High-expressibility Quantum Neural Networks using only classical resources

TL;DR

This work shows that some desired properties attributed to QNN models can be efficiently reproduced without necessarily resorting to quantum hardware, and assess the level of primary quantum resources, entanglement and non-stabilizerness in random ensembles of such quantum states, tracking their convergence towards the Haar distribution.

Abstract

Quantum neural networks (QNNs), as currently formulated, are near-term quantum machine learning architectures that leverage parameterized quantum circuits with the aim of improving upon the performance of their classical counterparts. In this work, we show that some desired properties attributed to these models can be efficiently reproduced without necessarily resorting to quantum hardware. We indeed study the expressibility of parametrized quantum circuit commonly used in QNN applications and contrast it to those of two classes of states that can be efficiently simulated classically: matrix-product states (MPS), and Clifford-enhanced MPS (CMPS), obtained by applying a set of Clifford gates to MPS. In addition to expressibility, we assess the level of primary quantum resources, entanglement and non-stabilizerness (a.k.a. "magic"), in random ensembles of such quantum states, tracking their convergence towards the Haar distribution. While MPS require a large number of parameters to effectively reproduce an arbitrary quantum state, we find that CMPS approach the Haar distribution more rapidly, in terms of both entanglement and magic. Our results on states with up to 20 qubits indicate that high expressibility in QNNs is attainable with purely classical resources.

Figures (6)

• Figure 1: Qualitative representation of the phase space defined by non-stabilizerness (magic) and entanglement, illustrating the regions typically explored by different classes of wave function Ansätze. Tensor networks (TNs), shown in blue, can access areas with relatively low entanglement, but potentially high magic chen2024magic_mps. Parameterized quantum circuits (PQCs), depicted in red, generally have a layered structure that allows them to simultaneously increase both magic and entanglement, when the circuit is made deeper. They tend to reach the quantum resources of the Haar distribution (region bounded in gray) more efficiently than TNs. Stabilizer states, shown in yellow, have zero magic.
• Figure 2: A Quantum states samples in the entanglement-magic phase space ${\widetilde{\mathcal{S}}}-{\widetilde{\mathcal{M}}}$. The results are obtained for $n=10$ qubits. States sampled from the Haar measure form an extremely localized distribution szombathy2025independent centered at $(1,1)$ (marked by the gold diamond). The other colors correspond to the various architectures shown in B,C,D. Red colors indicate fQNN states (B) with different number of layers, blue colors indicate MPS states (C) with different bond dimension, while green colors denote CMPS (D) results. Empty circles correspond to the average values of ${\widetilde{\mathcal{S}}}$ and ${\widetilde{\mathcal{M}}}$ for the various sets of samples.
• Figure 3: A Average values of $\widetilde{\mathcal{M}}$ (golden triangles) and $\widetilde{\mathcal{S}}$ (violet circles) for fQNN, MPS and CMPS ensembles, as a function of the number of layers $\mathcal{L}$ or the bond dimension $\chi$. Different shades of the colors indicate different numbers of qubits $n$. B Number of parameters $\mathcal{P}$ that are needed such that magic and entanglement of the various ensembles reach 90% of the asymptotic values of the Haar distribution, namely $\widetilde{\mathcal{M}}_{n, \mu}=0.9$ and $\widetilde{\mathcal{S}}_{n, \mu}=0.9$. Triangles and circles represent magic and entanglement, respectively. Different colors indicate different classes of states. Here $\widetilde{\mathcal{M}}_{n, {\rm CMPS}}$ is omitted since $\widetilde{\mathcal{M}}_{n, {\rm CMPS}} = \widetilde{\mathcal{M}}_{n, {\rm MPS}}$ by construction.
• Figure 4: Rescaled frame potentials $\widetilde{\mathcal{F}}^{(t)}_\mu$ in function of the $t$, numerically estimated with $10^7$ samples of fidelities $F$. In panel A we show results for $n=10$ qubits and different classes of states. The Welch bound is marked by the black horizontal line. In panel B we show a zoom in a region close to the Welch bound for $t=4,5,6$. The results for CMPS are compatible with the Haar moments distribution even for a bond dimension of $\chi = 2$. For a fair comparison, we show, for instance, the case of fQNN with $\mathcal{L} = 12$ layers because it has the same number of parameters, $\mathcal{P} = 360$, as a CMPS with bond dimension $\chi = 3$.
• Figure 5: Kullback-Leibler divergence $D_{\rm KL}$ between numerically estimated fidelity distributions $\hat{P}_{\mu}(F)$, for different classes of states, and the exact Haar fidelity distribution $P_{\text{H}}(F)$. Error bars are computed using jackknife resampling. Lower $D_{\rm KL}$ divergence signifies higher expressibility of the architecture in approximating the Haar distribution on pure quantum states. The gray shaded region represents the KL divergence between the true Haar fidelity distribution $P_{\text{H}}(F)$ and the one obtained by sampling $10^5$ Haar-distributed fidelities for $n = 10,16,20$ qubits, accounting for the errors in this estimate. In the inset, the number of parameters, normalized by the number of qubits, required by different architectures to reach a threshold of $D_{\rm KL}=10^{-3}$ (black dashed line in the main plot) is shown.