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Quantum Simulation of Non-unitary Dynamics via Contour-based Matrix Decomposition

Chao Wang, Huan-Yu Liu, Cheng Xue, Xi-Ning Zhuang, Menghan Dou, Zhao-Yun Chen, Guo-Ping Guo

TL;DR

This work introduces contour-based matrix decomposition (CBMD), a framework to efficiently simulate non-unitary dynamics by generalizing matrix-valued residue calculus to decompose non-Hermitian propagators into a linear combination of Hermitian operations, implementable with quantum singular value transformation (QSVT). By coupling CBMD with an eigenvalue-shifting technique (EST), the authors achieve optimal query complexity that scales with the system's spectral range rather than its operator norm, and extend applicability to general matrix polynomials via Runge's theorem and polynomial approximations. The key contributions include a fast-converging LCU series for non-unitary evolution, a rigorous matrix-series identity, and demonstrated connections to and unification of prior LCHS/Schrödingerization approaches, all while avoiding assumptions about diagonalizability or conditioning. Overall, CBMD provides a unifying, scalable approach for a broad class of non-unitary problems on quantum hardware with potential impact across quantum simulation, control, and quantum-enabled numerical methods.

Abstract

We introduce contour-based matrix decomposition (CBMD), a framework for scalable simulation of non-unitary dynamics. Unlike existing methods that follow the ``integrate-then-discretize" paradigm and rely heavily on numerical quadrature, CBMD generalizes Cauchy's residue theorem to matrix-valued functions and directly decomposes a non-Hermitian function into a linear combination of Hermitian ones, which can be implemented efficiently using techniques such as quantum singular value transformation (QSVT). For non-Hermitian dynamics, CBMD achieves optimal query complexity. With an additional eigenvalue-shifting technique, the improved complexity depends on the spectral range of the system instead of its spectral norm. For more general dynamics that can be approximated by non-Hermitian polynomials, where algorithms like QSVT face significant difficulties, CBMD remains applicable and avoids the assumptions of diagonalizability as well as the dependence on condition numbers that limit other approaches.

Quantum Simulation of Non-unitary Dynamics via Contour-based Matrix Decomposition

TL;DR

This work introduces contour-based matrix decomposition (CBMD), a framework to efficiently simulate non-unitary dynamics by generalizing matrix-valued residue calculus to decompose non-Hermitian propagators into a linear combination of Hermitian operations, implementable with quantum singular value transformation (QSVT). By coupling CBMD with an eigenvalue-shifting technique (EST), the authors achieve optimal query complexity that scales with the system's spectral range rather than its operator norm, and extend applicability to general matrix polynomials via Runge's theorem and polynomial approximations. The key contributions include a fast-converging LCU series for non-unitary evolution, a rigorous matrix-series identity, and demonstrated connections to and unification of prior LCHS/Schrödingerization approaches, all while avoiding assumptions about diagonalizability or conditioning. Overall, CBMD provides a unifying, scalable approach for a broad class of non-unitary problems on quantum hardware with potential impact across quantum simulation, control, and quantum-enabled numerical methods.

Abstract

We introduce contour-based matrix decomposition (CBMD), a framework for scalable simulation of non-unitary dynamics. Unlike existing methods that follow the ``integrate-then-discretize" paradigm and rely heavily on numerical quadrature, CBMD generalizes Cauchy's residue theorem to matrix-valued functions and directly decomposes a non-Hermitian function into a linear combination of Hermitian ones, which can be implemented efficiently using techniques such as quantum singular value transformation (QSVT). For non-Hermitian dynamics, CBMD achieves optimal query complexity. With an additional eigenvalue-shifting technique, the improved complexity depends on the spectral range of the system instead of its spectral norm. For more general dynamics that can be approximated by non-Hermitian polynomials, where algorithms like QSVT face significant difficulties, CBMD remains applicable and avoids the assumptions of diagonalizability as well as the dependence on condition numbers that limit other approaches.

Paper Structure

This paper contains 14 sections, 5 theorems, 58 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Non-unitary dynamics
  3. Contour-based matrix decomposition
  4. Implementation on non-unitary dynamics
  5. Connection to LCHS Methods
  6. General Matrix Dynamics
  7. Discussion
  8. Proof of Matrix Series Identity Theorem \ref{['Matrix_Series']}
  9. Proof of Fast-Converging LCU Series Theorem \ref{['Query_Analyse']}
  10. Proof of Corollary \ref{['corEST']}
  11. Connection to LCHS Methods
  12. Original LCHS
  13. Improved LCHS
  14. Optimal LCHS

Key Result

Proposition 1

Let $\Gamma$ be a simple closed contour and $M(z)$ a matrix-valued function whose entries are meromorphic inside $\Gamma$. If $\{z_k\}$ are the simple poles of $M(z)$ enclosed by $\Gamma$, then where $\mathrm{Res}(M, z_k)$ denotes the matrix of element-wise residues at pole $z_k$.

Figures (3)

  • Figure 1: Schematic diagram of the matrix contour integral in Eq. \ref{['int_main']}.
  • Figure 2: Schematic diagram of the matrix contour integral in Eq. \ref{['improved_lchs_int']}.
  • Figure 3: Schematic diagram of the matrix contour integral in Eq. \ref{['eqs_optimal_lchs']}.

Theorems & Definitions (8)

  • Proposition 1: Matrix Residue Theorem
  • Theorem 1: Matrix Series Identity
  • Theorem 2: Fast-Converging LCU Series
  • Theorem 3: Non-Hermitian dynamics simulation via CBMD
  • Corollary 4
  • proof
  • proof
  • proof