Andrea Rocchetto

Carlo Ciliberto, Andrea Rocchetto, Alessandro Rudi et al.

Within the framework of statistical learning theory it is possible to bound the minimum number of samples required by a learner to reach a target accuracy. We show that if the bound on the accuracy is taken into account, quantum machine learning algorithms for supervised learning---for which statistical guarantees are available---cannot achieve polylogarithmic runtimes in the input dimension. We conclude that, when no further assumptions on the problem are made, quantum machine learning algorithms for supervised learning can have at most polynomial speedups over efficient classical algorithms, even in cases where quantum access to the data is naturally available.

Varun Kanade, Andrea Rocchetto, Simone Severini

We show that DNF formulae can be quantum PAC-learned in polynomial time under product distributions using a quantum example oracle. The best classical algorithm (without access to membership queries) runs in superpolynomial time. Our result extends the work by Bshouty and Jackson (1998) that proved that DNF formulae are efficiently learnable under the uniform distribution using a quantum example oracle. Our proof is based on a new quantum algorithm that efficiently samples the coefficients of a μ-biased Fourier transform.

Andrea Rocchetto, Scott Aaronson, Simone Severini et al.

The number of parameters describing a quantum state is well known to grow exponentially with the number of particles. This scaling clearly limits our ability to do tomography to systems with no more than a few qubits and has been used to argue against the universal validity of quantum mechanics itself. However, from a computational learning theory perspective, it can be shown that, in a probabilistic setting, quantum states can be approximately learned using only a linear number of measurements. Here we experimentally demonstrate this linear scaling in optical systems with up to 6 qubits. Our results highlight the power of computational learning theory to investigate quantum information, provide the first experimental demonstration that quantum states can be "probably approximately learned" with access to a number of copies of the state that scales linearly with the number of qubits, and pave the way to probing quantum states at new, larger scales.

Carlo Ciliberto, Mark Herbster, Alessandro Davide Ialongo et al.

Recently, increased computational power and data availability, as well as algorithmic advances, have led machine learning techniques to impressive results in regression, classification, data-generation and reinforcement learning tasks. Despite these successes, the proximity to the physical limits of chip fabrication alongside the increasing size of datasets are motivating a growing number of researchers to explore the possibility of harnessing the power of quantum computation to speed-up classical machine learning algorithms. Here we review the literature in quantum machine learning and discuss perspectives for a mixed readership of classical machine learning and quantum computation experts. Particular emphasis will be placed on clarifying the limitations of quantum algorithms, how they compare with their best classical counterparts and why quantum resources are expected to provide advantages for learning problems. Learning in the presence of noise and certain computationally hard problems in machine learning are identified as promising directions for the field. Practical questions, like how to upload classical data into quantum form, will also be addressed.

Andrea Rocchetto

The exponential scaling of the wave function is a fundamental property of quantum systems with far reaching implications in our ability to process quantum information. A problem where these are particularly relevant is quantum state tomography. State tomography, whose objective is to obtain a full description of a quantum system, can be analysed in the framework of computational learning theory. In this model, quantum states have been shown to be Probably Approximately Correct (PAC)-learnable with sample complexity linear in the number of qubits. However, it is conjectured that in general quantum states require an exponential amount of computation to be learned. Here, using results from the literature on the efficient classical simulation of quantum systems, we show that stabiliser states are efficiently PAC-learnable. Our results solve an open problem formulated by Aaronson [Proc. R. Soc. A, 2088, (2007)] and propose learning theory as a tool for exploring the power of quantum computation.