Complexity in two-point measurement schemes

TL;DR

This paper introduces a novel approach to quantify the complexity of state evolution in two-point measurement schemes, linking it to circuit complexity and analyzing its behavior in chaotic versus integrable spin chains.

Contribution

It develops a framework connecting the spread of evolved states to circuit complexity measures and applies it to quench dynamics in spin chains, revealing chaos-dependent complexity saturation.

Findings

Complexity saturates only for chaotic pre-quench Hamiltonians.

Lanczos coefficients relate to geometric circuit complexity measures.

Analytical and numerical analysis of complexity evolution in spin chains.

Abstract

We show that the characteristic function of the probability distribution associated with the change of an observable in a two-point measurement protocol with a perturbation can be written as an auto-correlation function between an initial state and a certain unitary evolved state by an effective unitary operator. Using this identification, we probe how the evolved state spreads in the corresponding conjugate space, by defining a notion of the complexity of the spread of this evolved state. For a sudden quench scenario, where the parameters of an initial Hamiltonian (taken as the observable measured in the two-point measurement protocol) are suddenly changed to a new set of values, we first obtain the corresponding Krylov basis vectors and the associated Lanczos coefficients for an initial pure state, and obtain the spread complexity. Interestingly, we find that in such a protocol, the Lanczos coefficients can be related to various cost functions used in the geometric formulation of circuit complexity, for example the one used to define Fubini-Study complexity. We illustrate the evolution of spread complexity both analytically, by using Lie algebraic techniques, and by performing numerical computations. This is done for cases when the Hamiltonian before and after the quench are taken as different combinations of chaotic and integrable spin chains. We show that the complexity saturates for large values of the parameter only when the pre-quench Hamiltonian is chaotic. Further, in these examples we also discuss the important role played by the initial state which is determined by the time-evolved perturbation operator.