Spread complexity for measurement-induced non-unitary dynamics and Zeno effect

TL;DR

This paper investigates non-unitary quantum dynamics using spread complexity and entropy, extending computational methods for complex Hamiltonians, and explores measurement-induced effects including the quantum Zeno phenomenon.

Contribution

It introduces an efficient algorithm for non-hermitian Hamiltonians, extending the bi-Lanczos method, and applies it to study measurement effects on quantum complexity and the Zeno effect.

Findings

Spread complexity initially grows then saturates over time.

Measurement frequency influences the growth time of complexity.

Quantum Zeno effect emerges as measurement intervals approach zero.

Abstract

Using spread complexity and spread entropy, we study non-unitary quantum dynamics. For non-hermitian Hamiltonians, we extend the bi-Lanczos construction for the Krylov basis to the Schr\"odinger picture. Moreover, we implement an algorithm adapted to complex symmetric Hamiltonians. This reduces the computational memory requirements by half compared to the bi-Lanczos construction. We apply this construction to the one-dimensional tight-binding Hamiltonian subject to repeated measurements at fixed small time intervals, resulting in effective non-unitary dynamics. We find that the spread complexity initially grows with time, followed by an extended decay period and saturation. The choice of initial state determines the saturation value of complexity and entropy. In analogy to measurement-induced phase transitions, we consider a quench between hermitian and non-hermitian Hamiltonian evolution induced by turning on regular measurements at different frequencies. We find that as a function of the measurement frequency, the time at which the spread complexity starts growing increases. This time asymptotes to infinity when the time gap between measurements is taken to zero, indicating the onset of the quantum Zeno effect, according to which measurements impede time evolution.