Using variance of gradients to explain quantum phase recognition algorithms

Phase transitions separate different phases of quantum matter and are indicated by discontinuities in the thermodynamic limit (the limit of infinite system size). In contrast, adiabatic state preparation prepares the ground state of one Hamiltonian from the ground state of another. This is efficient if the two Hamiltonians can be deformed into each other without the gap between the ground state and first excited state closing. Using these properties we study the accuracy of the quantum convolutional neural network (QCNN) proposed by Cong et al [Nature Physics volume 15, pages 1273–1278 (2019)] at determining the phase of matter of a state.

We show that, with the increasing depth of the QCNN, the change in the QCNN’s output with respect to perturbations of the parent Hamiltonian concentrates around zero. Using this we deduce that the QCNN output converges to a constant value within each phase of matter as long as the QCNN the depth is sufficient. We show the transition in QCNN output exactly matches the phase boundary in the thermodynamic limit (for a logarithmic depth QCNN). Using these insights we generate an infinite family of QCNNs similar to the Cong QCNN, and discuss how these results could be generalised to other Hamiltonian families.