Complexity of operators generated by quantum mechanical Hamiltonians

TL;DR

This paper develops a framework for computing the complexity of operators generated by Hamiltonians in quantum field theory and quantum mechanics, revealing geometric and spacetime connections, including an equivalence to AdS3 and implications for complexity bounds.

Contribution

It introduces new principles and methods for complexity in QFT/QM, establishing geometric interpretations and linking complexity to spacetime properties and bounds.

Findings

Complexity geometry for 1D quadratic Hamiltonians is equivalent to AdS3 spacetime.

Complexity of operators from free Hamiltonians is zero, aligning with expectations.

In low energy limits, the critical spacetime dimension for nonnegative complexity is 3+1.

Abstract

We propose how to compute the complexity of operators generated by Hamiltonians in quantum field theory (QFT) and quantum mechanics (QM). The Hamiltonians in QFT/QM and quantum circuit have a few essential differences, for which we introduce new principles and methods for complexity. We show that the complexity geometry corresponding to one-dimensional quadratic Hamiltonians is equivalent to AdS$_3$ spacetime. Here, the requirement that the complexity is nonnegative corresponds to the fact that the Hamiltonian is lower bounded and the speed of a particle is not superluminal. Our proposal proves the complexity of the operator generated by a free Hamiltonian is zero, as expected. By studying a non-relativistic particle in compact Riemannian manifolds we find the complexity is given by the global geometric property of the space. In particular, we show that in low energy limit the critical spacetime dimension to ensure the "nonnegative" complexity is the 3+1 dimension.