Spectral, information-theoretic, and perturbative methods for quantum learning and error mitigation

Abstract

We present spectral and information-theoretic characterizations of learning tasks involving quantum systems, and develop new perturbative error mitigation techniques for near-term devices. In the first part of this thesis, we explore connections between quantum information and learning theory. We demonstrate theoretically that kernel bandwidth enables quantum kernel methods associated with a high dimensional quantum feature space to generalize. We then characterize quantum machine learning models that generalize despite overfitting their training data, contradicting standard expectations from learning theory. In such learning tasks, the learner may fail due to noise in the input data. So we next consider a setting where the learner has access to correlated auxiliary noise, a resource that contains information about an otherwise unknown noise source corrupting input data. We use classical Shannon theory to relate the strength of these correlations to the classical capacity of a bit flip channel with correlated auxiliary noise, and we extend this analysis to derive the quantum capacity of a quantum bit flip channel given access to an auxiliary system entangled with the environmental source of the noise. Finally, we derive an information-theoretic guarantee for the learnability of data by an optimal learner and, extending this technique to a quantum setting, we introduce and characterize an entanglement manipulation task that generalizes the notion of classical learning.

The second part of this thesis introduces techniques for error mitigation on near-term quantum processors and provides guarantees in the perturbative limit. We introduce a technique for mitigating measurement errors using truncated matrix operations. We then propose and characterize a technique that uses the time-reversibility of a quantum circuit to measure the quality of a subset of qubits, and we apply this technique to assign logical circuits to qubits on a physical device in a nearly optimal manner using a simulated annealing optimization algorithm.