Quantum Algorithms for Inverse Participation Ratio Estimation in multi-qubit and multi-qudit systems

TL;DR

This paper introduces three quantum algorithms for estimating Inverse Participation Ratios (IPRs) in multi-qubit and multi-qudit systems, providing tools to analyze quantum state spread and system properties.

Contribution

The paper presents novel quantum algorithms for IPR estimation in different bases and system types, with resource analysis and benchmarking on various quantum models.

Findings

Algorithms successfully estimate IPRs in multiple systems.

Resource requirements are analyzed for practical implementation.

Benchmarking demonstrates effectiveness on physical quantum models.

Abstract

Inverse Participation Ratios (IPRs) and the related Participation Entropies quantify the spread of a quantum state over a selected basis of the Hilbert space, offering insights into the equilibrium and non-equilibrium properties of the system. In this work, we propose three quantum algorithms to estimate IPRs on multi-qubit and multi-qudit quantum devices. The first algorithm allows for the estimation of IPRs in the computational basis by single-qubit measurements, while the second one enables measurement of IPR in the eigenbasis of a selected Hamiltonian, without the knowledge about the eigenstates of the system. Next, we provide an algorithm for IPR in the computational basis for a multi-qudit system. We discuss resources required by the algorithms and benchmark them by investigating the one-axis twisting protocol, the thermalization in a deformed PXP model, and the ground state of a spin-$1$ AKLT chain in a transverse field.