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Efficient simulation of sparse Markovian quantum dynamics

Andrew M. Childs, Tongyang Li

TL;DR

This work addresses the challenge of efficiently simulating Markovian open-system dynamics described by Lindbladians that need not be local, extending quantum simulation techniques beyond Hamiltonian evolution. The authors develop two complementary frameworks: a sparse Stinespring isometry approach for invariantly sparse Lindbladians and a short-time evolution method for sparse Lindblad operators, enabling efficient treatment of a broad class of nonlocal dissipative dynamics. They classify five sparse Lindbladian families via an overcomplete GKS Gram matrix and provide concrete circuit constructions and complexity analyses, with applications to a truncated damped quantum harmonic oscillator and open quantum walks. A no-fast-forwarding theorem establishes fundamental time-scaling limits in black-box models, illustrating that, in general, Lindbladian dynamics cannot be arbitrarily fast-forwarded.

Abstract

Quantum algorithms for simulating Hamiltonian dynamics have been extensively developed, but there has been much less work on quantum algorithms for simulating the dynamics of open quantum systems. We give the first efficient quantum algorithms for simulating Markovian quantum dynamics generated by Lindbladians that are not necessarily local. We introduce two approaches to simulating sparse Lindbladians. First, we show how to simulate Lindbladians that act within small invariant subspaces using a quantum algorithm to implement sparse Stinespring isometries. Second, we develop a method for simulating sparse Lindblad operators by concatenating a sequence of short-time evolutions. We also show limitations on Lindbladian simulation by proving a no-fast-forwarding theorem for simulating sparse Lindbladians in black-box models.

Efficient simulation of sparse Markovian quantum dynamics

TL;DR

This work addresses the challenge of efficiently simulating Markovian open-system dynamics described by Lindbladians that need not be local, extending quantum simulation techniques beyond Hamiltonian evolution. The authors develop two complementary frameworks: a sparse Stinespring isometry approach for invariantly sparse Lindbladians and a short-time evolution method for sparse Lindblad operators, enabling efficient treatment of a broad class of nonlocal dissipative dynamics. They classify five sparse Lindbladian families via an overcomplete GKS Gram matrix and provide concrete circuit constructions and complexity analyses, with applications to a truncated damped quantum harmonic oscillator and open quantum walks. A no-fast-forwarding theorem establishes fundamental time-scaling limits in black-box models, illustrating that, in general, Lindbladian dynamics cannot be arbitrarily fast-forwarded.

Abstract

Quantum algorithms for simulating Hamiltonian dynamics have been extensively developed, but there has been much less work on quantum algorithms for simulating the dynamics of open quantum systems. We give the first efficient quantum algorithms for simulating Markovian quantum dynamics generated by Lindbladians that are not necessarily local. We introduce two approaches to simulating sparse Lindbladians. First, we show how to simulate Lindbladians that act within small invariant subspaces using a quantum algorithm to implement sparse Stinespring isometries. Second, we develop a method for simulating sparse Lindblad operators by concatenating a sequence of short-time evolutions. We also show limitations on Lindbladian simulation by proving a no-fast-forwarding theorem for simulating sparse Lindbladians in black-box models.

Paper Structure

This paper contains 32 sections, 25 theorems, 219 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Overcomplete GKS form
  4. Sparse Stinespring isometries
  5. Product formulas for Markovian quantum dynamics
  6. Sparse Stinespring isometry framework
  7. A brief introduction
  8. Efficient implementation of $d$-sparse Stinespring isometries
  9. Proof of \ref{['thm:sparse-Stinespring-isometry']}
  10. Notations
  11. Proof
  12. Applications of the sparse Stinespring isometry framework
  13. Local Lindbladians
  14. Identical-coordinate Lindbladians
  15. Diagonal Lindbladians
  16. ...and 17 more sections

Key Result

Theorem 1

$\mathcal{L}$ is a Lindbladian if and only if there exists an $N^{2}\times N^{2}$ positive semidefinite matrix $A$ with entries $A_{(k,l),(k',l')}$ for $k,l,k',l' \in [N]$ such that

Figures (6)

  • Figure 1: The gate $\wedge_{x}(U)$ with $x=101$.
  • Figure 2: The gate $\wedge_{x,|\phi\rangle}(U)$.
  • Figure 3: The quantum circuit for $\tilde{U}_{x,k}$.
  • Figure 4: The quantum circuit for $U_{\rightarrow}$.
  • Figure 5: The quantum circuit for $U_{\leftarrow}$.
  • ...and 1 more figures

Theorems & Definitions (52)

  • Theorem 1
  • proof
  • Definition 1
  • Definition 2
  • Example 1
  • Proposition 1
  • proof
  • Definition 3
  • Definition 4: kitaev1997quantum
  • Proposition 2
  • ...and 42 more