Quantum-Classical Separations in Shallow-Circuit-Based Learning with and without Noises

TL;DR

This paper demonstrates a quantum advantage over classical neural networks in shallow circuits for certain classification tasks, highlighting the role of quantum nonlocality and analyzing noise effects on this separation.

Contribution

It constructs a specific quantum-classical separation for shallow circuits, establishes noise thresholds for practical quantum advantage, and shows limitations under constant noise levels.

Findings

Quantum circuits outperform classical neural networks in shallow-depth classification tasks.

Noise thresholds are identified where quantum advantage persists or vanishes.

No super-polynomial separation exists for shallow Clifford circuits under constant noise.

Abstract

We study quantum-classical separations between classical and quantum supervised learning models based on constant depth (i.e., shallow) circuits, in scenarios with and without noises. We construct a classification problem defined by a noiseless shallow quantum circuit and rigorously prove that any classical neural network with bounded connectivity requires logarithmic depth to output correctly with a larger-than-exponentially-small probability. This unconditional near-optimal quantum-classical separation originates from the quantum nonlocality property that distinguishes quantum circuits from their classical counterparts. We further derive the noise thresholds for demonstrating such a separation on near-term quantum devices under the depolarization noise model. We prove that this separation will persist if the noise strength is upper bounded by an inverse polynomial with respect to the system size, and vanish if the noise strength is greater than an inverse polylogarithmic function. In addition, for quantum devices with constant noise strength, we prove that no super-polynomial classical-quantum separation exists for any classification task defined by shallow Clifford circuits, independent of the structures of the circuits that specify the learning models.