$TimeEvolver$: A Program for Time Evolution With Improved Error Bound

TL;DR

TimeEvolver is a software tool that efficiently computes quantum system time evolution with rigorous error bounds, enabling high-precision simulations in large Hilbert spaces.

Contribution

It introduces a new error bound for Krylov subspace methods applied to Hermitian matrices, improving accuracy and reliability over existing software.

Findings

Allows computation in Hilbert spaces >10^6 dimensions

Provides rigorous a posteriori error bounds

Enables adjustable precision in quantum simulations

Abstract

We present $TimeEvolver$, a program for computing time evolution in a generic quantum system. It relies on well-known Krylov subspace techniques to tackle the problem of multiplying the exponential of a large sparse matrix $i H$, where $H$ is the Hamiltonian, with an initial vector $v$. The fact that $H$ is Hermitian makes it possible to provide an easily computable bound on the accuracy of the Krylov approximation. Apart from effects of numerical roundoff, the resulting a posteriori error bound is rigorous, which represents a crucial novelty as compared to existing software packages such as $Expokit$ (R. Sidje, ACM Trans. Math. Softw. 24 (1) 1998). On a standard notebook, $TimeEvolver$ allows to compute time evolution with adjustable precision in Hilbert spaces of dimension greater than $10^6$. Additionally, we provide routines for deriving the matrix $H$ from a more abstract representation of the Hamiltonian operator.