arXiv subjects

François Le Gall

Publications and source records attributed to François Le Gall.

At least 73 records · Page 4Linked to original sources

arXiv2019

Average-Case Quantum Advantage with Shallow Circuits

Recently Bravyi, Gosset and König (Science 2018) proved an unconditional separation between the computational powers of small-depth quantum and classical circuits for a relation. In this paper we show a similar separation in the average-case setting that gives stronger evidence of the superiority of small-depth quantum computation: we construct a computational task that can be solved on all inputs by a quantum circuit of constant depth with bounded-fanin gates (a "shallow" quantum circuit) and show that any classical circuit with bounded-fanin gates solving this problem on a non-negligible fraction of the inputs must have logarithmic depth. Our results are obtained by introducing a technique to create quantum states exhibiting global quantum correlations from any graph, via a construction that we call the \emph{extended graph}. Similar results have been very recently (and independently) obtained by Coudron, Stark and Vidick (arXiv:1810.04233), and Bene Watts, Kothari, Schaeffer and Tal (STOC 2019).

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arXiv2018

Quantum Advantage for the LOCAL Model in Distributed Computing

There are two central models considered in (fault-free synchronous) distributed computing: the CONGEST model, in which communication channels have limited bandwidth, and the LOCAL model, in which communication channels have unlimited bandwidth. Very recently, Le Gall and Magniez (PODC 2018) showed the superiority of quantum distributed computing over classical distributed computing in the CONGEST model. In this work we show the superiority of quantum distributed computing in the LOCAL model: we exhibit a computational task that can be solved in a constant number of rounds in the quantum setting but requires $Ω(n)$ rounds in the classical setting, where $n$ denotes the size of the network.

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arXiv2018

Interactive Proofs with Polynomial-Time Quantum Prover for Computing the Order of Solvable Groups

In this paper we consider what can be computed by a user interacting with a potentially malicious server, when the server performs polynomial-time quantum computation but the user can only perform polynomial-time classical (i.e., non-quantum) computation. Understanding the computational power of this model, which corresponds to polynomial-time quantum computation that can be efficiently verified classically, is a well-known open problem in quantum computing. Our result shows that computing the order of a solvable group, which is one of the most general problems for which quantum computing exhibits an exponential speed-up with respect to classical computing, can be realized in this model.

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arXiv2017

Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor

In the past few years, successive improvements of the asymptotic complexity of square matrix multiplication have been obtained by developing novel methods to analyze the powers of the Coppersmith-Winograd tensor, a basic construction introduced thirty years ago. In this paper we show how to generalize this approach to make progress on the complexity of rectangular matrix multiplication as well, by developing a framework to analyze powers of tensors in an asymmetric way. By applying this methodology to the fourth power of the Coppersmith-Winograd tensor, we succeed in improving the complexity of rectangular matrix multiplication. Let $α$ denote the maximum value such that the product of an $n\times n^α$ matrix by an $n^α\times n$ matrix can be computed with $O(n^{2+ε})$ arithmetic operations for any $ε>0$. By analyzing the fourth power of the Coppersmith-Winograd tensor using our methods, we obtain the new lower bound $α>0.31389$, which improves the previous lower bound $α>0.30298$ obtained five years ago by Le Gall (FOCS'12) from the analysis of the second power of the Coppersmith-Winograd tensor. More generally, we give faster algorithms computing the product of an $n\times n^k$ matrix by an $n^k\times n$ matrix for any value $k\neq 1$. (In the case $k=1$, we recover the bounds recently obtained for square matrix multiplication). These improvements immediately lead to improvements in the complexity of a multitude of fundamental problems for which the bottleneck is rectangular matrix multiplication, such as computing the all-pair shortest paths in directed graphs with bounded weights.

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arXiv2017

Triangle Finding and Listing in CONGEST Networks

Triangle-free graphs play a central role in graph theory, and triangle detection (or triangle finding) as well as triangle enumeration (triangle listing) play central roles in the field of graph algorithms. In distributed computing, algorithms with sublinear round complexity for triangle finding and listing have recently been developed in the powerful CONGEST clique model, where communication is allowed between any two nodes of the network. In this paper we present the first algorithms with sublinear complexity for triangle finding and triangle listing in the standard CONGEST model, where the communication topology is the same as the topology of the network. More precisely, we give randomized algorithms for triangle finding and listing with round complexity $O(n^{2/3}(\log n)^{2/3})$ and $O(n^{3/4}\log n)$, respectively, where $n$ denotes the number of nodes of the network. We also show a lower bound $Ω(n^{1/3}/\log n)$ on the round complexity of triangle listing, which also holds for the CONGEST clique model.

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arXiv2016

On the Group and Color Isomorphism Problems

In this paper, we prove results on the relationship between the complexity of the group and color isomorphism problems. The difficulty of color isomorphism problems is known to be closely linked to the the composition factors of the permutation group involved. Previous works are primarily concerned with applying color isomorphism to bou nded degree graph isomorphism, and have therefore focused on the alternating composit ion factors, since those are the bottleneck in the case of graph isomorphism. We consider the color isomorphism problem with composition factors restricted to those other than the alternating group, show that group isomorphism reduces in n^(O(log log n)) time to this problem, and, conversely, that a special case of this color isomorphism problem reduces to a slight generalization of group isomorphism. We then sharpen our results by identifying the projective special linear group as the main obstacle to faster algorithms for group isomorphism and prove that the aforementioned reduc tion from group isomorphism to color isomorphism in fact produces only cyclic and projective special linear factors. Our results demonstrate that, just as the alternatin g group was a barrier to faster algorithms for graph isomorphism for three decades, the projective special linear group is an obstacle to faster algorithms for group isomorphism.

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arXiv2016

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 $\widetildeΘ(n\sqrt{\ell})$, 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 $\mathbb{F}_2$. 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.

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arXiv2016

Further Algebraic Algorithms in the Congested Clique Model and Applications to Graph-Theoretic Problems

Censor-Hillel et al. [PODC'15] recently showed how to efficiently implement centralized algebraic algorithms for matrix multiplication in the congested clique model, a model of distributed computing that has received increasing attention in the past few years. This paper develops further algebraic techniques for designing algorithms in this model. We present deterministic and randomized algorithms, in the congested clique model, for efficiently computing multiple independent instances of matrix products, computing the determinant, the rank and the inverse of a matrix, and solving systems of linear equations. As applications of these techniques, we obtain more efficient algorithms for the computation, again in the congested clique model, of the all-pairs shortest paths and the diameter in directed and undirected graphs with small weights, improving over Censor-Hillel et al.'s work. We also obtain algorithms for several other graph-theoretic problems such as computing the number of edges in a maximum matching and the Gallai-Edmonds decomposition of a simple graph, and computing a minimum vertex cover of a bipartite graph.

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arXiv2016

Solving Laplacian Systems in Logarithmic Space

We investigate the space complexity of solving linear systems of equations. While all known deterministic or randomized algorithms solving a square system of $n$ linear equations in $n$ variables require $Ω(\log^2 n)$ space, Ta-Shma (STOC 2013) recently showed that on a quantum computer an approximate solution can be computed in logarithmic space, giving the first explicit computational task for which quantum computation seems to outperform classical computation with respect to space complexity. In this paper we show that for systems of linear equations in the Laplacian matrix of graphs, the same logarithmic space complexity can actually be achieved by a classical (i.e., non-quantum) algorithm. More precisely, given a system of linear equations $Lx=b$, where $L$ is the (normalized) Laplacian matrix of a graph on $n$ vertices and $b$ is a unit-norm vector, our algorithm outputs a vector $\tilde x$ such that $\left\lVert\tilde x -x\right\rVert\le 1/\mathrm{poly}(n)$ and uses only $O(\log n)$ space if the underlying graph has polynomially bounded weights. We also show how to estimate, again in logarithmic space, the smallest non-zero eigenvalue of $L$.

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arXiv2015

Quantum Algorithm for Triangle Finding in Sparse Graphs

This paper presents a quantum algorithm for triangle finding over sparse graphs that improves over the previous best quantum algorithm for this task by Buhrman et al. [SIAM Journal on Computing, 2005]. Our algorithm is based on the recent $\tilde O(n^{5/4})$-query algorithm given by Le Gall [FOCS 2014] for triangle finding over dense graphs (here $n$ denotes the number of vertices in the graph). We show in particular that triangle finding can be solved with $O(n^{5/4-ε})$ queries for some constant $ε>0$ whenever the graph has at most $O(n^{2-c})$ edges for some constant $c>0$.

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arXiv2014

Fast Matrix Multiplication: Limitations of the Laser Method

Until a few years ago, the fastest known matrix multiplication algorithm, due to Coppersmith and Winograd (1990), ran in time $O(n^{2.3755})$. Recently, a surge of activity by Stothers, Vassilevska-Williams, and Le Gall has led to an improved algorithm running in time $O(n^{2.3729})$. These algorithms are obtained by analyzing higher and higher tensor powers of a certain identity of Coppersmith and Winograd. We show that this exact approach cannot result in an algorithm with running time $O(n^{2.3725})$, and identify a wide class of variants of this approach which cannot result in an algorithm with running time $O(n^{2.3078})$; in particular, this approach cannot prove the conjecture that for every $ε> 0$, two $n\times n$ matrices can be multiplied in time $O(n^{2+ε})$. We describe a new framework extending the original laser method, which is the method underlying the previously mentioned algorithms. Our framework accommodates the algorithms by Coppersmith and Winograd, Stothers, Vassilevska-Williams and Le Gall. We obtain our main result by analyzing this framework. The framework is also the first to explain why taking tensor powers of the Coppersmith-Winograd identity results in faster algorithms.

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arXiv2014

The quantum query complexity of certification

We study the quantum query complexity of finding a certificate for a d-regular, k-level balanced NAND formula. Up to logarithmic factors, we show that the query complexity is Theta(d^{(k+1)/2}) for 0-certificates, and Theta(d^{k/2}) for 1-certificates. In particular, this shows that the zero-error quantum query complexity of evaluating such formulas is O(d^{(k+1)/2}) (again neglecting a logarithmic factor). Our lower bound relies on the fact that the quantum adversary method obeys a direct sum theorem.

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arXiv2014

Privacy in Quantum Communication Complexity

In two-party quantum communication complexity, Alice and Bob receive some classical inputs and wish to compute some function that depends on both these inputs, while minimizing the communication. This model has found numerous applications in many areas of computer science. One question that has received a lot of attention recently is whether it is possible to perform such protocols in a private way. We show that defining privacy for quantum protocols is not so straightforward and it depends on whether we assume that the registers where Alice and Bob receive their classical inputs are in fact classical registers (and hence unentangled with the rest of the protocol) or quantum registers (and hence can be entangled with the rest of the protocol or the environment). We provide new quantum protocols for the Inner Product function and for Private Information Retrieval, and show that the privacy assuming classical input registers can be exponentially smaller than the privacy assuming quantum input registers. We also argue that the right notion of privacy of a communication protocol is the one assuming classical input registers, since otherwise the players can deviate considerably from the protocol.

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arXiv2014

Generalized Quantum Arthur-Merlin Games

This paper investigates the role of interaction and coins in public-coin quantum interactive proof systems (also called quantum Arthur-Merlin games). While prior works focused on classical public coins even in the quantum setting, the present work introduces a generalized version of quantum Arthur-Merlin games where the public coins can be quantum as well: the verifier can send not only random bits, but also halves of EPR pairs. First, it is proved that the class of two-turn quantum Arthur-Merlin games with quantum public coins, denoted qq-QAM in this paper, does not change by adding a constant number of turns of classical interactions prior to the communications of the qq-QAM proof systems. This can be viewed as a quantum analogue of the celebrated collapse theorem for AM due to Babai. To prove this collapse theorem, this paper provides a natural complete problem for qq-QAM: deciding whether the output of a given quantum circuit is close to a totally mixed state. This complete problem is on the very line of the previous studies investigating the hardness of checking the properties related to quantum circuits, and is of independent interest. It is further proved that the class qq-QAM_1 of two-turn quantum-public-coin quantum Arthur-Merlin proof systems with perfect completeness gives new bounds for standard well-studied classes of two-turn interactive proof systems. Finally, the collapse theorem above is extended to comprehensively classify the role of interaction and public coins in quantum Arthur-Merlin games: it is proved that, for any constant m>1, the class of problems having an m-turn quantum Arthur-Merlin proof system is either equal to PSPACE or equal to the class of problems having a two-turn quantum Arthur-Merlin game of a specific type, which provides a complete set of quantum analogues of Babai's collapse theorem.

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arXiv2014

Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments

In this paper we present a quantum algorithm solving the triangle finding problem in unweighted graphs with query complexity $\tilde O(n^{5/4})$, where $n$ denotes the number of vertices in the graph. This improves the previous upper bound $O(n^{9/7})=O(n^{1.285...})$ recently obtained by Lee, Magniez and Santha. Our result shows, for the first time, that in the quantum query complexity setting unweighted triangle finding is easier than its edge-weighted version, since for finding an edge-weighted triangle Belovs and Rosmanis proved that any quantum algorithm requires $Ω(n^{9/7}/\sqrt{\log n})$ queries. Our result also illustrates some limitations of the non-adaptive learning graph approach used to obtain the previous $O(n^{9/7})$ upper bound since, even over unweighted graphs, any quantum algorithm for triangle finding obtained using this approach requires $Ω(n^{9/7}/\sqrt{\log n})$ queries as well. To bypass the obstacles characterized by these lower bounds, our quantum algorithm uses combinatorial ideas exploiting the graph-theoretic properties of triangle finding, which cannot be used when considering edge-weighted graphs or the non-adaptive learning graph approach.

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arXiv2014

Quantum Algorithms for Finding Constant-sized Sub-hypergraphs

We develop a general framework to construct quantum algorithms that detect if a $3$-uniform hypergraph given as input contains a sub-hypergraph isomorphic to a prespecified constant-sized hypergraph. This framework is based on the concept of nested quantum walks recently proposed by Jeffery, Kothari and Magniez [SODA'13], and extends the methodology designed by Lee, Magniez and Santha [SODA'13] for similar problems over graphs. As applications, we obtain a quantum algorithm for finding a $4$-clique in a $3$-uniform hypergraph on $n$ vertices with query complexity $O(n^{1.883})$, and a quantum algorithm for determining if a ternary operator over a set of size $n$ is associative with query complexity $O(n^{2.113})$.

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arXiv2014

Powers of Tensors and Fast Matrix Multiplication

This paper presents a method to analyze the powers of a given trilinear form (a special kind of algebraic constructions also called a tensor) and obtain upper bounds on the asymptotic complexity of matrix multiplication. Compared with existing approaches, this method is based on convex optimization, and thus has polynomial-time complexity. As an application, we use this method to study powers of the construction given by Coppersmith and Winograd [Journal of Symbolic Computation, 1990] and obtain the upper bound $ω<2.3728639$ on the exponent of square matrix multiplication, which slightly improves the best known upper bound.

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arXiv2013

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}).

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