Hybrid Quantum-classical Neural Networks for Recognizing Topological Phases of Matter

With recent advances in quantum processors, standard methods for characterizing their output quantum states based on direct measurements and classical post-processing are becoming increasingly impractical due to large measurement costs. Quantum neural networks could directly process quantum states to identify underlying characteristics with reduced measurement efforts, but they often require deep quantum circuits that cannot be implemented on existing noisy intermediate-scale devices. To overcome these challenges, we introduce hybrid quantum-classical neural networks that consist of a short-depth parametrized quantum circuit, measurements, and a classical neural network for post-processing. The parametrized quantum circuit performs a nonlocal transformation of the measurement basis that is trained, jointly with the classical neural network, to variationally maximize the statistical distance between data obtained by measuring quantum states. Using supervised learning, we train these hybrid neural networks to detect a topological phase of the surface code perturbed by an external field and demonstrate that they distinguish the topological phase from all product states. We compare the sample complexity of these hybrid neural networks and a classical neural network trained on the outcomes of randomized Pauli measurements. These hybrid neural networks feature short quantum circuits that can be readily implemented on existing quantum processors and thus open the way for the efficient characterization of quantum states.