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arXiv:2107.04605 · Quantum

Heisenberg-limited quantum phase estimation of multiple eigenvalues with few control qubits

Alicja Dutkiewicz, Barbara M. Terhal, Thomas E. O apos Brien

Published Oct 6, 2022• 33 citations
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Abstract

Quantum phase estimation is a cornerstone in quantum algorithm design, allowing for the inference of eigenvalues of exponentially-large 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. Single-control 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, a l s o when one is unable to prepare eigenstates of the system. Given a quantum subroutine which provides samples of a `phase function' g ( k ) = ∑ j A j e i ϕ j k with unknown eigenphases ϕ j and overlaps A j at quantum cost O ( k ) , we show how to estimate the phases { ϕ j } with (root-mean-square) error δ for total quantum cost T = O ( δ − 1 ) . Our scheme combines the idea of Heisenberg-limited multi-order quantum phase estimation for a single eigenvalue phase [Higgins et al (2009) and Kimmel et al (2015)] with subroutines with so-called dense quantum phase estimation which uses classical processing via time-series 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 Heisenberg-limited scaling when we use the time-series/QEEP subroutine. We present numerical evidence that using the matrix pencil technique the algorithm can achieve Heisenberg-limited scaling as well.

DOI
10.22331/q-2022-10-06-830

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Quantum phase estimation is a cornerstone in quantum algorithm design, allowing for the inference of eigenvalues of exponentially-large 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. Single-control 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, a l s o when one is unable to prepare eigenstates of the system. Given a quantum subroutine which provides samples of a `phase function' g ( k ) = ∑ j A j e i ϕ j k with unknown eigenphases ϕ j and overlaps A j at quantum cost O ( k ) , we show how to estimate the phases { ϕ j } with (root-mean-square) error δ for total quantum cost T = O ( δ − 1 ) . Our scheme combines the idea of Heisenberg-limited multi-order quantum phase estimation for a single eigenvalue phase [Higgins et al (2009) and Kimmel et al (2015)] with subroutines with so-called dense quantum phase estimation which uses classical processing via time-series 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 Heisenberg-limited scaling when we use the time-series/QEEP subroutine. We present numerical evidence that using the matrix pencil technique the algorithm can achieve Heisenberg-limited scaling as well.
Quantum phase estimation is a cornerstone in quantum algorithm design, allowing for the inference of eigenvalues of exponentially-large sparse matrices. The maximum rate at which these eigenvalues may be learned, --known as the Heisenberg limit--, is constrained by bounds on the circuit depth required to simulate an arbitrary Hamiltonian. Single-control qubit variants of quantum phase estimation 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, {\em also} when one is unable to prepare eigenstates of the system. Given a quantum subroutine which provides samples of a `phase function' g(k)=∑jAjeiϕjkg(k)=\sum_j A_j e^{i \phi_j k}g(k)=∑j​Aj​eiϕj​k with unknown eigenvalue phases ϕj\phi_jϕj​ and probabilities AjA_jAj​ at quantum cost O(k)O(k)O(k), we show how to estimate the phases {ϕj}\{\phi_j\}{ϕj​} with accuracy (root-mean-square) error δ\deltaδ for total quantum cost T=O(δ−1)T=O(\delta^{-1})T=O(δ−1). Our scheme combines the idea of Heisenberg-limited multi-order quantum phase estimation for a single eigenvalue phase \cite{Higgins09Demonstrating,Kimmel15Robust} with subroutines with so-called dense quantum phase estimation which uses classical processing via time-series analysis for the QEEP problem \cite{Somma19Quantum} or the matrix pencil method. For our algorithm which adaptively fixes the choice for kkk in g(k)g(k)g(k) we prove Heisenberg-limited scaling when we use the time-series/QEEP subroutine. We present numerical evidence that using the matrix pencil technique the algorithm can achieve Heisenberg-limited scaling as well.
This paper is an algorithmic study of quantum phase estimation with multiple eigenvalues. We present robust multiple-phase estimation (RMPE) algorithms with Heisenberg-limited scaling. The proposed algorithms improve significantly from the idea of single-phase estimation methods by combining carefully designed signal processing routines and an adaptive determination of runtime amplifying factors. They address both the integer-power model, where the unitary UUU is given as a black box with only integer runtime accessible, and the real-power model, where UUU is defined through a Hamiltonian HHH by $U=exp(\ensuremath{-}2\ensuremath{\pi}\mathrm{i}H)$ with any real runtime allowed. These algorithms are particularly suitable for early fault-tolerant quantum computers in the following senses: (1) a minimal number of ancilla qubits are used, (2) an imperfect initial state with a significant residual is allowed, (3) the prefactor in the maximum runtime can be arbitrarily small given that the residual is sufficiently small and a gap among the dominant eigenvalues is known in advance. Even if the eigenvalue gap does not exist, the proposed RMPE algorithms can achieve the Heisenberg limit while maintaining (1) and (2).
Quantum phase estimation (QPE) is the workhorse behind any quantum algorithm and a promising method for determining ground state energies of strongly correlated quantum systems. Low-cost QPE techniques make use of circuits which only use a single ancilla qubit, requiring classical post-processing to extract eigenvalue details of the system. We investigate choices for phase estimation for a unitary matrix with low-depth noise-free or noisy circuits, varying both the phase estimation circuits themselves as well as the classical post-processing to determine the eigenvalue phases. We work in the scenario when the input state is not an eigenstate of the unitary matrix. We develop a new post-processing technique to extract eigenvalues from phase estimation data based on a classical time-series (or frequency) analysis and contrast this to an analysis via Bayesian methods. We calculate the variance in estimating single eigenvalues via the time-series analysis analytically, finding that it scales to first order in the number of experiments performed, and to first or second order (depending on the experiment design) in the circuit depth. Numerical simulations confirm this scaling for both estimators. We attempt to compensate for the noise with both classical post-processing techniques, finding good results in the presence of depolarizing noise, but smaller improvements in 9-qubit circuit-level simulations of superconducting qubits aimed at resolving the electronic ground state of a H 4 -molecule.
Quantum phase estimation (QPE) is a central algorithmic primitive that estimates eigenvalues of a Hamiltonian up to precision εεε in Heisenberg-limited time T=Θ(1/ε)T=Θ(1/ε)T=Θ(1/ε). Standard gate-based implementations of QPE require deep controlled time-evolution circuits and are not native to analog hardware. Here, we present a simple adiabatic protocol for QPE that achieves (up to logarithmic factors) the optimal Heisenberg-limited scaling T=O(1εlog⁡(δ−1))T = O\left( \frac{1}ε \log\left(δ^{-1}\right)\right)T=O(ε1​log(δ−1)) in both the precision εεε and failure probability δδδ. By encoding eigenvalues in populations of computational basis states rather than complex phases, our approach is naturally robust against certain dephasing errors. The adiabatic protocol only requires the ability to couple a single ancilla qubit to the system Hamiltonian as well as pairwise couplings within the ancilla register.
This paper is an algorithmic study of quantum phase estimation with multiple eigenvalues. We present robust multiple-phase estimation (RMPE) algorithms with Heisenberg-limited scaling. The proposed algorithms improve significantly from the idea of single-phase estimation methods by combining carefully designed signal processing routines and an adaptive determination of runtime amplifying factors. They address both the {\em integer-power} model, where the unitary UUU is given as a black box with only integer runtime accessible, and the {\em real-power} model, where UUU is defined through a Hamiltonian HHH by U=exp⁡(−2πiH)U = \exp(-2π\mathrm{i} H)U=exp(−2πiH) with any real runtime allowed. These algorithms are particularly suitable for early fault-tolerant quantum computers in the following senses: (1) a minimal number of ancilla qubits are used, (2) an imperfect initial state with a significant residual is allowed, (3) the prefactor in the maximum runtime can be arbitrarily small given that the residual is sufficiently small and a gap among the dominant eigenvalues is known in advance. Even if the eigenvalue gap does not exist, the proposed RMPE algorithms can achieve the Heisenberg limit while maintaining (1) and (2).
Quantum phase estimation (QPE) is fundamental to quantum algorithms for eigenvalue problems, chemistry simulation, and number-theoretic applications. However, canonical QPE implementations employ controlled unitary operations and quantum Fourier transform, incurring substantial circuit depth and control overhead that severely limit applicability on nearterm quantum hardware. This work develops a novel class of control-free phase estimation algorithms that eliminate both controlled unitaries and QFT by leveraging signal processing principles. Our approach applies uncontrolled powers of the unitary U = e −iHτ to an eigenstate or eigenstate-rich state, measures in multiple bases to capture interference patterns, and reconstructs eigenphases via classical post-processing techniques including Prony's method, compressed sensing, and harmonic inversion. We establish that for eigenstates or states with sparse eigendecomposition, our algorithms achieve phase estimation precision ϵ using Õ(1/ϵ) applications of U , matching standard QPE query complexity up to logarithmic factors while dramatically reducing circuit depth and eliminating control overhead. We provide detailed error analysis characterizing robustness to measurement noise, finite sampling, and state preparation errors. Applications to electronic structure estimation in small molecules and variational quantum eigensolvers demonstrate the practical viability of control-free QPE on both NISQ and early fault-tolerant quantum computers. Our results suggest that signal-processing-based phase estimation may offer more efficient implementations of quantum algorithms on realistic hardware.