On adaptive low-depth quantum algorithms for robust multiple-phase estimation

TL;DR

This paper introduces robust, adaptive quantum algorithms for multiple-phase estimation that achieve Heisenberg-limited scaling, suitable for early fault-tolerant quantum computers, with minimal ancilla qubits and tolerance to imperfect initial states.

Contribution

The paper develops new adaptive algorithms for multi-phase quantum estimation that improve upon existing methods by combining signal processing and runtime adaptation, achieving optimal scaling.

Findings

Achieve Heisenberg-limited scaling in multi-phase estimation.

Require minimal ancilla qubits and tolerate imperfect initial states.

Work effectively even without a known eigenvalue gap.

Abstract

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 $U$ is given as a black box with only integer runtime accessible, and the {\em real-power} model, where $U$ is defined through a Hamiltonian $H$ by $U = \exp(-2\pi\mathrm{i} H)$ 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).