Information-theoretic bounds on quantum advantage in machine learning

Abstract: We compare the complexity of training classical and quantum machine learning (ML) models for predicting outcomes of physical experiments. The experiments depend on an input parameter x and involve the execution of a (possibly unknown) quantum process $E$. Our figure of merit is the number of runs of $E$ needed during training, disregarding other measures of complexity. A classical ML performs a measurement and records the classical outcome after each run of $E$, while a quantum ML can access $E$ coherently to acquire quantum data; the classical or quantum data is then used to predict outcomes of future experiments. We prove that, for any input distribution $D(x)$, a classical ML can provide accurate predictions on average by accessing $E$ a number of times comparable to the optimal quantum ML. In contrast, for achieving accurate prediction on all inputs, we show that exponential quantum advantage exists in certain tasks. For example, to predict expectation values of all Pauli observables in an $n-$qubit system, we present a quantum ML using only $O(n)$ data and prove that a classical ML requires $2^{\Omega(n)}$ data.

data structures and algorithmsmachine learningmathematical physicsinformation theoryoptimization and controldata analysis, statistics and probability

Audience: researchers in the topic

Mathematics, Physics and Machine Learning (IST, Lisbon)

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