Quantum Phase Processing and its Applications in Estimating Phase and Entropies

TL;DR

This paper introduces quantum phase processing, a new framework for applying transformations to eigenphases of unitary operators, enabling efficient quantum algorithms for phase estimation, Hamiltonian simulation, and entropy measurement.

Contribution

The paper develops quantum phase processing as a novel, efficient method for phase transformations and eigen-information extraction, improving quantum algorithms for various phase-related problems.

Findings

Proposes a quantum phase estimation algorithm without quantum Fourier transform.

Demonstrates improvements in Hamiltonian simulation and entanglement spectroscopy.

Achieves optimal or near-optimal performance in quantum entropy estimation.

Abstract

Quantum computing can provide speedups in solving many problems as the evolution of a quantum system is described by a unitary operator in an exponentially large Hilbert space. Such unitary operators change the phase of their eigenstates and make quantum algorithms fundamentally different from their classical counterparts. Based on this unique principle of quantum computing, we develop a new algorithmic toolbox "quantum phase processing" that can directly apply arbitrary trigonometric transformations to eigenphases of a unitary operator. The quantum phase processing circuit is constructed simply, consisting of single-qubit rotations and controlled-unitaries, typically using only one ancilla qubit. Besides the capability of phase transformation, quantum phase processing in particular can extract the eigen-information of quantum systems by simply measuring the ancilla qubit, making it naturally compatible with indirect measurement. Quantum phase processing complements another powerful framework known as quantum singular value transformation and leads to more intuitive and efficient quantum algorithms for solving problems that are particularly phase-related. As a notable application, we propose a new quantum phase estimation algorithm without quantum Fourier transform, which requires the fewest ancilla qubits and matches the best performance so far. We further exploit the power of our method by investigating a plethora of applications in Hamiltonian simulation, entanglement spectroscopy and quantum entropies estimation, demonstrating improvements or optimality for almost all cases.