Renormalisation Through The Lens Of QCNNs

The cluster-Ising model is an example of a quantum model with a symmetry protected topological (SPT) phase. The efficiency of phase recognition has recently been improved over measuring string order parameter (SOP) by the use of a particular quantum convolutional neural network (QCNN) that was motivated by renormalisation theory.

Unlike most neural networks, the function of the QCNN used here is relatively straightforward to explain. First, each layer of the QCNN performs a process analogous to both renormalisation and quantum error correction. Then second, the remainder of the circuit simply determines if we are in the ground state of a stabiliser Hamiltonian. If the energy is sufficiently low we consider this to be in the target phase.

This QCNN also has a second feature, it is exactly equivalent to a constant depth quantum circuit + post-processing. Beyond just providing a cheaper circuit, this also points to the generalisation of phase recognising QCNNs beyond the cluster-ising model. Combining these with the fidelity view of quantum phases, I will discuss the potential of phase recognising QCNNs as a quantum information theory construction of renormalisation and phase transitions, as well as how this approach may be extended to transitions arising from incoherent errors.