Wave Matrix Lindbladization I: Quantum Programs for Simulating Markovian Dynamics

TL;DR

This paper introduces Wave Matrix Lindbladization, a quantum algorithm for simulating Markovian dynamics governed by Lindblad equations, using quantum states encoding Lindblad operators, with a sample complexity of O(t^2/ε).

Contribution

It presents a novel quantum algorithm for simulating Lindblad dynamics from encoded Lindblad operators, extending density matrix exponentiation techniques.

Findings

Uses O(t^2/ε) samples of the encoded state

Achieves dynamics simulation with O(ε) approximation error

Provides analysis of sample complexity for the algorithm

Abstract

Density Matrix Exponentiation is a technique for simulating Hamiltonian dynamics when the Hamiltonian to be simulated is available as a quantum state. In this paper, we present a natural analogue to this technique, for simulating Markovian dynamics governed by the well known Lindblad master equation. For this purpose, we first propose an input model in which a Lindblad operator $L$ is encoded into a quantum state $\psi$. Then, given access to $n$ copies of the state $\psi$, the task is to simulate the corresponding Markovian dynamics for time $t$. We propose a quantum algorithm for this task, called Wave Matrix Lindbladization, and we also investigate its sample complexity. We show that our algorithm uses $n = O(t^2/\varepsilon)$ samples of $\psi$ to achieve the target dynamics, with an approximation error of $O(\varepsilon)$.