Standard Quantum Phase Estimation Detects All Eigenvalues via Randomized Initial States

TL;DR

This paper demonstrates that standard quantum phase estimation, when combined with randomized initial states, can detect all eigenvalues without circuit modifications, supported by theoretical proofs and numerical validation.

Contribution

It shows that using random initial states from a 1-design enables standard QPE to access all eigenvalues, overcoming previous limitations.

Findings

Random initial states equalize mode weights in expectation.

All eigenvalues produce peaks in the QPE distribution.

Numerical experiments confirm theoretical predictions.

Abstract

Standard quantum phase estimation (QPE) has often been regarded as unsuitable for simultaneous detection of all eigenvalues, because it requires initial states with sufficient overlap with the target eigenstates. In this paper, we show that this limitation is not inherent to the QPE circuit itself. The output distribution of standard QPE can be written as a superposition of Fej\'er kernels weighted by the squared overlaps with the eigenmodes. We prove that, if the initial state is independently drawn at each shot from a 1-design (in particular, by random selection of computational basis states), these mode weights are equalized in expectation, yielding a state-averaged QPE distribution that exhibits peaks at every eigenphase location. In this sense, all eigenvalues become accessible without any modification of the standard QPE circuit; repeated eigenvalues appear through the aggregated weight of their eigenspaces. For distinct eigenphases satisfying a separation condition, we further establish a rigorous peak-detection theory and derive a sufficient shot-count estimate for detecting all peaks. We validate the theory through numerical experiments on a finite element method (FEM) matrix with 1,008 degrees of freedom arising from computer-aided engineering (CAE).