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Infinite-dimensional Extension of the Linear Combination of Hamiltonian Simulation: Theorems and Applications

Rundi Lu, Hao-En Li, Zhengwei Liu, Jin-Peng Liu

TL;DR

The paper develops Infinite-dimensional Extension of the Linear Combination of Hamiltonian Simulation (Inf-LCHS), enabling time-evolution simulation for unbounded operators and PDEs by expressing non-unitary dynamics as a linear combination of unitary propagators. The Inf-LCHS theorem is established under two key conditions, guaranteeing analyticity and derivative existence via BCH/Trotter machinery, with two sampling schemes—Inf-LCHS-Gaussian and Inf-LCHS-MC—providing quantum-implementation pathways. It provides cost-scalings for these schemes and demonstrates applicability to diverse non-Hermitian dynamics, including linear parabolic PDEs, birth-death queues, Schrödinger equations with complex potentials, Lindblad equations, and black hole thermal field equations. By bridging finite- and infinite-dimensional quantum dynamics, the work sets a formal framework for quantum simulation of broad linear dynamics and offers concrete resource estimates for PDE-related quantum algorithms.

Abstract

We generalize the Linear Combination of Hamiltonian Simulation (LCHS) formula [An, Liu, Lin, Phys. Rev. Lett. 2023] to simulate time-evolution operators in infinite-dimensional spaces, including scenarios involving unbounded operators. This extension, named Inf-LCHS for short, bridges the gap between finite-dimensional quantum simulations and the broader class of infinite-dimensional quantum dynamics governed by partial differential equations (PDEs). Furthermore, we propose two sampling methods by integrating the infinite-dimensional LCHS with Gaussian quadrature schemes (Inf-LCHS-Gaussian) or Monte Carlo integration schemes (Inf-LCHS-MC). We demonstrate the applicability of the Inf-LCHS theorem to a wide range of non-Hermitian dynamics, including linear parabolic PDEs, queueing models (birth-or-death processes), Schrödinger equations with complex potentials, Lindblad equations, and black hole thermal field equations. Our analysis provides insights into simulating general linear dynamics using a finite number of quantum dynamics and includes cost estimates for the corresponding quantum algorithms.

Infinite-dimensional Extension of the Linear Combination of Hamiltonian Simulation: Theorems and Applications

TL;DR

The paper develops Infinite-dimensional Extension of the Linear Combination of Hamiltonian Simulation (Inf-LCHS), enabling time-evolution simulation for unbounded operators and PDEs by expressing non-unitary dynamics as a linear combination of unitary propagators. The Inf-LCHS theorem is established under two key conditions, guaranteeing analyticity and derivative existence via BCH/Trotter machinery, with two sampling schemes—Inf-LCHS-Gaussian and Inf-LCHS-MC—providing quantum-implementation pathways. It provides cost-scalings for these schemes and demonstrates applicability to diverse non-Hermitian dynamics, including linear parabolic PDEs, birth-death queues, Schrödinger equations with complex potentials, Lindblad equations, and black hole thermal field equations. By bridging finite- and infinite-dimensional quantum dynamics, the work sets a formal framework for quantum simulation of broad linear dynamics and offers concrete resource estimates for PDE-related quantum algorithms.

Abstract

We generalize the Linear Combination of Hamiltonian Simulation (LCHS) formula [An, Liu, Lin, Phys. Rev. Lett. 2023] to simulate time-evolution operators in infinite-dimensional spaces, including scenarios involving unbounded operators. This extension, named Inf-LCHS for short, bridges the gap between finite-dimensional quantum simulations and the broader class of infinite-dimensional quantum dynamics governed by partial differential equations (PDEs). Furthermore, we propose two sampling methods by integrating the infinite-dimensional LCHS with Gaussian quadrature schemes (Inf-LCHS-Gaussian) or Monte Carlo integration schemes (Inf-LCHS-MC). We demonstrate the applicability of the Inf-LCHS theorem to a wide range of non-Hermitian dynamics, including linear parabolic PDEs, queueing models (birth-or-death processes), Schrödinger equations with complex potentials, Lindblad equations, and black hole thermal field equations. Our analysis provides insights into simulating general linear dynamics using a finite number of quantum dynamics and includes cost estimates for the corresponding quantum algorithms.

Paper Structure

This paper contains 16 sections, 16 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Main Theorem
  3. Main Theorem Conditions: Derivation and Proof
  4. Inf-LCHS Implementation and Sampling Methods
  5. Inf-LCHS-Gaussian: Composite Gaussian Quadrature
  6. Inf-LCHS-MC: Monte Carlo Integration
  7. Applications
  8. Linear Parabolic PDEs
  9. Queueing Models (Birth-death Processes)
  10. Schrödinger Equations with Complex Potentials
  11. Lindblad Equations
  12. Black Hole Thermal Field Equations
  13. Discussion and Outlook
  14. Preliminaries
  15. Example Demonstrating the Validity of Inf-LCHS Assumptions
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose $A(t) = L(t) + \mathrm{i}H(t)$, $L(t)$ and $H(t)$ satisfy adjoint_assum. We define $Y(t)=\int_0^t L(s)\mathrm{~d}s$ and $X(t)=-\mathrm{i}\int_0^t H(s)\mathrm{~d}s$. Suppose that $f(z)$ is a function of $z \in \mathbb{C}$ that satisfies assum: func. If operators $L(t)$ and $H(t)$ satisfy one and $L(s) \succeq \lambda_0>0$ for all $0 \leq s \leq t$, then $\forall \psi_0\in X_0$ defined in a

Theorems & Definitions (34)

  • Theorem 1: Inf-LCHS Theorem
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2: Inf-LCHS-Gaussian
  • proof
  • Corollary 1
  • Theorem 3: Inf-LCHS-MC
  • ...and 24 more