Risk-minimizing states for the quantum-phase-estimation algorithm

TL;DR

This paper develops a method to find risk-minimizing input states for quantum phase estimation, demonstrating that these states outperform traditional uniform states and achieve optimal quantum advantage, even under noise conditions.

Contribution

It introduces a general approach to identify optimal input states for QPEA based on eigenvectors of Toeplitz matrices, improving performance over uniform states.

Findings

Risk-minimizing states outperform uniform states in QPEA.

Optimal states are well approximated by cosine forms.

Methods to mitigate depolarizing noise effects.

Abstract

The quantum-phase-estimation algorithm (QPEA) is widely used to find estimates of unknown phases. The original algorithm relied on an input state in a uniform superposition of all possible bit strings. However, it is known that other input states can reduce certain Bayesian risks of the final estimate. Here, we derive a method to find the risk-minimizing input state for any risk. These states are represented by an eigenvector of a Toeplitz matrix with elements given by the Fourier coefficients of the loss function of interest. We show that, while the true optimal state does not have a closed form for a general loss function, it is well approximated by a state with a cosine form. When the cosine frequency is chosen appropriately, these states outperform the original QPEA and achieve the optimal theoretical quantum-advantage scaling for three common risks. Furthermore, we prove that the uniform input state is suboptimal for any reasonable loss function. Finally, we design methods to mitigate the impact of depolarizing noise on the performance of QPEA.