Quantum Dissipative Search via Lindbladians

TL;DR

This paper investigates the efficiency of dissipative quantum walks governed by Lindblad equations, revealing conditions under which they match classical search and highlighting the role of coherence in quantum speedup.

Contribution

It provides a detailed analysis of convergence and speed in dissipative quantum walks, clarifies when they replicate classical behavior, and explores the impact of coherence on quantum search speedup.

Findings

Certain jump operators make quantum walks replicate classical ones

OQRWs are not more efficient than classical search

Coherence is crucial for quadratic speedup

Abstract

Closed quantum systems follow a unitary time evolution that can be simulated on quantum computers. By incorporating non-unitary effects via, e.g., measurements on ancilla qubits, these algorithms can be extended to open-system dynamics, such as Markovian processes described by the Lindblad master equation. In this paper, we analyze the convergence criteria and speed of a Markovian, purely dissipative quantum random walk on an unstructured classical search space. Notably, we show that certain jump operators make the quantum process replicate a classical one, while others yield differences between open quantum (OQRW) and classical random walks. We also clarify a previously observed quadratic speedup, demonstrating that OQRWs are no more efficient than classical search. Finally, we analyze a dissipative discrete-time ground-state preparation algorithm with a lower implementation cost. This allows us to interpolate between the dissipative and the unitary domain and thereby illustrate the important role of coherence for the quadratic speedup.