Hybrid quantum-classical neural networks for the recognition of topological phases

With the increasing complexity of quantum computers, standard methods for characterizing quantum states based on direct measurements and classical post-processing have become impractical due to large measurement costs. Quantum neural networks can 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 devices. To overcome this challenge, we introduce hybrid quantum-classical neural networks that consist of a short-depth parametrized quantum circuit, measurement and a classical neural network. Using supervised learning, we train these hybrid neural networks to detect a topological phase of the surface code perturbed by an external field, and to recognize two symmetry-protected topological phases of a generalized cluster-Ising model from one another. The parametrized quantum circuit performs a nonlocal transformation of the measurement basis that is trained to variationally maximize the statistical distance between data obtained by measuring quantum states. This significantly reduces the sample complexity required to recognize the topological phases using the classical neural network. Furthermore, we employ unsupervised learning by confusion to identify the boundaries between these phases and show the hybrid neural networks approximate the fidelity susceptibility, which is known to detect a wide range of quantum phase transitions but is hard to measure directly. These hybrid neural networks feature short quantum circuits that can be readily implemented on existing quantum computers and thus open the way for the efficient characterization of quantum states.