At the beginning of the project, we finished the study of explicitly constructed error-tolerant quantum convolutional neural networks (QCNNs) using multiscale entanglement renormalization ansatz that was initiated before the start of this project. During the revision of the manuscript, we provided a further analytical analysis, explained a general approach for achieving error tolerance in QCNNs and outlined its extension for two- and higher-dimensional systems. This manuscript was published in Phys. Rev. Research in month 11.
Together with my colleagues, I finished the study of explicitly constructed QCNNs for the two-dimensional toric code model, which was initially posted on the arXiv [arXiv:2407.04114] in month 11 of the outgoing phase. During the return phase of HyNNet NISQ, we substantially revised the manuscript to extend the QCNN to capture correlations between all stabilizer elements of the toric code, which cannot be accessed by direct measurements. This increases the probability of correlated incoherent errors that can be tolerated by the network and allows for the correct identification of the topological phase in the presence of strong correlated errors. This is not possible with methods based on direct measurements, showing the advantage achieved via the processing of quantum states on the quantum processor compared to direct measurements. To this end, we designed a new convolutional layer of the QCNN that takes correlations between error syndromes into account. This manuscript was published in Phys. Rev. Research in month 25.
We showed that these explicitly constructed QCNNs can be implemented as a short-depth quantum circuit, measurement and classical post-processing of measurement results. This motivates the design of trainable hybrid neural networks (HNNs), which are the main subject of this project. Therefore, these QCNNs provide an important input for the investigation of HNNs.
In the HyNNet NISQ project, we designed trainable HNNs depicted in the figure below. Analogously to the explicitly constructed QCNNs, they consist of a quantum circuit, measurement and classical-post processing. In contrast to the previous works, where the quantum circuits and classical post-processing procedures were explicitly constructed, here we studied their parametrization and training. Note that in this project, we considered small system sizes up to 25 qubits, and we, thus, did not impose translational invariance in these HNNs.
We designed HNNs for the detection of the topological phase of the surface code in a magnetic field, which features intrinsic topological order. The parametrized quantum circuit depicted in the figure below consists of native gates of the superconducting quantum processor in the group of Prof. Andreas Wallraff at ETH Zurich. This facilitated the implementation of these HNNs on the quantum processor, which complements the theoretical investigation performed in this project. We focused on supervised learning, where these HNNs are trained to uncover characteristics of the topological phase from training data. The trained HNN distinguishes the topological phase not only from a topologically trivial magnetic phase but also from product states.
To investigate the practical advantages of these HNNs, we compared their sample complexity to alternative methods for quantum phase recognition using direct randomized measurements of quantum states in conjunction with classical shadow methods. In particular, we showed that the HNN learns the features of the topological phase with smaller sample complexity than a classical neural network trained with the outcomes of such local randomized measurements. We are currently preparing a manuscript presenting these results.
In parallel to this theoretical effort, we realized, together with the group of Prof. Andreas Wallraff at ETH Zurich, these HNNs on a superconducting quantum processor to test their training and performance in the presence of noise due to decoherence and experimental imperfections. In this collaboration, we optimized the hybrid two-loop training procedure for their quantum processor, significantly reducing the classical neural-network training time to mitigate latency in the exploitation of the quantum device. Furthermore, we devised variational circuits for the preparation of the surface code ground states in a magnetic field on the quantum processor. We optimized these variational circuits in classical preprocessing using state vector simulations and estimated their performance taking into account detailed noise models of the superconducting chip via Kraus vector simulations. We are currently preparing a manuscript presenting these results.
