Wave Matrix Lindbladization I: Quantum Programs for Simulating Markovian Dynamics

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 $n=O(t^2/\epsilon )$ samples of $\psi$ to achieve the target dynamics, with an approximation error of $O(\epsilon )$.

arXiv comment29 pages, 7 figures, published in the journal special issue dedicated to the memory of Göran Lindblad

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• The wave operator representation of quantum and classical dynamics
• Quantum Dynamic Programming
• Sample-based Hamiltonian and Lindbladian simulation: Non-asymptotic analysis of sample complexity
• Simulating quantum collision models with Hamiltonian simulations using early fault-tolerant quantum computers
• Digital Quantum Simulations of the Non-Resonant Open Tavis-Cummings Model
• Quantum Imaginary-Time Evolution with Polynomial Resources in Evolution Time