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

TL;DR

The paper investigates unconditional quantum-classical separations in learning tasks implemented by shallow circuits, with and without noise. It constructs a relation $R^*$ realizable by a constant-depth quantum circuit and proves that any classical bounded-connectivity network requires depth $\Omega(\epsilon \log n)$ to reach comparable accuracy, driven by quantum nonlocality. Under depolarizing noise, the separation persists only if the noise scales as $p=O(n^{-\epsilon})$ and collapses for $p=\Omega(1/\sqrt{\log n})$, while constant-noise Clifford circuits do not yield a super-polynomial advantage. These results illuminate the regime and limitations for near-term quantum learning, guiding experimental thresholds and highlighting the role of nonlocality and Clifford structure in quantum-classical gaps.

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.

Figures (3)

• Figure 1: (a) An illustration of classical and quantum supervised learning models using shallow quantum circuits and neural networks with bounded connectivity. The data samples are first encoded into bit strings $\bm{x}$. These bit strings are input into a variational quantum circuit in the quantum-enhanced scenario and a classical neural network in the classical scenario. Each neuron in the classical neural network has bounded connectivity to the neurons in the previous layer. The outputs for two classifiers are distributions on all possible labels encoded in bit strings $\bm{y}$. (b) An implementation of a noisy quantum circuit, where an identical amount $p$ of depolarizing noise is added to each qubit at each step. (c) The characterization of noise rate regimes for the existence and absence of quantum-classical separation in learning classification tasks defined by shallow quantum circuits. Here, classification tasks $R^*$ and $R_C$ are relations defined in \ref{['thm:IdealSep']} and \ref{['thm:NoisyBound']}, respectively. The classification task $R^*$ can provide an exponential separation in the prediction accuracy when the noise rate is bounded above by $O(n^{-\epsilon})$, while such separation will vanish in the noise regime of $p>\Omega(1/\sqrt{\log n})$. In addition, any classification defined by a shallow Clifford circuit possesses no super-polynomial quantum-classical separation when the quantum circuit is implemented with a constant noise rate.
• Figure S1: Schematics of supervised learning with variational quantum circuits. After feeding the training samples into the model and obtaining the outputs, the optimizer will update the variational parameters in order to minimize the loss function.
• Figure S2: Depth complexity in simulating shallow Clifford $+$ T quantum circuit using classical circuits. As the quantum circuit only contains single- and two-qubit gates, the light cone of a gate can contain at most $O(1)$ qubits in the $\sqrt{c\log n/c_1}\times \sqrt{c\log n/c_1}$ block. Therefore, the interacting qubits on one border can at most affect the output qubits of size $\sqrt{c\log n/c_1}\times O(1)$ as shown in (a). As shown in (b), all the output qubits in the block can thus be divided into qubits (the outer region) interacting with the qubits outside the block and the qubits (the inner region) independent of qubits outside the block.

Theorems & Definitions (15)

• Theorem 1
• proof
• Theorem 2
• proof
• Definition S1
• Lemma S1: Theorem 2, Ref. Gall2018Average
• Lemma S2: Theorem 10, Ref. Gall2018Average
• Proposition 1
• Theorem S1
• proof