  The generation of certifiable randomness is the most fundamental
information-theoretic task that meaningfully separates quantum devices from
their classical counterparts. We propose a protocol for exponential certified
randomness expansion using a single quantum device. The protocol calls for the
device to implement a simple quantum circuit of constant depth on a 2D lattice
of qubits. The output of the circuit can be verified classically in linear
time, and is guaranteed to contain a polynomial number of certified random bits
assuming that the device used to generate the output operated using a
(classical or quantum) circuit of sub-logarithmic depth. This assumption
contrasts with the locality assumption used for randomness certification based
on Bell inequality violation or computational assumptions. To demonstrate
randomness generation it is sufficient for a device to sample from the ideal
output distribution within constant statistical distance.
  Our procedure is inspired by recent work of Bravyi et al. (Science 2018), who
introduced a relational problem that can be solved by a constant-depth quantum
circuit, but provably cannot be solved by any classical circuit of
sub-logarithmic depth. We develop the discovery of Bravyi et al. into a
framework for robust randomness expansion. Our proposal does not rest on any
complexity-theoretic conjectures, but relies on the physical assumption that
the adversarial device being tested implements a circuit of sub-logarithmic
depth. Success on our task can be easily verified in classical linear time.
Finally, our task is more noise-tolerant than most other existing proposals
that can only tolerate multiplicative error, or require additional conjectures
from complexity theory; in contrast, we are able to allow a small constant
additive error in total variation distance between the sampled and ideal
distributions.
