Heisenberglimited quantum phase estimation of multiple eigenvalues with few control qubits
Abstract
Quantum phase estimation is a cornerstone in quantum algorithm design, allowing for the inference of eigenvalues of exponentiallylarge sparse matrices.The maximum rate at which these eigenvalues may be learned, known as the Heisenberg limit, is constrained by bounds on the circuit complexity required to simulate an arbitrary Hamiltonian. Singlecontrol qubit variants of quantum phase estimation that do not require coherence between experiments have garnered interest in recent years due to lower circuit depth and minimal qubit overhead. In this work we show that these methods can achieve the Heisenberg limit, also when one is unable to prepare eigenstates of the system. Given a quantum subroutine which provides samples of a `phase function' g(k)=∑jAjeiϕjk with unknown eigenphases ϕj and overlaps Aj at quantum cost O(k), we show how to estimate the phases {ϕj} with (rootmeansquare) error δ for total quantum cost T=O(δ1). Our scheme combines the idea of Heisenberglimited multiorder quantum phase estimation for a single eigenvalue phase [Higgins et al (2009) and Kimmel et al (2015)] with subroutines with socalled dense quantum phase estimation which uses classical processing via timeseries analysis for the QEEP problem [Somma (2019)] or the matrix pencil method. For our algorithm which adaptively fixes the choice for k in g(k) we prove Heisenberglimited scaling when we use the timeseries/QEEP subroutine. We present numerical evidence that using the matrix pencil technique the algorithm can achieve Heisenberglimited scaling as well.
 Publication:

Quantum
 Pub Date:
 October 2022
 DOI:
 10.22331/q20221006830
 arXiv:
 arXiv:2107.04605
 Bibcode:
 2022Quant...6..830D
 Keywords:

 Quantum Physics
 EPrint:
 25 pages + 5 page appendix, 3 figures, accepted in Quantum