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Liming Zhao, Aman Agrawal, Patrick Rebentrost

Restricted Boltzmann Machines (RBMs) are widely used probabilistic undirected graphical models with visible and latent nodes, playing an important role in statistics and machine learning. The task of structure learning for RBMs involves inferring the underlying graph by using samples from the visible nodes. Specifically, learning the two-hop neighbors of each visible node allows for the inference of the graph structure. Prior research has addressed the structure learning problem for specific classes of RBMs, namely ferromagnetic and locally consistent RBMs. In this paper, we extend the scope to the quantum computing domain and propose corresponding quantum algorithms for this problem. Our study demonstrates that the proposed quantum algorithms yield a polynomial speedup compared to the classical algorithms for learning the structure of these two classes of RBMs.

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Liming Zhao, Naixu Guo, Ming-Xing Luo, Patrick Rebentrost

The complete learning of an $n$-qubit quantum state requires samples exponentially in $n$. Several works consider subclasses of quantum states that can be learned in polynomial sample complexity such as stabilizer states or high-temperature Gibbs states. Other works consider a weaker sense of learning, such as PAC learning and shadow tomography. In this work, we consider learning states that are close to neural network quantum states, which can efficiently be represented by a graphical model called restricted Boltzmann machines (RBMs). To this end, we exhibit robustness results for efficient provable two-hop neighborhood learning algorithms for ferromagnetic and locally consistent RBMs. We consider the $L_p$-norm as a measure of closeness, including both total variation distance and max-norm distance in the limit. Our results allow certain quantum states to be learned with a sample complexity \textit{exponentially} better than naive tomography. We hence provide new classes of efficiently learnable quantum states and apply new strategies to learn them.

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Po-Wei Huang, Patrick Rebentrost

Quantum computing has the potential to provide substantial computational advantages over current state-of-the-art classical supercomputers. However, current hardware is not advanced enough to execute fault-tolerant quantum algorithms. An alternative of using hybrid quantum-classical computing with variational algorithms can exhibit barren plateau issues, causing slow convergence of gradient-based optimization techniques. In this paper, we discuss "post-variational strategies", which shift tunable parameters from the quantum computer to the classical computer, opting for ensemble strategies when optimizing quantum models. We discuss various strategies and design principles for constructing individual quantum circuits, where the resulting ensembles can be optimized with convex programming. Further, we discuss architectural designs of post-variational quantum neural networks and analyze the propagation of estimation errors throughout such neural networks. Lastly, we show that our algorithm can be applied to real-world applications such as image classification on handwritten digits, producing a 96% classification accuracy.

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Yihui Quek, Clement Canonne, Patrick Rebentrost

We consider the problem of finding the minimum element in a list of length $N$ using a noisy comparator. The noise is modelled as follows: given two elements to compare, if the values of the elements differ by at least $\alpha$ by some metric defined on the elements, then the comparison will be made correctly; if the values of the elements are closer than $\alpha$, the outcome of the comparison is not subject to any guarantees. We demonstrate a quantum algorithm for noisy quantum minimum-finding that preserves the quadratic speedup of the noiseless case: our algorithm runs in time $\tilde O(\sqrt{N (1+\Delta)})$, where $\Delta$ is an upper-bound on the number of elements within the interval $\alpha$, and outputs a good approximation of the true minimum with high probability. Our noisy comparator model is motivated by the problem of hypothesis selection, where given a set of $N$ known candidate probability distributions and samples from an unknown target distribution, one seeks to output some candidate distribution $O(\varepsilon)$-close to the unknown target. Much work on the classical front has been devoted to speeding up the run time of classical hypothesis selection from $O(N^2)$ to $O(N)$, in part by using statistical primitives such as the Scheff\'{e} test. Assuming a quantum oracle generalization of the classical data access and applying our noisy quantum minimum-finding algorithm, we take this run time into the sublinear regime. The final expected run time is $\tilde O( \sqrt{N(1+\Delta)})$, with the same $O(\log N)$ sample complexity from the unknown distribution as the classical algorithm. We expect robust quantum minimum-finding to be a useful building block for algorithms in situations where the comparator (which may be another quantum or classical algorithm) is resolution-limited or subject to some uncertainty.

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Yassine Hamoudi, Patrick Rebentrost, Ansis Rosmanis, Miklos Santha

Submodular functions are set functions mapping every subset of some ground set of size $n$ into the real numbers and satisfying the diminishing returns property. Submodular minimization is an important field in discrete optimization theory due to its relevance for various branches of mathematics, computer science and economics. The currently fastest strongly polynomial algorithm for exact minimization [LSW15] runs in time $\widetilde{O}(n^3 \cdot \mathrm{EO} + n^4)$ where $\mathrm{EO}$ denotes the cost to evaluate the function on any set. For functions with range $[-1,1]$, the best $\epsilon$-additive approximation algorithm [CLSW17] runs in time $\widetilde{O}(n^{5/3}/\epsilon^{2} \cdot \mathrm{EO})$. In this paper we present a classical and a quantum algorithm for approximate submodular minimization. Our classical result improves on the algorithm of [CLSW17] and runs in time $\widetilde{O}(n^{3/2}/\epsilon^2 \cdot \mathrm{EO})$. Our quantum algorithm is, up to our knowledge, the first attempt to use quantum computing for submodular optimization. The algorithm runs in time $\widetilde{O}(n^{5/4}/\epsilon^{5/2} \cdot \log(1/\epsilon) \cdot \mathrm{EO})$. The main ingredient of the quantum result is a new method for sampling with high probability $T$ independent elements from any discrete probability distribution of support size $n$ in time $O(\sqrt{Tn})$. Previous quantum algorithms for this problem were of complexity $O(T\sqrt{n})$.

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Zhikuan Zhao, Alejandro Pozas-Kerstjens, Patrick Rebentrost, Peter Wittek

Bayesian methods in machine learning, such as Gaussian processes, have great advantages compared to other techniques. In particular, they provide estimates of the uncertainty associated with a prediction. Extending the Bayesian approach to deep architectures has remained a major challenge. Recent results connected deep feedforward neural networks with Gaussian processes, allowing training without backpropagation. This connection enables us to leverage a quantum algorithm designed for Gaussian processes and develop a new algorithm for Bayesian deep learning on quantum computers. The properties of the kernel matrix in the Gaussian process ensure the efficient execution of the core component of the protocol, quantum matrix inversion, providing an at least polynomial speedup over the classical algorithm. Furthermore, we demonstrate the execution of the algorithm on contemporary quantum computers and analyze its robustness with respect to realistic noise models.

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Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, Seth Lloyd

Fuelled by increasing computer power and algorithmic advances, machine learning techniques have become powerful tools for finding patterns in data. Since quantum systems produce counter-intuitive patterns believed not to be efficiently produced by classical systems, it is reasonable to postulate that quantum computers may outperform classical computers on machine learning tasks. The field of quantum machine learning explores how to devise and implement concrete quantum software that offers such advantages. Recent work has made clear that the hardware and software challenges are still considerable but has also opened paths towards solutions.

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Zhikuan Zhao, Vedran Dunjko, Jack K. Fitzsimons, Patrick Rebentrost, Joseph F. Fitzsimons

The intersection between the fields of machine learning and quantum information processing is proving to be a fruitful field for the discovery of new quantum algorithms, which potentially offer an exponential speed-up over their classical counterparts. However, many such algorithms require the ability to produce states proportional to vectors stored in quantum memory. Even given access to quantum databases which store exponentially long vectors, the construction of which is considered a one-off overhead, it has been argued that the cost of preparing such amplitude-encoded states may offset any exponential quantum advantage. Here we argue that specifically in the context of machine learning applications it suffices to prepare a state close to the ideal state only in the $\infty$-norm, and that this can be achieved with only a constant number of memory queries.

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Patrick Rebentrost, Masoud Mohseni, Seth Lloyd

Supervised machine learning is the classification of new data based on already classified training examples. In this work, we show that the support vector machine, an optimized binary classifier, can be implemented on a quantum computer, with complexity logarithmic in the size of the vectors and the number of training examples. In cases when classical sampling algorithms require polynomial time, an exponential speed-up is obtained. At the core of this quantum big data algorithm is a non-sparse matrix exponentiation technique for efficiently performing a matrix inversion of the training data inner-product (kernel) matrix.

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