Integrability and complexity in quantum spin chains

TL;DR

This paper establishes a quantitative link between integrability and reduced complexity in quantum spin chains by relating conserved quantities to bounds on the quantum evolution operator's Nielsen complexity.

Contribution

It introduces a matrix construction connecting eigenvectors to conserved quantities and shows how eigenvalue magnitudes bound the complexity of quantum evolution in integrable systems.

Findings

Conserved quantities correspond to null eigenvalues of a specific matrix.

Eigenvalue magnitudes provide bounds on Nielsen complexity.

Application to various quantum spin chains demonstrates the theory.

Abstract

There is a widespread perception that dynamical evolution of integrable systems should be simpler in a quantifiable sense than the evolution of generic systems, though demonstrating this relation between integrability and reduced complexity in practice has remained elusive. We provide a connection of this sort by constructing a specific matrix in terms of the eigenvectors of a given quantum Hamiltonian. The null eigenvalues of this matrix are in one-to-one correspondence with conserved quantities that have simple locality properties (a hallmark of integrability). The typical magnitude of the eigenvalues, on the other hand, controls an explicit bound on Nielsen's complexity of the quantum evolution operator, defined in terms of the same locality specifications. We demonstrate how this connection works in a few concrete examples of quantum spin chains that possess diverse arrays of highly structured conservation laws mandated by integrability.