# Dynamical phase transitions in sampling complexity

## TL;DR

This paper investigates how the complexity of approximately simulating quantum systems via sampling changes over time, revealing a dynamical phase transition in sampling difficulty related to physical phases.

## Contribution

It establishes bounds on sampling complexity over time for bosonic systems and identifies a phase transition in classical simulability linked to physical phases.

## Key findings

- Sampling complexity has an upper bound scaling with system size.
- A lower bound indicates when sampling becomes computationally hard.
- Systems in the Anderson-localized phase are always easy to sample from.

## Abstract

We make the case for studying the complexity of approximately simulating (sampling) quantum systems for reasons beyond that of quantum computational supremacy, such as diagnosing phase transitions. We consider the sampling complexity as a function of time $t$ due to evolution generated by spatially local quadratic bosonic Hamiltonians. We obtain an upper bound on the scaling of $t$ with the number of bosons $n$ for which approximate sampling is classically efficient. We also obtain a lower bound on the scaling of $t$ with $n$ for which any instance of the boson sampling problem reduces to this problem and hence implies that the problem is hard, assuming the conjectures of Aaronson and Arkhipov [Proc. 43rd Annu. ACM Symp. Theory Comput. STOC '11]. This establishes a dynamical phase transition in sampling complexity. Further, we show that systems in the Anderson-localized phase are always easy to sample from at arbitrarily long times. We view these results in the light of classifying phases of physical systems based on parameters in the Hamiltonian. In doing so, we combine ideas from mathematical physics and computational complexity to gain insight into the behavior of condensed matter, atomic, molecular and optical systems.