Fault-tolerant quantum speedup from constant depth quantum circuits

Abstract

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, based on nonadaptive measurements on regular graph states composed of m qubits. We show that these sampling problems can be implemented fault tolerantly using quantum circuits of constant depth. We present two constructions, each taking poly(m) 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-neighbor 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-neighbor 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(m) 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 concepts such as constant depth magic state distillation and constant depth output routing, which arise naturally in measurement based quantum computation, but have no constant-depth analog in the circuit model.