Dynamical Phase Transitions in Sampling Complexity

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 [Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing (ACM Press, New York, New York, USA, 2011), p. 333]. 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 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.

Figure 1

An example of the initial state in $d=2$ dimensions. Here $m=96$, $n=4$, $\beta =3$, and $c_1=3/2$. The black circles represent sites with a single boson. The cyan circles represent the ancillae.

Figure 2

Complexity phase diagram of free bosons, illustrating that sampling complexity can delineate boundaries of a physical system as a function of system parameters, including time. Our results indicate that $(\beta -1)/d\le c\le (\beta +d)/d$, where $c$ is the transition point for the scaling exponent of $t$ with $n$.