Abstract
We characterize the expressive power of quantum circuits with the pseudo-dimension, a measure of complexity for probabilistic concept classes. We prove pseudo-dimension bounds on the output probability distributions of quantum circuits; the upper bounds are polynomial in circuit depth and number of gates. Using these bounds, we exhibit a class of circuit output states out of which at least one has exponential gate complexity of state preparation, and moreover demonstrate that quantum circuits of known polynomial size and depth are PAC-learnable.
Highlights
An important line of research in classical learning theory is characterizing the expressive power of function classes using complexity measures
In this work we describe a new way of applying complexity measures from classical learning, pseudo-dimension, to quantum information
Aaronson (2007) showed that using the framework of PAC learning, one can introduce a variant of quantum state tomography and prove an upper bound on the required number of copies of the unknown state
Summary
An important line of research in classical learning theory is characterizing the expressive power of function classes using complexity measures. Aaronson (2007) related a variant of state tomography to a classical learning task whose fat-shattering dimension can be bounded using a particular function class. Gate complexity of unitary implementation and state preparation are yet another example of how one may capture the richness of a function class that corresponds to a quantum computational process (see, e.g., Aaronson 2016). We associate with a quantum circuit a natural probabilistic function class describing the outcome probabilities of measurements performed on the circuit output In this way, a function class corresponding to a quantum circuit can be studied with the classical tool of pseudo-dimension. We show that the pseudo-dimension of such a class can be bounded in
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