsilon})$ for a target accuracy of $\varepsilon$ when the dimension $d\geq 2$. This improvement results from the Hamiltonian simulation strategy applied to the Schrödingerized preconditioning dynamics, coupled with the smoothing of initial data in the extended space.

Quantum preconditioning method for linear systems problems via Schrödingerization

TL;DR

The paper presents a quantum algorithm for solving linear systems by transforming convergent stationary iterations into Schrödinger-type dynamics through Schrödingerization, enabling direct quantum-state preparation of the solution. By incorporating a BPX multilevel preconditioner, it achieves near-polylogarithmic dependence on the target accuracy and reduces sensitivity to discretization size in high dimensions. The approach hinges on block-encoding techniques for the BPX substructures and a carefully crafted augmented ODE-to-Hamiltonian mapping, yielding an input model and quantum-simulation procedure that can realize with high probability. This work thus provides a path to quantum acceleration for high-dimensional Poisson-like problems and other multilevel elliptic systems, with potential impact on numerical PDE workflows and quantum linear algebra. $

Abstract

We present a quantum computational framework that systematically converts classical linear iterative algorithms with fixed iteration operators into their quantum counterparts using the Schrödingerization technique [Shi Jin, Nana Liu and Yue Yu, Phys. Rev. Lett., vol. 133 No. 230602,2024]. This is achieved by capturing the steady state of the associated differential equations. The Schrödingerization technique transforms linear partial and ordinary differential equations into Schrödinger-type systems, making them suitable for quantum computing. This is accomplished through the so-called warped phase transformation, which maps the equation into a higher-dimensional space. Building on this framework, we develop a quantum preconditioning algorithm that leverages the well-known BPX multilevel preconditioner for the finite element discretization of the Poisson equation. The algorithm achieves a near-optimal dependence on the number of queries to our established input models, with a complexity of for a target accuracy of when the dimension . This improvement results from the Hamiltonian simulation strategy applied to the Schrödingerized preconditioning dynamics, coupled with the smoothing of initial data in the extended space.
Paper Structure (14 sections, 11 theorems, 119 equations, 6 figures, 2 algorithms)

This paper contains 14 sections, 11 theorems, 119 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Iteration and preconditioning for linear system problems
  3. Schrödingerization for symmetric preconditioned system
  4. Steady-state solution of linear ODEs
  5. Schrödingerization method for linear ODEs
  6. Implementation of the Hamiltonian simulation
  7. Complexity of general quantum preconditioned system
  8. Multilevel BPX preconditioning
  9. Blocking encoding of the preconditioned system
  10. The issue of preconditioning
  11. The block encoding of \ref{['eq:ODE square root']}
  12. The block encoding of prolongation matrices
  13. Numerical examples
  14. Conclusions

Key Result

Lemma 2.1

The stationary iterative algorithm alg:SIM converges for any initial guess if and only if the spectral radius $\rho(I - BA)<1$.

Figures (6)

  • Figure 1: Quantum circuit for block encoding of sparse matrices.
  • Figure 1: The exact and numerical solutions for $\mathbb{P}_1$-Lagrange elements
  • Figure 2: Block encoding of $S_{j,e}$
  • Figure 2: The convergence rates for $\mathbb{P}_1$-Lagrange elements in $L^2$ and $H^1$ norms
  • Figure 3: The controlled-$S_{j,e}$
  • ...and 1 more figures

Theorems & Definitions (28)

  • Lemma 2.1
  • Lemma 3.1
  • Theorem 3.2
  • Proof 1
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.5
  • Proof 2
  • Remark 3.6
  • Definition 3.7
  • ...and 18 more