Regression relation for pure quantum states and its implications for efficient computing

TL;DR

This paper introduces a modified Onsager regression relation for pure quantum states, enabling efficient computation of high-temperature correlation functions in many-body quantum systems without full Hamiltonian diagonalization.

Contribution

It presents a new regression relation and a computational method based on quantum typicality, reducing memory requirements and allowing simulations of larger quantum systems.

Findings

Successfully computed correlation functions for 29-spin chains

Validated the spin diffusion hypothesis with numerical results

Demonstrated exponential memory reduction in simulations

Abstract

We obtain a modified version of the Onsager regression relation for the expectation values of quantum-mechanical operators in pure quantum states of isolated many-body quantum systems. We use the insights gained from this relation to show that high-temperature time correlation functions in many-body quantum systems can be controllably computed without complete diagonalization of the Hamiltonians, using instead the direct integration of the Schroedinger equation for randomly sampled pure states. This method is also applicable to quantum quenches and other situations describable by time-dependent many-body Hamiltonians. The method implies exponential reduction of the computer memory requirement in comparison with the complete diagonalization. We illustrate the method by numerically computing infinite-temperature correlation functions for translationally invariant Heisenberg chains of up to 29 spins 1/2. Thereby, we also test the spin diffusion hypothesis and find it in a satisfactory agreement with the numerical results. Both the derivation of the modified regression relation and the justification of the computational method are based on the notion of quantum typicality.