P. A. M. Casares | Searcharxiv

arXiv2020

A quantum active learning algorithm for sampling against adversarial attacks

Adversarial attacks represent a serious menace for learning algorithms and may compromise the security of future autonomous systems. A theorem by Khoury and Hadfield-Menell (KH), provides sufficient conditions to guarantee the robustness of machine learning algorithms, but comes with a caveat: it is crucial to know the smallest distance among the classes of the corresponding classification problem. We propose a theoretical framework that allows us to think of active learning as sampling the most promising new points to be classified, so that the minimum distance between classes can be found and the theorem KH used. Additionally, we introduce a quantum active learning algorithm that makes use of such framework and whose complexity is polylogarithmic in the dimension of the space, $m$, and the size of the initial training data $n$, provided the use of qRAMs; and polynomial in the precision, achieving an exponential speedup over the equivalent classical algorithm in $n$ and $m$. This algorithm may be nevertheless `dequantized' reducing the advantage to polynomial.

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arXiv2020

A Quantum Interior-Point Predictor-Corrector Algorithm for Linear Programming

We introduce a new quantum optimization algorithm for dense Linear Programming problems, which can be seen as the quantization of the Interior Point Predictor-Corrector algorithm \cite{Predictor-Corrector} using a Quantum Linear System Algorithm \cite{DenseHHL}. The (worst case) work complexity of our method is, up to polylogarithmic factors, $O(L\sqrt{n}(n+m)\overline{||M||_F}\barκ^2ε^{-2})$ for $n$ the number of variables in the cost function, $m$ the number of constraints, $ε^{-1}$ the target precision, $L$ the bit length of the input data, $\overline{||M||_F}$ an upper bound to the Frobenius norm of the linear systems of equations that appear, $||M||_F$, and $\barκ$ an upper bound to the condition number $κ$ of those systems of equations. This represents a quantum speed-up in the number $n$ of variables in the cost function with respect to the comparable classical Interior Point algorithms when the initial matrix of the problem $A$ is dense: if we substitute the quantum part of the algorithm by classical algorithms such as Conjugate Gradient Descent, that would mean the whole algorithm has complexity $O(L\sqrt{n}(n+m)^2\barκ \log(ε^{-1}))$, or with exact methods, at least $O(L\sqrt{n}(n+m)^{2.373})$. Also, in contrast with any Quantum Linear System Algorithm, the algorithm described in this article outputs a classical description of the solution vector, and the value of the optimal solution.

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