Quantum and approximation algorithms for maximum witnesses of Boolean matrix products

The problem of finding maximum (or minimum) witnesses of the Boolean product of two Boolean matrices (MW for short) has a number of important applications, in particular the all-pairs lowest common ancestor (LCA) problem in directed acyclic graphs (dags). The best known upper time-bound on the MW problem for n\times n Boolean matrices of the form O(n^{2.575}) has not been substantially improved since 2006. In order to obtain faster algorithms for this problem, we study quantum algorithms for MW and approximation algorithms for MW (in the standard computational model). Some of our quantum algorithms are input or output sensitive. Our fastest quantum algorithm for the MW problem, and consequently for the related problems, runs in time \tilde{O}(n^{2+\lambda/2})=\tilde{O}(n^{2.434}), where \lambda satisfies the equation \omega(1, \lambda, 1) = 1 + 1.5 \, \lambda and \omega(1, \lambda, 1) is the exponent of the multiplication of an n \times n^{\lambda}$ matrix by an n^{\lambda} \times n matrix. Next, we consider a relaxed version of the MW problem (in the standard model) asking for reporting a witness of bounded rank (the maximum witness has rank 1) for each non-zero entry of the matrix product. First, by adapting the fastest known algorithm for maximum witnesses, we obtain an algorithm for the relaxed problem that reports for each non-zero entry of the product matrix a witness of rank at most \ell in time \tilde{O}((n/\ell)n^{\omega(1,\log_n \ell,1)}). Then, by reducing the relaxed problem to the so called k-witness problem, we provide an algorithm that reports for each non-zero entry C[i,j] of the product matrix C a witness of rank O(\lceil W_C(i,j)/k\rceil ), where W_C(i,j) is the number of witnesses for C[i,j], with high probability. The algorithm runs in \tilde{O}(n^{\omega}k^{0.4653} +n^2k) time, where \omega=\omega(1,1,1).Comment: 14 pages, 3 figure

Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision

The quantum query complexity of Boolean matrix multiplication is typically studied as a function of the matrix dimension, n, as well as the number of 1s in the output, \ell. We prove an upper bound of O (n\sqrt{\ell}) for all values of \ell. This is an improvement over previous algorithms for all values of \ell. On the other hand, we show that for any \eps < 1 and any \ell <= \eps n^2, there is an \Omega(n\sqrt{\ell}) lower bound for this problem, showing that our algorithm is essentially tight. We first reduce Boolean matrix multiplication to several instances of graph collision. We then provide an algorithm that takes advantage of the fact that the underlying graph in all of our instances is very dense to find all graph collisions efficiently

Quantum Algorithms for Matrix Products over Semirings

In this paper we construct quantum algorithms for matrix products over several algebraic structures called semirings, including the (max,min)-matrix product, the distance matrix product and the Boolean matrix product. In particular, we obtain the following results. We construct a quantum algorithm computing the product of two n x n matrices over the (max,min) semiring with time complexity O(n^{2.473}). In comparison, the best known classical algorithm for the same problem, by Duan and Pettie, has complexity O(n^{2.687}). As an application, we obtain a O(n^{2.473})-time quantum algorithm for computing the all-pairs bottleneck paths of a graph with n vertices, while classically the best upper bound for this task is O(n^{2.687}), again by Duan and Pettie. We construct a quantum algorithm computing the L most significant bits of each entry of the distance product of two n x n matrices in time O(2^{0.64L} n^{2.46}). In comparison, prior to the present work, the best known classical algorithm for the same problem, by Vassilevska and Williams and Yuster, had complexity O(2^{L}n^{2.69}). Our techniques lead to further improvements for classical algorithms as well, reducing the classical complexity to O(2^{0.96L}n^{2.69}), which gives a sublinear dependency on 2^L. The above two algorithms are the first quantum algorithms that perform better than the $\tilde O(n^{5/2})$-time straightforward quantum algorithm based on quantum search for matrix multiplication over these semirings. We also consider the Boolean semiring, and construct a quantum algorithm computing the product of two n x n Boolean matrices that outperforms the best known classical algorithms for sparse matrices. For instance, if the input matrices have O(n^{1.686...}) non-zero entries, then our algorithm has time complexity O(n^{2.277}), while the best classical algorithm has complexity O(n^{2.373}).Comment: 19 page

Computing the Boolean product of two n\times n Boolean matrices using O(n^2) mechanical operation

We study the problem of determining the Boolean product of two n\times n Boolean matrices in an unconventional computational model allowing for mechanical operations. We show that O(n^2) operations are sufficient to compute the product in this model.Comment: 11 pages, 7 figure

Synthesis and Optimization of Reversible Circuits - A Survey

Reversible logic circuits have been historically motivated by theoretical research in low-power electronics as well as practical improvement of bit-manipulation transforms in cryptography and computer graphics. Recently, reversible circuits have attracted interest as components of quantum algorithms, as well as in photonic and nano-computing technologies where some switching devices offer no signal gain. Research in generating reversible logic distinguishes between circuit synthesis, post-synthesis optimization, and technology mapping. In this survey, we review algorithmic paradigms --- search-based, cycle-based, transformation-based, and BDD-based --- as well as specific algorithms for reversible synthesis, both exact and heuristic. We conclude the survey by outlining key open challenges in synthesis of reversible and quantum logic, as well as most common misconceptions.Comment: 34 pages, 15 figures, 2 table

Quantum Communication Complexity of Distributed Set Joins

Computing set joins of two inputs is a common task in database theory. Recently, Van Gucht, Williams, Woodruff and Zhang [PODS 2015] considered the complexity of such problems in the natural model of (classical) two-party communication complexity and obtained tight bounds for the complexity of several important distributed set joins. In this paper we initiate the study of the quantum communication complexity of distributed set joins. We design a quantum protocol for distributed Boolean matrix multiplication, which corresponds to computing the composition join of two databases, showing that the product of two n times n Boolean matrices, each owned by one of two respective parties, can be computed with widetilde-O(sqrt{n} ell^{3/4}) qubits of communication, where ell denotes the number of non-zero entries of the product. Since Van Gucht et al. showed that the classical communication complexity of this problem is widetilde-Theta(n sqrt{ell}), our quantum algorithm outperforms classical protocols whenever the output matrix is sparse. We also show a quantum lower bound and a matching classical upper bound on the communication complexity of distributed matrix multiplication over F_2. Besides their applications to database theory, the communication complexity of set joins is interesting due to its connections to direct product theorems in communication complexity. In this work we also introduce a notion of all-pairs product theorem, and relate this notion to standard direct product theorems in communication complexity

On Nondeterministic Derandomization of Freivalds\u27 Algorithm: Consequences, Avenues and Algorithmic Progress

Motivated by studying the power of randomness, certifying algorithms and barriers for fine-grained reductions, we investigate the question whether the multiplication of two n x n matrices can be performed in near-optimal nondeterministic time O~(n^2). Since a classic algorithm due to Freivalds verifies correctness of matrix products probabilistically in time O(n^2), our question is a relaxation of the open problem of derandomizing Freivalds\u27 algorithm. We discuss consequences of a positive or negative resolution of this problem and provide potential avenues towards resolving it. Particularly, we show that sufficiently fast deterministic verifiers for 3SUM or univariate polynomial identity testing yield faster deterministic verifiers for matrix multiplication. Furthermore, we present the partial algorithmic progress that distinguishing whether an integer matrix product is correct or contains between 1 and n erroneous entries can be performed in time O~(n^2) - interestingly, the difficult case of deterministic matrix product verification is not a problem of "finding a needle in the haystack", but rather cancellation effects in the presence of many errors. Our main technical contribution is a deterministic algorithm that corrects an integer matrix product containing at most t errors in time O~(sqrt{t} n^2 + t^2). To obtain this result, we show how to compute an integer matrix product with at most t nonzeroes in the same running time. This improves upon known deterministic output-sensitive integer matrix multiplication algorithms for t = Omega(n^{2/3}) nonzeroes, which is of independent interest

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