Time-dependent Hamiltonians and Geometry of Operators Generated by Them

TL;DR

This paper develops a geometric framework to quantify the complexity of time-dependent quantum Hamiltonians, providing analytical cost expressions and exploring their relation to information-theoretic measures in specific quantum systems.

Contribution

It introduces a method to compute complexity geometry for time-dependent Hamiltonians using Nielsen's approach, with analytical expressions and comparisons to entropy.

Findings

Derived analytical expressions for complexity costs in time-dependent systems.

Established equivalence between costs of different harmonic oscillator models.

Compared complexity evolution with Shannon entropy in a quench protocol.

Abstract

We obtain the complexity geometry associated with the Hamiltonian of a quantum mechanical system, specifically in cases where the Hamiltonian is explicitly time-dependent. Using Nielsen's geometric formulation of circuit complexity, we calculate the bi-invariant cost associated with these time-dependent Hamiltonians by suitably regularising their norms and obtain analytical expressions of the costs for several well-known time-dependent quantum mechanical systems. Specifically, we show that an equivalence exists between the total costs of obtaining an operator through time evolution generated by a unit mass harmonic oscillator whose frequency depends on time, and a harmonic oscillator whose both mass and frequency are functions of time. These results are illustrated with several examples, including a specific smooth quench protocol where the comparison of time variation of the cost with other information theoretic quantities, such as the Shannon entropy, is discussed.