Abstract
We quantify the role of scrambling in quantum machine learning. We characterize a quantum neural network’s (QNNs) error in terms of the network’s scrambling properties via the out-of-time-ordered correlator (OTOC). A network can be trained by minimizing a loss function. We show that the loss function can be bounded by the OTOC. We prove that the gradient of the loss function can be bounded by the gradient of the OTOC. This demonstrates that the OTOC landscape regulates the trainability of a QNN. We show numerically that this landscape is flat for maximally scrambling QNNs, which can pose a challenge to training. Our results pave the way for the exploration of quantum chaos in quantum neural networks.
Highlights
A quantum neural network (QNN) [1–6] is a quantum generalization of a classical neural network [7–9] used to learn or optimize functions
We show that when the QNN is maximally scrambling, the of-time-ordered correlator (OTOC) landscape is flat, which can pose a challenge to training
We have shown that training error is bounded by the OTOC, a scrambling measure
Summary
A quantum neural network (QNN) [1–6] is a quantum generalization of a classical neural network [7–9] used to learn or optimize functions. We relate chaos to QNNs by establishing upper and lower bounds on training error in terms of quantum scrambling. QNNs themselves may have chaotic properties which characterize their learning ability These properties have been investigated through scrambling measures such as the tripartite mutual information [28] and operator size [29]. Numerical evidence correlating the tripartite mutual information to the network’s empirical training error has been demonstrated in [28]. We demonstrate that training error can be bounded by the out-of-time-ordered correlator (OTOC), defined . This correlator is an essential tool in the study of chaos, as it can characterize fast scramblers [30–34] and has even been used to decode the Hayden-Preskill protocol [35, 36]. We show that when the QNN is maximally scrambling, the OTOC landscape is flat, which can pose a challenge to training
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