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Efficient Quantum Algorithms for Simulating Lindblad Evolution

Richard Cleve, Chunhao Wang

TL;DR

This work addresses simulating Lindblad evolution, a generalization of Schrödinger dynamics to Markovian open quantum systems, by constructing a quantum circuit that implements the channel $e^{\mathcal{L}t}$ on $n$ qubits to error $\epsilon$. The core advance is a new variant of the linear combination of unitaries (LCU) method tailored to channels, together with an oblivious amplitude amplification (OAA) scheme for isometries, which together avoid the inefficiencies of reducing Lindblad evolution to large-system Hamiltonian evolution. The main result provides near-optimal gate and query complexities for general, local, and sparse Lindbladians, with the key quantity $\tau = t\|\mathcal{L}\|_{\textsf{pauli}}$ and norms ensuring stable error control; for polynomial-sized Pauli decompositions, the gate count scales as $O(\tau\,\mathrm{polylog}(\tau/\epsilon)^2/\log\log(\tau/\epsilon))$ up to poly$(n)$ factors. This yields substantial improvements over prior approaches that incurred overheads like $O(t^2/\epsilon)$, and it broadens the practical reach of quantum simulation to open quantum systems, including local and sparse Lindbladians. The work thus has potential impact on quantum chemistry, quantum thermodynamics, and noise-aware quantum information processing by enabling more efficient, state-independent simulation of dissipative dynamics.

Abstract

We consider the natural generalization of the Schrödinger equation to Markovian open system dynamics: the so-called the Lindblad equation. We give a quantum algorithm for simulating the evolution of an $n$-qubit system for time $t$ within precision $ε$. If the Lindbladian consists of $\mathrm{poly}(n)$ operators that can each be expressed as a linear combination of $\mathrm{poly}(n)$ tensor products of Pauli operators then the gate cost of our algorithm is $O(t\, \mathrm{polylog}(t/ε)\mathrm{poly}(n))$. We also obtain similar bounds for the cases where the Lindbladian consists of local operators, and where the Lindbladian consists of sparse operators. This is remarkable in light of evidence that we provide indicating that the above efficiency is impossible to attain by first expressing Lindblad evolution as Schrödinger evolution on a larger system and tracing out the ancillary system: the cost of such a \textit{reduction} incurs an efficiency overhead of $O(t^2/ε)$ even before the Hamiltonian evolution simulation begins. Instead, the approach of our algorithm is to use a novel variation of the "linear combinations of unitaries" construction that pertains to channels.

Efficient Quantum Algorithms for Simulating Lindblad Evolution

TL;DR

This work addresses simulating Lindblad evolution, a generalization of Schrödinger dynamics to Markovian open quantum systems, by constructing a quantum circuit that implements the channel on qubits to error . The core advance is a new variant of the linear combination of unitaries (LCU) method tailored to channels, together with an oblivious amplitude amplification (OAA) scheme for isometries, which together avoid the inefficiencies of reducing Lindblad evolution to large-system Hamiltonian evolution. The main result provides near-optimal gate and query complexities for general, local, and sparse Lindbladians, with the key quantity and norms ensuring stable error control; for polynomial-sized Pauli decompositions, the gate count scales as up to poly factors. This yields substantial improvements over prior approaches that incurred overheads like , and it broadens the practical reach of quantum simulation to open quantum systems, including local and sparse Lindbladians. The work thus has potential impact on quantum chemistry, quantum thermodynamics, and noise-aware quantum information processing by enabling more efficient, state-independent simulation of dissipative dynamics.

Abstract

We consider the natural generalization of the Schrödinger equation to Markovian open system dynamics: the so-called the Lindblad equation. We give a quantum algorithm for simulating the evolution of an -qubit system for time within precision . If the Lindbladian consists of operators that can each be expressed as a linear combination of tensor products of Pauli operators then the gate cost of our algorithm is . We also obtain similar bounds for the cases where the Lindbladian consists of local operators, and where the Lindbladian consists of sparse operators. This is remarkable in light of evidence that we provide indicating that the above efficiency is impossible to attain by first expressing Lindblad evolution as Schrödinger evolution on a larger system and tracing out the ancillary system: the cost of such a \textit{reduction} incurs an efficiency overhead of even before the Hamiltonian evolution simulation begins. Instead, the approach of our algorithm is to use a novel variation of the "linear combinations of unitaries" construction that pertains to channels.

Paper Structure

This paper contains 17 sections, 6 theorems, 92 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Previous work
  3. New results
  4. Brief summary of novel techniques
  5. The standard LCU method performs poorly on Stinespring dilations
  6. New LCU method for channels and completely positive maps
  7. Overview of the main result, Theorem \ref{['thm:main_thm']}
  8. A simple mapping $\mathcal{M}_{\delta}$ that approximates $e^{\mathcal{L}\delta}$ for small $\delta$
  9. Approximating $\mathcal{M}_{\delta}$ by a quantum circuit via the new LCU method
  10. Simulating $r$ iterations of $\mathcal{M}_{O(1/r)}$ with constant success probability
  11. Oblivious amplitude amplification for isometries
  12. Concentration bound and encoding scheme
  13. Total number of gates and proof of the main theorem
  14. Lindbladians with sparse Hamiltonian and Lindblad operators
  15. Acknowledgments
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\mathcal{L}$ be a Lindbladian presented as a linear combination of $q$ Paulis. Then, for any $t > 0$ and $\epsilon > 0$, there exists a quantum circuit of size that implements a quantum channel $\mathcal{N}$, such that $\left\|\mathcal{N} - e^{\mathcal{L}t}\right\|_\diamond \le \epsilon$, where $\tau = t\,\|\mathcal{L}\|_{\textrm{\sf pauli}}$ and $m$ is the number of jump operators in $\math

Figures (4)

  • Figure 1: Lindblad evolution for time $t$ approximated by unitary operations. There are $N$ iterations and $\delta = t/N$. This converges to Lindblad evolution as $N \rightarrow \infty$.
  • Figure 2: The circuit $W$ for simulating a channel using the new LCU method.
  • Figure 3: $N$-stage $\epsilon$-precision discretization of the trajectory resulting from $\mathcal{L}$. For each $k \in \{1,\dots,N\}$, after $k$ stages, the channel should be within $\epsilon$ of $\exp\bigl(\frac{kT}{N}\mathcal{L}\bigr)$.
  • Figure 4: The Local Hamiltonian Approximation Lemma. The first register is $d$-dimensional, the second register contains $n$ qubits, and the approximation is within $O(\delta^2)$ (independent of $d$ and $n$).

Theorems & Definitions (13)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • proof : Proof of Lemma \ref{['lemma:OAA']}
  • proof : Proof of Theorem \ref{['thm:main_thm']}
  • Definition 3
  • Theorem 4
  • ...and 3 more