Error-tolerant quantum convolutional neural networks for the recognition of symmetry-protected topological phases

The development and application of quantum computers require tools to evaluate noisy quantum data produced by near-term quantum hardware. Quantum neural networks based on parametrized quantum circuits, measurements and feedforward can process large quantum data, to detect non-local quantum correlations with reduced measurement and computational efforts compared to standard characterization techniques. Here we construct quantum convolutional neural networks (QCNNs) based on the multiscale entanglement renormalization ansatz that can recognize different symmetry-protected topological phases of generalized cluster-Ising Hamiltonians in the presence of incoherent errors, simulating the effects of decoherence under NISQ conditions. Using matrix product state simulations, we show that the QCNN output is robust against symmetry-preserving errors if the error channel is invertible. Moreover, the QCNNs tolerate symmetry-breaking errors below a threshold error probability in contrast to previous QCNN designs and string order parameters (SOPs), which are significantly suppressed for any non-vanishing error probability. Even though the error tolerance is limited close to phase boundaries due to a diverging correlation length, the QCNNs can precisely determine critical values of Hamiltonian parameters. To facilitate the implementation of QCNNs on near-term quantum computers, we show how to shorten a general class of QCNN circuits from logarithmic to constant depth in system size by performing a large part of computation in classical post-processing after the measurement of all qubits. The QCNN with a constant-depth quantum circuit reduces sample complexity exponentially with system size in comparison to the direct sampling of the QCNN output using local Pauli measurements with only logarithmic classical computational costs.