Decoherent histories of quantum searching

TL;DR

This paper uses the decoherent histories framework to analyze Grover's quantum search algorithm, showing how environmental decoherence can cause quantum speedup to degrade into classical behavior.

Contribution

It provides an exactly solvable model linking decoherence with the transition from quantum to classical search performance, offering an alternative derivation of Grover's algorithm results.

Findings

Decoherence parameter controls the transition from quantum to classical search times.

Environmental influence can significantly impair quantum computational advantage.

The model demonstrates emergence of classical physics from quantum laws through decoherence.

Abstract

The theory of decoherent histories is an attempt to derive classical physics from positing only quantum laws at the fundamental level without notions of a classical apparatus or collapse of the wave-function. Searching for a marked target in a list of N items requires \Omega(N) oracle queries when using a classical computer, while a quantum computer can accomplish the same task in O{\sqrt{N}} queries using Grover's quantum algorithm. We study a closed quantum system executing Grover algorithm in the framework of decoherent histories and find it to be an exactly solvable model, thus yielding an alternate derivation of Grover's famous result. We also subject the Grover-executing computer to a generic external influence without needing to know the specifics of the Hamiltonian insofar as the histories decohere. Depending on the amount of decoherence, which is captured in our model by a single parameter related to the amount of information obtained by the environment, the search time can range from quantum to classical. Thus, we identify a key effect induced by the environment that can adversely affect a quantum computer's performance and demonstrate exactly how classical computing can emerge from quantum laws.