Fault-tolerant quantum speedup from constant depth quantum circuits

TL;DR

This paper demonstrates fault-tolerant quantum circuits of constant depth that maintain quantum speedup in sampling problems, even under realistic noise models, by developing novel constructions involving noisy magic states and classical-quantum interaction.

Contribution

It introduces two constant-depth quantum circuit constructions that preserve quantum speedup under noise, utilizing new concepts like constant depth magic state distillation and output routing.

Findings

Quantum circuits of constant depth can achieve quantum speedup under noise.

Classical algorithms cannot efficiently simulate these quantum circuits assuming standard conjectures.

New techniques like constant depth magic state distillation are developed for fault-tolerance.

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, 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.