A classical limit of Grover's algorithm induced by dephasing: Coherence vs entanglement

TL;DR

This paper investigates how rapid dephasing affects Grover's algorithm, showing that loss of quantum coherence shifts its efficiency from quadratic to linear, highlighting coherence's role over entanglement in quantum speedup.

Contribution

It introduces a classical limit of Grover's algorithm under rapid dephasing, emphasizing the importance of quantum coherence over entanglement for quantum speedup.

Findings

Rapid dephasing reduces Grover's speedup from √N to N steps.

Quantum coherence is essential for the quadratic speedup.

Loss of coherence in noisy environments leads to classical-like amplification.

Abstract

A new approach to the classical limit of Grover's algorithm is discussed by assuming a very rapid dephasing of a system between consecutive Grover's unitary operations, which drives pure quantum states to decohered mixed states. One can identify a specific element among $N$ unsorted elements by a probability of the order of unity after $k\sim N$ steps of classical amplification, which is realized by a combination of Grover's unitary operation and rapid dephasing, in contrast to $k\sim \pi \sqrt{N}/4$ steps in quantum mechanical amplification. The initial two-state system with enormously unbalanced existence probabilities, which is realized by a chosen specific state and a superposition of all the rest of states among $N$ unsorted states, is crucial in the present analysis of classical amplification. This analysis illustrates Grover's algorithm in extremely noisy circumstances. A similar increase from $k\sim \sqrt{N}$ to $k\sim N$ steps due to the loss of quantum coherence takes place in the {\em analog} model of Farhi and Gutmann where the entanglement does not play an obvious role. This supports a view that entanglement is crucial in quantum computation to describe quantum states by a set of qubits, but the actual speedup of the quantum computation is based on quantum coherence.