Digital-Analog Quantum Neural Network with Long-Range Interactions

Abstract

Protocols combining single-qubit digital operations with analog quantum evolutions bypass large numbers of two-qubit entangling gates, offering a promising approach to variational algorithms and hybrid quantum simulation. Focusing on the digital-analog quantum neural network (DA-QNN) as a representative framework, this work demonstrates that incorporating long-range interactions enhances the efficiency of such hybrid algorithms. Within the analog blocks, an Ising model with power-law interactions is employed to generate entanglement, and this protocol is benchmarked across supervised and unsupervised learning tasks. Beyond discussing the effects of the transverse field and evolution time, the primary analysis focuses on how the interaction range governs the expressibility and trainability of the ansatz. For supervised image classification and many-body phase detection, long-range ansatzes match the performance of their short-range counterparts with fewer DA-QNN layers or shorter evolution times, owing to the rapid information propagation across the system enabled by nonlocal couplings. This resource reduction alleviates the classical optimization overhead and eases experimental implementation. In unsupervised phase detection, although the long-range ansatz does not outperform its short-range counterpart in every regime, the two ansatzes provide complementary signatures that jointly reconstruct the complete phase diagram. These findings establish long-range interacting platforms as a compelling resource for advancing practical, large-scale digital-analog quantum computing.

The paper asks how interaction range affects the resources and performance of digital–analog quantum neural networks. The ansatz alternates arbitrary single-qubit rotations with evolution under a Kac-normalized transverse-field Ising Hamiltonian with couplings proportional to $|i-j|^{-\alpha}$. A depth-$\ell$ circuit has $3N(\ell+1)$ trainable parameters; $\alpha$, transverse field, and analog evolution time are fixed hyperparameters. Exact-state simulations benchmark $\alpha=0.2$–$6$ and the nearest-neighbor limit on binary MNIST classification, supervised cluster-SPT phase classification, and unsupervised XXZ phase characterization, using systems of up to ten qubits. Results generally average over 10–20 optimization seeds. For eight-qubit MNIST 3-versus-8 classification, the $\alpha=0.2$ ansatz reaches a target accuracy of $0.70$ with one analog layer at both tested fields, whereas short-range ansatzes require approximately $1.5$ layers averaged over seeds. Long-range blocks also reach this target with shorter evolution times. Their shallow-circuit accuracy advantage increases over the tested system sizes and is less sensitive to feature placement and readout location. Cluster-SPT classification shows a similar size-dependent advantage. These trends are attributed to faster information propagation through nonlocal couplings. The advantage is conditional: with four analog layers, MNIST performance becomes essentially independent of interaction range for all tested sizes. Increasing qubit count also lowers MNIST accuracy, attributed to additional low-variance features and harder optimization. In the eight-qubit XXZ task, long- and short-range ansatzes yield complementary loss-map signatures rather than uniform long-range improvement. The study is numerical; large-system benefits and noise resilience remain prospective, and long-range interactions are explicitly not claimed to overcome barren-plateau scaling.