A quantum algorithm for modular flow

TL;DR

This paper introduces a quantum algorithm for modular flow, enabling the estimation of modular operators related to entanglement, with applications in topological systems and holography, and establishes a fundamental complexity lower bound.

Contribution

It develops the first quantum algorithm for modular flow using the QSVT framework and analyzes its complexity and potential applications.

Findings

Algorithm successfully estimates modular flow operators.

Application to extracting topological invariants like chiral central charge.

Proven query complexity lower bound for modular flow estimation.

Abstract

Entanglement is a defining property of quantum systems. For a subsystem of a larger quantum system, one can formally define an operator known as the modular Hamiltonian, which is closely linked to the entanglement properties of that subsystem, and a corresponding operator flow called the modular flow. Algorithms for estimating the von Neumann entropy, the best-known entanglement measure, are well-established, but no equivalent procedures have been previously described for the modular flow. In this work, we briefly review the quantum singular value transform (QSVT) framework for developing quantum algorithms, and then discuss the implementation of modular flow within this framework. We conclude by describing select applications of our modular flow algorithm, such as extracting the chiral central charge of a topologically ordered system and simulating the experience of the bulk observer in holography. We also prove a query complexity lower bound for modular flow, which shows that our method cannot be improved further substantially.