Noise effect on Grover algorithm

TL;DR

This paper investigates how decoherence affects Grover's quantum search algorithm, finding an exponential damping law and an error threshold, and demonstrating fault-tolerant encoding to improve success probability.

Contribution

It introduces a noise model for decoherence in Grover's algorithm, derives an error threshold law, and explores fault-tolerant encoding to mitigate noise effects.

Findings

Success probability decays exponentially with noise over time.

Error threshold for the algorithm scales as approximately 1/N^{1.1}.

Fault-tolerant encoding improves success probability for small qubit systems.

Abstract

The decoherence effect on Grover algorithm has been studied numerically through a noise modelled by a depolarizing channel. Two types of error are introduced characterizing the qubit time evolution and gate application, so the noise is directly related to the quantum network construction. The numerical simulation concludes an exponential damping law for the successive probability of the maxima as time increases. We have obtained an allowed-error law for the algorithm: the error threshold for the allowed noise behaves as Eth(N) ~ 1/N1.1 (N being the size of the data set). As the power of N is almost one, we consider the Grover algorithm as robust to a certain extent against decoherence. This law also provides an absolute threshold: if the free evolution error is greater than 0.043, Grover algorithm does not work for any number of qubits affected by the present error model. The improvement in the probability of success, in the case of two qubits has been illustrated by using a fault-tolerant encoding of the initial state by means of the [[7,1,3]] quantum code.