Theoretical Analyses of Quantum Counting against Decoherence Errors

TL;DR

This paper provides a theoretical analysis of quantum counting under decoherence, comparing two circuit implementations and revealing that the ascending-order circuit is more robust against errors, with implications for phase estimation tasks.

Contribution

It introduces a comparative analysis of quantum counting circuits under decoherence, highlighting the robustness of the ascending-order circuit through theoretical and numerical methods.

Findings

Ascending-order circuit is more robust against decoherence.

Probability distributions differ between the two circuits under errors.

Results are applicable to phase estimation and factoring.

Abstract

In this paper, we analyze the quantum counting under the decoherence, which can find the number of solutions satisfying a given oracle. We investigate probability distributions related to the first order term of the error rate on the quantum counting with the depolarizing channel. We also implement two circuits for the quantum counting -- the ascending-order circuit and the descending-order circuit -- by reversing ordering of application of controlled-Grover operations. By theoretical and numerical calculations for probability distributions, we reveal the difference of probability distributions on two circuits in the presence of decoherence and show that the ascending-order circuit is more robust against the decoherence than the descending-order circuit. This property of the robustness is applicable to the phase estimation such as the factoring.