  A defining feature in the field of quantum computing is the potential of a
quantum device to outperform its classical counterpart for a specific
computational task. By now, several proposals exist showing that certain
sampling problems can be done efficiently quantumly, but are not possible
efficiently classically, assuming strongly held conjectures in complexity
theory. A feature dubbed quantum speedup. However, the effect of noise on these
proposals is not well understood in general, and in certain cases it is known
that simple noise can destroy the quantum speedup.
  Here we develop a fault-tolerant version of one family of these sampling
problems, which we show can be implemented using quantum circuits of constant
depth. We present two constructions, each taking $poly(n)$ physical qubits,
some of which are prepared in noisy magic states. The first of our
constructions is a constant depth quantum circuit composed of single and
two-qubit nearest neighbour Clifford gates in four dimensions. This circuit has
one layer of interaction with a classical computer before final measurements.
Our second construction is a constant depth quantum circuit with single and
two-qubit nearest neighbour Clifford gates in three dimensions, but with two
layers of interaction with a classical computer before the final measurements.
  For each of these constructions, we show that there is no classical algorithm
which can sample according to its output distribution in $poly(n)$ time,
assuming two standard complexity theoretic conjectures hold. The noise model we
assume is the so-called local stochastic quantum noise. Along the way, we
introduce various new concepts such as constant depth magic state distillation
(MSD), and constant depth output routing, which arise naturally in measurement
based quantum computation (MBQC), but have no constant-depth analogue in the
circuit model.
