A Survey of Quantum Learning Theory

This paper surveys quantum learning theory: the theoretical aspects of machine learning using quantum computers. We describe the main results known for three models of learning: exact learning from membership queries, and Probably Approximately Correct (PAC) and agnostic learning from classical or quantum examples.Comment: 26 pages LaTeX. v2: many small changes to improve the presentation. This version will appear as Complexity Theory Column in SIGACT News in June 2017. v3: fixed a small ambiguity in the definition of gamma(C) and updated a referenc

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Tohoku University青木孝文課

Violating Constant Degree Hypothesis Requires Breaking Symmetry

The Constant Degree Hypothesis was introduced by Barrington et. al. (1990) to study some extensions of $q$-groups by nilpotent groups and the power of these groups in a certain computational model. In its simplest formulation, it establishes exponential lower bounds for $\mathrm{AND}_d \circ \mathrm{MOD}_m \circ \mathrm{MOD}_q$ circuits computing AND of unbounded arity $n$ (for constant integers $d,m$ and a prime $q$). While it has been proved in some special cases (including $d=1$), it remains wide open in its general form for over 30 years. In this paper we prove that the hypothesis holds when we restrict our attention to symmetric circuits with $m$ being a prime. While we build upon techniques by Grolmusz and Tardos (2000), we have to prove a new symmetric version of their Degree Decreasing Lemma and apply it in a highly non-trivial way. Moreover, to establish the result we perform a careful analysis of automorphism groups of $\mathrm{AND} \circ \mathrm{MOD}_m$ subcircuits and study the periodic behaviour of the computed functions. Finally, our methods also yield lower bounds when $d$ is treated as a function of $n$

Anytime Algorithms for ROBDD Symmetry Detection and Approximation

Reduced Ordered Binary Decision Diagrams (ROBDDs) provide a dense and memory efficient representation of Boolean functions. When ROBDDs are applied in logic synthesis, the problem arises of detecting both classical and generalised symmetries. State-of-the-art in symmetry detection is represented by Mishchenko's algorithm. Mishchenko showed how to detect symmetries in ROBDDs without the need for checking equivalence of all co-factor pairs. This work resulted in a practical algorithm for detecting all classical symmetries in an ROBDD in O(|G|3) set operations where |G| is the number of nodes in the ROBDD. Mishchenko and his colleagues subsequently extended the algorithm to find generalised symmetries. The extended algorithm retains the same asymptotic complexity for each type of generalised symmetry. Both the classical and generalised symmetry detection algorithms are monolithic in the sense that they only return a meaningful answer when they are left to run to completion. In this thesis we present efficient anytime algorithms for detecting both classical and generalised symmetries, that output pairs of symmetric variables until a prescribed time bound is exceeded. These anytime algorithms are complete in that given sufficient time they are guaranteed to find all symmetric pairs. Theoretically these algorithms reside in O(n3+n|G|+|G|3) and O(n3+n2|G|+|G|3) respectively, where n is the number of variables, so that in practice the advantage of anytime generality is not gained at the expense of efficiency. In fact, the anytime approach requires only very modest data structure support and offers unique opportunities for optimisation so the resulting algorithms are very efficient. The thesis continues by considering another class of anytime algorithms for ROBDDs that is motivated by the dearth of work on approximating ROBDDs. The need for approximation arises because many ROBDD operations result in an ROBDD whose size is quadratic in the size of the inputs. Furthermore, if ROBDDs are used in abstract interpretation, the running time of the analysis is related not only to the complexity of the individual ROBDD operations but also the number of operations applied. The number of operations is, in turn, constrained by the number of times a Boolean function can be weakened before stability is achieved. This thesis proposes a widening that can be used to both constrain the size of an ROBDD and also ensure that the number of times that it is weakened is bounded by some given constant. The widening can be used to either systematically approximate an ROBDD from above (i.e. derive a weaker function) or below (i.e. infer a stronger function). The thesis also considers how randomised techniques may be deployed to improve the speed of computing an approximation by avoiding potentially expensive ROBDD manipulation

Constant-time discrete Gaussian sampling

© 2018 IEEE. Sampling from a discrete Gaussian distribution is an indispensable part of lattice-based cryptography. Several recent works have shown that the timing leakage from a non-constant-time implementation of the discrete Gaussian sampling algorithm could be exploited to recover the secret. In this paper, we propose a constant-time implementation of the Knuth-Yao random walk algorithm for performing constant-time discrete Gaussian sampling. Since the random walk is dictated by a set of input random bits, we can express the generated sample as a function of the input random bits. Hence, our constant-time implementation expresses the unique mapping of the input random-bits to the output sample-bits as a Boolean expression of the random-bits. We use bit-slicing to generate multiple samples in batches and thus increase the throughput of our constant-time sampling manifold. Our experiments on an Intel i7-Broadwell processor show that our method can be as much as 2.4 times faster than the constant-time implementation of cumulative distribution table based sampling and consumes exponentially less memory than the Knuth-Yao algorithm with shuffling for a similar level of security