arXiv subjects

François Le Gall

Publications and source records attributed to François Le Gall.

At least 55 records · Page 3Linked to original sources

arXiv2022

Distributed Quantum Interactive Proofs

The study of distributed interactive proofs was initiated by Kol, Oshman, and Saxena [PODC 2018] as a generalization of distributed decision mechanisms (proof-labeling schemes, etc.), and has received a lot of attention in recent years. In distributed interactive proofs, the nodes of an $n$-node network $G$ can exchange short messages (called certificates) with a powerful prover. The goal is to decide if the input (including $G$ itself) belongs to some language, with as few turns of interaction and as few bits exchanged between nodes and the prover as possible. There are several results showing that the size of certificates can be reduced drastically with a constant number of interactions compared to non-interactive distributed proofs. In this paper, we introduce the quantum counterpart of distributed interactive proofs: certificates can now be quantum bits, and the nodes of the network can perform quantum computation. The first result of this paper shows that by using quantum distributed interactive proofs, the number of interactions can be significantly reduced. More precisely, our result shows that for any constant~$k$, the class of languages that can be decided by a $k$-turn classical (i.e., non-quantum) distributed interactive protocol with $f(n)$-bit certificate size is contained in the class of languages that can be decided by a $5$-turn distributed quantum interactive protocol with $O(f(n))$-bit certificate size. We also show that if we allow to use shared randomness, the number of turns can be reduced to 3-turn. Since no similar turn-reduction \emph{classical} technique is currently known, our result gives evidence of the power of quantum computation in the setting of distributed interactive proofs as well.

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arXiv2022

An optimal oracle separation of classical and quantum hybrid schemes

Recently, Chia, Chung and Lai (STOC 2020) and Coudron and Menda (STOC 2020) have shown that there exists an oracle $\mathcal{O}$ such that $\mathsf{BQP}^\mathcal{O} \neq (\mathsf{BPP^{BQNC}})^\mathcal{O} \cup (\mathsf{BQNC^{BPP}})^\mathcal{O}$. In fact, Chia et al. proved a stronger statement: for any depth parameter $d$, there exists an oracle that separates quantum depth $d$ and $2d+1$, when polynomial-time classical computation is allowed. This implies that relative to an oracle, doubling quantum depth gives classical and quantum hybrid schemes more computational power. In this paper, we show that for any depth parameter $d$, there exists an oracle that separates quantum depth $d$ and $d+1$, when polynomial-time classical computation is allowed. This gives an optimal oracle separation of classical and quantum hybrid schemes. To prove our result, we consider $d$-Bijective Shuffling Simon's Problem (which is a variant of $d$-Shuffling Simon's Problem considered by Chia et al.) and an oracle inspired by an "in-place" permutation oracle.

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arXiv2022

Quantum Meets Fine-grained Complexity: Sublinear Time Quantum Algorithms for String Problems

Longest common substring (LCS), longest palindrome substring (LPS), and Ulam distance (UL) are three fundamental string problems that can be classically solved in near linear time. In this work, we present sublinear time quantum algorithms for these problems along with quantum lower bounds. Our results shed light on a very surprising fact: Although the classic solutions for LCS and LPS are almost identical (via suffix trees), their quantum computational complexities are different. While we give an exact $\tilde O(\sqrt{n})$ time algorithm for LPS, we prove that LCS needs at least time $\tilde Ω(n^{2/3})$ even for 0/1 strings.

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arXiv2022

Quantum Advantage with Shallow Circuits Under Arbitrary Corruption

Recent works by Bravyi, Gosset and König (Science 2018), Bene Watts et al. (STOC 2019), Coudron, Stark and Vidick (QIP 2019) and Le Gall (CCC 2019) have shown unconditional separations between the computational powers of shallow (i.e., small-depth) quantum and classical circuits: quantum circuits can solve in constant depth computational problems that require logarithmic depth to solve with classical circuits. Using quantum error correction, Bravyi, Gosset, König and Tomamichel (Nature Physics 2020) further proved that a similar separation still persists even if quantum circuits are subject to local stochastic noise. In this paper, we consider the case where any constant fraction of the qubits (for instance, huge blocks of qubits) may be arbitrarily corrupted at the end of the computation. We make a first step forward towards establishing a quantum advantage even in this extremely challenging setting: we show that there exists a computational problem that can be solved in constant depth by a quantum circuit but such that even solving any large subproblem of this problem requires logarithmic depth with bounded fan-in classical circuits. This gives another compelling evidence of the computational power of quantum shallow circuits. In order to show our result, we consider the Graph State Sampling problem (which was also used in prior works) on expander graphs. We exploit the "robustness" of expander graphs against vertex corruption to show that a subproblem hard for small-depth classical circuits can still be extracted from the output of the corrupted quantum circuit.

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arXiv2022

Quantum Approximate Counting for Markov Chains and Application to Collision Counting

In this paper we show how to generalize the quantum approximate counting technique developed by Brassard, Høyer and Tapp [ICALP 1998] to a more general setting: estimating the number of marked states of a Markov chain (a Markov chain can be seen as a random walk over a graph with weighted edges). This makes it possible to construct quantum approximate counting algorithms from quantum search algorithms based on the powerful "quantum walk based search" framework established by Magniez, Nayak, Roland and Santha [SIAM Journal on Computing 2011]. As an application, we apply this approach to the quantum element distinctness algorithm by Ambainis [SIAM Journal on Computing 2007]: for two injective functions over a set of $N$ elements, we obtain a quantum algorithm that estimates the number $m$ of collisions of the two functions within relative error $ε$ by making $\tilde{O}\left(\frac{1}{ε^{25/24}}\big(\frac{N}{\sqrt{m}}\big)^{2/3}\right)$ queries, which gives an improvement over the $Θ\big(\frac{1}ε\frac{N}{\sqrt{m}}\big)$-query classical algorithm based on random sampling when $m\ll N$.

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arXiv2022

Quantum Distributed Algorithms for Detection of Cliques

The possibilities offered by quantum computing have drawn attention in the distributed computing community recently, with several breakthrough results showing quantum distributed algorithms that run faster than the fastest known classical counterparts, and even separations between the two models. A prime example is the result by Izumi, Le Gall, and Magniez [STACS 2020], who showed that triangle detection by quantum distributed algorithms is easier than triangle listing, while an analogous result is not known in the classical case. In this paper we present a framework for fast quantum distributed clique detection. This improves upon the state-of-the-art for the triangle case, and is also more general, applying to larger clique sizes. Our main technical contribution is a new approach for detecting cliques by encapsulating this as a search task for nodes that can be added to smaller cliques. To extract the best complexities out of our approach, we develop a framework for nested distributed quantum searches, which employ checking procedures that are quantum themselves. Moreover, we show a circuit-complexity barrier on proving a lower bound of the form $Ω(n^{3/5+ε})$ for $K_p$-detection for any $p \geq 4$, even in the classical (non-quantum) distributed CONGEST setting.

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arXiv2021

Lower Bounds for Induced Cycle Detection in Distributed Computing

The distributed subgraph detection asks, for a fixed graph $H$, whether the $n$-node input graph contains $H$ as a subgraph or not. In the standard CONGEST model of distributed computing, the complexity of clique/cycle detection and listing has received a lot of attention recently. In this paper we consider the induced variant of subgraph detection, where the goal is to decide whether the $n$-node input graph contains $H$ as an \emph{induced} subgraph or not. We first show a $\tildeΩ(n)$ lower bound for detecting the existence of an induced $k$-cycle for any $k\geq 4$ in the CONGEST model. This lower bound is tight for $k=4$, and shows that the induced variant of $k$-cycle detection is much harder than the non-induced version. This lower bound is proved via a reduction from two-party communication complexity. We complement this result by showing that for $5\leq k\leq 7$, this $\tildeΩ(n)$ lower bound cannot be improved via the two-party communication framework. We then show how to prove stronger lower bounds for larger values of $k$. More precisely, we show that detecting an induced $k$-cycle for any $k\geq 8$ requires $\tildeΩ(n^{2-Θ{(1/k)}})$ rounds in the CONGEST model, nearly matching the known upper bound $\tilde{O}(n^{2-Θ{(1/k)}})$ of the general $k$-node subgraph detection (which also applies to the induced version) by Eden, Fiat, Fischer, Kuhn, and Oshman~[DISC 2019]. Finally, we investigate the case where $H$ is the diamond (the diamond is obtained by adding an edge to a 4-cycle, or equivalently removing an edge from a 4-clique), and show non-trivial upper and lower bounds on the complexity of the induced version of diamond detecting and listing.

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arXiv2021

Test of Quantumness with Small-Depth Quantum Circuits

Recently Brakerski, Christiano, Mahadev, Vazirani and Vidick (FOCS 2018) have shown how to construct a test of quantumness based on the learning with errors (LWE) assumption: a test that can be solved efficiently by a quantum computer but cannot be solved by a classical polynomial-time computer under the LWE assumption. This test has lead to several cryptographic applications. In particular, it has been applied to producing certifiable randomness from a single untrusted quantum device, self-testing a single quantum device and device-independent quantum key distribution. In this paper, we show that this test of quantumness, and essentially all the above applications, can actually be implemented by a very weak class of quantum circuits: constant-depth quantum circuits combined with logarithmic-depth classical computation. This reveals novel complexity-theoretic properties of this fundamental test of quantumness and gives new concrete evidence of the superiority of small-depth quantum circuits over classical computation.

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arXiv2021

Quantum Logarithmic Space and Post-selection

Post-selection, the power of discarding all runs of a computation in which an undesirable event occurs, is an influential concept introduced to the field of quantum complexity theory by Aaronson (Proceedings of the Royal Society A, 2005). In the present paper, we initiate the study of post-selection for space-bounded quantum complexity classes. Our main result shows the identity $\sf PostBQL=PL$, i.e., the class of problems that can be solved by a bounded-error (polynomial-time) logarithmic-space quantum algorithm with post-selection ($\sf PostBQL$) is equal to the class of problems that can be solved by unbounded-error logarithmic-space classical algorithms ($\sf PL$). This result gives a space-bounded version of the well-known result $\sf PostBQP=PP$ proved by Aaronson for polynomial-time quantum computation. As a by-product, we also show that $\sf PL$ coincides with the class of problems that can be solved by bounded-error logarithmic-space quantum algorithms that have no time bound.

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arXiv2021

Fast Distributed Algorithms for Girth, Cycles and Small Subgraphs

In this paper we give fast distributed graph algorithms for detecting and listing small subgraphs, and for computing or approximating the girth. Our algorithms improve upon the state of the art by polynomial factors, and for girth, we obtain an constant-time algorithm for additive +1 approximation in the Congested Clique, and the first parametrized algorithm for exact computation in CONGEST. In the Congested Clique, we develop a technique for learning small neighborhoods, and apply it to obtain an $O(1)$-round algorithm that computes the girth with only an additive +1 error. Next, we introduce a new technique (the partition tree technique) allowing for efficiently and deterministically listing all copies of any subgraph, improving upon the state-of the-art for non-dense graphs. We give two applications of this technique: First we show that for constant $k$, $C_{2k}$-detection can be solved in $O(1)$ rounds in the Congested Clique, improving on prior work which used matrix multiplication and had polynomial round complexity. Second, we show that in triangle-free graphs, the girth can be exactly computed in time polynomially faster than the best known bounds for general graphs. In CONGEST, we describe a new approach for finding cycles, and apply it in two ways: first we show a fast parametrized algorithm for girth with round complexity $\tilde{O}(\min(g\cdot n^{1-1/Θ(g)},n))$ for any girth $g$; and second, we show how to find small even-length cycles $C_{2k}$ for $k = 3,4,5$ in $O(n^{1-1/k})$ rounds, which is a polynomial improvement upon the previous running times. Finally, using our improved $C_6$-freeness algorithm and the barrier on proving lower bounds on triangle-freeness of Eden et al., we show that improving the current $\tildeΩ(\sqrt{n})$ lower bound for $C_6$-freeness of Korhonen et al. by any polynomial factor would imply strong circuit complexity lower bounds.

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arXiv2020

Distributed Quantum Proofs for Replicated Data

The paper tackles the issue of $\textit{checking}$ that all copies of a large data set replicated at several nodes of a network are identical. The fact that the replicas may be located at distant nodes prevents the system from verifying their equality locally, i.e., by having each node consult only nodes in its vicinity. On the other hand, it remains possible to assign $\textit{certificates}$ to the nodes, so that verifying the consistency of the replicas can be achieved locally. However, we show that, as the data set is large, classical certification mechanisms, including distributed Merlin-Arthur protocols, cannot guarantee good completeness and soundness simultaneously, unless they use very large certificates. The main result of this paper is a distributed $\textit{quantum}$ Merlin-Arthur protocol enabling the nodes to collectively check the consistency of the replicas, based on small certificates, and in a single round of message exchange between neighbors, with short messages. In particular, the certificate-size is logarithmic in the size of the data set, which gives an exponential advantage over classical certification mechanisms.

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arXiv2020

Tight Distributed Listing of Cliques

Much progress has recently been made in understanding the complexity landscape of subgraph finding problems in the CONGEST model of distributed computing. However, so far, very few tight bounds are known in this area. For triangle (i.e., 3-clique) listing, an optimal $\tilde{O}(n^{1/3})$-round distributed algorithm has been constructed by Chang et al.~[SODA 2019, PODC 2019]. Recent works of Eden et al.~[DISC 2019] and of Censor-Hillel et al.~[PODC 2020] have shown sublinear algorithms for $K_p$-listing, for each $p \geq 4$, but still leaving a significant gap between the upper bounds and the known lower bounds of the problem. In this paper, we completely close this gap. We show that for each $p \geq 4$, there is an $\tilde{O}(n^{1 - 2/p})$-round distributed algorithm that lists all $p$-cliques $K_p$ in the communication network. Our algorithm is \emph{optimal} up to a polylogarithmic factor, due to the $\tildeΩ(n^{1 - 2/p})$-round lower bound of Fischer et al.~[SPAA 2018], which holds even in the CONGESTED CLIQUE model. Together with the triangle-listing algorithm by Chang et al.~[SODA 2019, PODC 2019], our result thus shows that the round complexity of $K_p$-listing, for all $p$, is the same in both the CONGEST and CONGESTED CLIQUE models, at $\tildeΘ(n^{1 - 2/p})$ rounds. For $p=4$, our result additionally matches the $\tildeΩ(n^{1/2})$ lower bound for $K_4$-\emph{detection} by Czumaj and Konrad [DISC 2018], implying that the round complexities for detection and listing of $K_4$ are equivalent in the CONGEST model.

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arXiv2020

Quantum Speedup for the Minimum Steiner Tree Problem

A recent breakthrough by Ambainis, Balodis, Iraids, Kokainis, Prūsis and Vihrovs (SODA'19) showed how to construct faster quantum algorithms for the Traveling Salesman Problem and a few other NP-hard problems by combining in a novel way quantum search with classical dynamic programming. In this paper, we show how to apply this approach to the minimum Steiner tree problem, a well-known NP-hard problem, and construct the first quantum algorithm that solves this problem faster than the best known classical algorithms. More precisely, the complexity of our quantum algorithm is $\mathcal{O}(1.812^k\poly(n))$, where $n$ denotes the number of vertices in the graph and $k$ denotes the number of terminals. In comparison, the best known classical algorithm has complexity $\mathcal{O}(2^k\poly(n))$.

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arXiv2020

On Distributed Listing of Cliques

We show an $\tilde{O}(n^{p/(p+2)})$-round algorithm in the \congest model for \emph{listing} of $K_p$ (a clique with $p$ nodes), for all $p =4, p\geq 6$. For $p = 5$, we show an $\tilde{O}(n^{3/4})$-round algorithm. For $p=4$ and $p=5$, our results improve upon the previous state-of-the-art of $O(n^{5/6+o(1)})$ and $O(n^{21/22+o(1)})$, respectively, by Eden et al. [DISC 2019]. For all $p\geq 6$, ours is the first sub-linear round algorithm for $K_p$ listing. We leverage the recent expander decomposition algorithm of Chang et al. [SODA 2019] to create clusters with a good mixing time. Three key novelties in our algorithm are: (1) we carefully iterate our listing process with coupled values of min-degree within the clusters and arboricity outside the clusters, (2) all the listing is done within the cluster, which necessitates new techniques for bringing into the cluster the information about \emph{all} edges that can potentially form $K_p$ instances with the cluster edges, and (3) within each cluster we use a sparsity-aware listing algorithm, which is faster than a general listing algorithm and which we can allow the cluster to use since we make sure to sparsify the graph as the iterations proceed. As a byproduct of our algorithm, we show an \emph{optimal} sparsity-aware algorithm for $K_p$ listing, which runs in $\tildeΘ(1 + m/n^{1 + 2/p})$ rounds in the \clique model. Previously, Pandurangan et al. [SPAA 2018], Chang et al. [SODA 2019], and Censor-Hillel et al. [TCS 2020] showed sparsity-aware algorithms for the case of $p = 3$, yet ours is the first such sparsity aware algorithm for $p \geq 4$.

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arXiv2020

Quantum-Inspired Classical Algorithms for Singular Value Transformation

A recent breakthrough by Tang (STOC 2019) showed how to "dequantize" the quantum algorithm for recommendation systems by Kerenidis and Prakash (ITCS 2017). The resulting algorithm, classical but "quantum-inspired", efficiently computes a low-rank approximation of the users' preference matrix. Subsequent works have shown how to construct efficient quantum-inspired algorithms for approximating the pseudo-inverse of a low-rank matrix as well, which can be used to (approximately) solve low-rank linear systems of equations. In the present paper, we pursue this line of research and develop quantum-inspired algorithms for a large class of matrix transformations that are defined via the singular value decomposition of the matrix. In particular, we obtain classical algorithms with complexity polynomially related (in most parameters) to the complexity of the best quantum algorithms for singular value transformation recently developed by Chakraborty, Gilyén and Jeffery (ICALP 2019) and Gilyén, Su, Low and Wiebe (STOC19).

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arXiv2020

Quantum Communication Complexity of Distribution Testing

The classical communication complexity of testing closeness of discrete distributions has recently been studied by Andoni, Malkin and Nosatzki (ICALP'19). In this problem, two players each receive $t$ samples from one distribution over $[n]$, and the goal is to decide whether their two distributions are equal, or are $ε$-far apart in the $l_1$-distance. In the present paper we show that the quantum communication complexity of this problem is $\tilde{O}(n/(tε^2))$ qubits when the distributions have low $l_2$-norm, which gives a quadratic improvement over the classical communication complexity obtained by Andoni, Malkin and Nosatzki. We also obtain a matching lower bound by using the pattern matrix method. Let us stress that the samples received by each of the parties are classical, and it is only communication between them that is quantum. Our results thus give one setting where quantum protocols overcome classical protocols for a testing problem with purely classical samples.

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arXiv2020

Quantum Distributed Algorithm for Triangle Finding in the CONGEST Model

This paper considers the triangle finding problem in the CONGEST model of distributed computing. Recent works by Izumi and Le Gall (PODC'17), Chang, Pettie and Zhang (SODA'19) and Chang and Saranurak (PODC'19) have successively reduced the classical round complexity of triangle finding (as well as triangle listing) from the trivial upper bound $O(n)$ to $\tilde O(n^{1/3})$, where~$n$ denotes the number of vertices in the graph. In this paper we present a quantum distributed algorithm that solves the triangle finding problem in $\tilde O(n^{1/4})$ rounds in the CONGEST model. This gives another example of quantum algorithm beating the best known classical algorithms in distributed computing. Our result also exhibits an interesting phenomenon: while in the classical setting the best known upper bounds for the triangle finding and listing problems are identical, in the quantum setting the round complexities of these two problems are now $\tilde O(n^{1/4})$ and $\tilde Θ(n^{1/3})$, respectively. Our result thus shows that triangle finding is easier than triangle listing in the quantum CONGEST model.

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arXiv2019

Quantum Distributed Algorithm for the All-Pairs Shortest Path Problem in the CONGEST-CLIQUE Model

The All-Pairs Shortest Path problem (APSP) is one of the most central problems in distributed computation. In the CONGEST-CLIQUE model, in which $n$ nodes communicate with each other over a fully connected network by exchanging messages of $O(\log n)$ bits in synchronous rounds, the best known general algorithm for APSP uses $\tilde O(n^{1/3})$ rounds. Breaking this barrier is a fundamental challenge in distributed graph algorithms. In this paper we investigate for the first time quantum distributed algorithms in the CONGEST-CLIQUE model, where nodes can exchange messages of $O(\log n)$ quantum bits, and show that this barrier can be broken: we construct a $\tilde O(n^{1/4})$-round quantum distributed algorithm for the APSP over directed graphs with polynomial weights in the CONGEST-CLIQUE model. This speedup in the quantum setting contrasts with the case of the standard CONGEST model, for which Elkin et al. (PODC 2014) showed that quantum communication does not offer significant advantages over classical communication. Our quantum algorithm is based on a relationship discovered by Vassilevska Williams and Williams (JACM 2018) between the APSP and the detection of negative triangles in a graph. The quantum part of our algorithm exploits the framework for quantum distributed search recently developed by Le Gall and Magniez (PODC 2018). Our main technical contribution is a method showing how to implement multiple quantum searches (one for each edge in the graph) in parallel without introducing congestions.

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