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Standard Quantum Phase Estimation Detects All Eigenvalues via Randomized Initial States

Yuki Izumi, Hitoshi Kawahara

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 Fejér 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).

Standard Quantum Phase Estimation Detects All Eigenvalues via Randomized Initial States

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 Fejér 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).

Paper Structure

This paper contains 37 sections, 9 theorems, 110 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Relation to Density of States Estimation
  4. Multi-Eigenvalue Phase Estimation
  5. Modified QPE for LDOS
  6. Preliminaries
  7. Notation and Basic Setup
  8. Quantum Phase Estimation (QPE)
  9. The Fejér Kernel
  10. Block Encoding
  11. 1-Design mele2024haar
  12. State-Averaged QPE
  13. QPE Measure and the State-Averaged Condition
  14. Realization of the State-Averaged Condition via 1-Designs
  15. Peak Detection and Shot Count Estimation
  16. ...and 22 more sections

Key Result

Proposition A.1

Let $U_{\textup{QPE}}$ denote the unitary operation implementing QPE on $|0^n\rangle \otimes |\psi\rangle$, and set $|\phi\rangle = U_{\textup{QPE}}\,(|0^n\rangle \otimes |\psi\rangle)$. Then The probability of measuring outcome $j = 0, 1, \dots, N-1$ is where $H_j = |j\rangle\langle j| \otimes I$. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Finite element mesh of the cantilever beam ($16 \times 6 \times 2$ hexahedral elements, $m_0 = 1{,}008$ DOFs after boundary conditions).
  • Figure 2: Fejér Kernel ($n = 8$)

Theorems & Definitions (21)

  • Definition 4.1: State-averaged
  • Remark 4.2
  • Proposition A.1: QPE Distribution
  • Definition A.2
  • Lemma A.3
  • proof
  • Theorem A.4
  • Remark A.5
  • Lemma A.6
  • proof
  • ...and 11 more