Non-unitary free boson dynamics and the boson sampling problem

TL;DR

This paper investigates how repeated measurements affect free boson dynamics, revealing a phase transition from localized to delocalized states and simplifying the boson sampling problem in certain regimes.

Contribution

It introduces a non-unitary quantum walk model for free bosons under measurements and maps it to directed polymers, revealing localization phenomena and phase transitions.

Findings

Bosons localize in real space at finite measurement strength.

Long-time wave functions share the same probability distribution.

Phase transition from localized to delocalized phase with varying measurement strength.

Abstract

We explore the free boson unitary dynamics subject to repeated random forced measurement. The input state is chosen as a Fock state in real space with the particle number conserved in the entire dynamics. We show that each boson is performing a non-unitary quantum walk in real space and its dynamics can be mapped to directed polymers in a random medium with complex amplitude. We numerically show that in the one dimensional system, when the measurement strength is finite, the system is in the frozen phase with all the bosons localized in the real space. Furthermore, these single particle wave functions take the same probability distribution in real space after long time evolution. Due to this property, the boson sampling for the output state becomes easy to solve. We further investigate circuit with non-local unitary dynamics and numerically demonstrate that there could exist a phase transition from a localized phase to a delocalized phase by varying the measurement strength.