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Sublinear quantum algorithms for training linear and kernelbased classifiers
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Quantum Attacks without Superposition Queries: the Offline Simon's Algorithm
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Quantum algorithms for matrix scaling and matrix balancing
Matrix scaling and matrix balancing are two basic linearalgebraic problems with a wide variety of applications, such as approximating the permanent, and preconditioning linear systems to make them more numerically stable. We study the power and limitations of quantum algorithms for these problems. We provide quantum implementations of two classical (in both senses of the word) methods: Sinkhorn's algorithm for matrix scaling and Osborne's algorithm for matrix balancing. Using amplitude estimation as our main tool, our quantum implementations both run in time Õ(√(mn)/ε^4) for scaling or balancing an n × n matrix (given by an oracle) with m nonzero entries to within ℓ_1error ε. Their classical analogs use time Õ(m/ε^2), and every classical algorithm for scaling or balancing with small constant ε requires Ω(m) queries to the entries of the input matrix. We thus achieve a polynomial speedup in terms of n, at the expense of a worse polynomial dependence on the obtained ℓ_1error ε. We emphasize that even for constant ε these problems are already nontrivial (and relevant in applications). Along the way, we extend the classical analysis of Sinkhorn's and Osborne's algorithm to allow for errors in the computation of marginals. We also adapt an improved analysis of Sinkhorn's algorithm for entrywisepositive matrices to the ℓ_1setting, leading to an Õ(n^1.5/ε^3)time quantum algorithm for εℓ_1scaling in this case. We also prove a lower bound, showing that our quantum algorithm for matrix scaling is essentially optimal for constant ε: every quantum algorithm for matrix scaling that achieves a constant ℓ_1error with respect to uniform marginals needs to make at least Ω(√(mn)) queries.
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