Dimensional Reductions for the Computation of Time-Dependent Quantum Expectations

TL;DR

This paper introduces a Chebyshev polynomial-based dimensional reduction method for efficiently computing expectation values in large, time-dependent quantum systems, outperforming existing techniques in scalability.

Contribution

The authors present a novel DEC method that enables direct and efficient expectation value calculations without losing information, suitable for large quantum systems.

Findings

DEC method shows superior scalability with system size

Compared to existing methods, DEC provides faster computations

Effective for autonomous quantum problems focusing on specific observables

Abstract

We consider dimensional reduction techniques for the Liouville-von Neumann equation for the evaluation of the expectation values in a mixed quantum system. In applications such as nuclear spin dynamics the main goal for simulations is being able to simulate a system with as many spins as possible, for this reason it is very important to have an efficient method that scales well with respect to particle numbers. We describe several existing methods that have appeared in the literature, pointing out their limitations particularly in the setting of large systems. We introduce a method for direct computation of expectations via Chebyshev polynomials (DEC) based on evaluation of a trace formula combined with expansion in modified Chebyshev polynomials. This reduction is highly efficient and does not destroy any information. We demonstrate the practical application of the scheme for a nuclear spin system and compare with several alternatives, focusing on the performance of the various methods with increasing system dimension. Our method may be applied to autonomous quantum problems where the desired outcome of quantum simulation, rather than being a full description of the system dynamics, is only the expectation value of some given observable.