Jérémie Roland | Université libre de Bruxelles (original) (raw)

Papers by Jérémie Roland

Symposium on Theoretical Aspects of Computer Science, 2013

arXiv (Cornell University), Sep 12, 2012

The polynomial method and the adversary method are the two main techniques to prove lower bounds ... more The polynomial method and the adversary method are the two main techniques to prove lower bounds on quantum query complexity, and they have so far been considered as unrelated approaches. Here, we show an explicit reduction from the polynomial method to the multiplicative adversary method. The proof goes by extending the polynomial method from Boolean functions to quantum state generation problems. In the process, the bound is even strengthened. We then show that this extended polynomial method is a special case of the multiplicative adversary method with an adversary matrix that is independent of the function. This new result therefore provides insight on the reason why in some cases the adversary method is stronger than the polynomial method. It also reveals a clear picture of the relation between the different lower bound techniques, as it implies that all known techniques reduce to the multiplicative adversary method.

Le développement de la Théorie du Calcul Quantique provient de l'idée qu'un ordinateur est avant ... more Le développement de la Théorie du Calcul Quantique provient de l'idée qu'un ordinateur est avant tout un système physique, de sorte que ce sont les lois de la Nature elles-mêmes qui constituent une limite ultime sur ce qui peutêtre calculé ou non. L'intérêt pour cette discipline fut stimulé par la découverte par Peter Shor d'un algorithme quantique rapide pour factoriser un nombre, alors qu'actuellement un tel algorithme n'est pas connu en Théorie du Calcul Classique. Un autre résultat important fut la construction par Lov Grover d'un algorithme capable de retrouver unélément dans une base de donnée non-structurée avec un gain de complexité quadratique par rapportà tout algorithme classique. Alors que ces algorithmes quantiques sont exprimés dans le modèle "standard" du Calcul Quantique, où le registreévolue de manière discrète dans le temps sous l'application successive de portes quantiques, un nouveau type d'algorithme aété récemment introduit, où le registré evolue continûment dans le temps sous l'action d'un Hamiltonien. Ainsi, l'idéeà la base du Calcul Quantique Adiabatique, proposée par Edward Farhi et ses collaborateurs, est d'utiliser un outil traditionnel de la Mécanique Quantique,à savoir le Théorème Adiabatique, pour concevoir des algorithmes quantiques où le registreévolue sous l'influence d'un Hamiltonien variant très lentement, assurant uneévolution adiabatique du système. Dans cette thèse, nous montrons tout d'abord comment reproduire le gain quadratique de l'algorithme de Grover au moyen d'un algorithme quantique adiabatique. Ensuite, nous montrons qu'il est possible de traduire ce nouvel algorithme adiabatique, ainsi qu'un autre algorithme de rechercheàévolution Hamiltonienne, dans le formalisme des circuits quantiques, de sorte que l'on obtient ainsi trois algorithmes quantiques de recherche très proches dans leur principe. Par la suite, nous utilisons ces résultats pour construire un algorithme adiabatique pour résoudre des problèmes avec structure, utilisant une technique, dite de "nesting", développée auparavant dans le cadre d'algorithmes quantiques de type circuit. Enfin, nous analysons la résistance au bruit de ces algorithmes adiabatiques, en introduisant un modèle de bruit utilisant la théorie des matrices aléatoires et enétudiant son effet par la théorie des perturbations. i ii Remerciements Je tiensà exprimer ma profonde gratitude pour mon promoteur Nicolas Cerf, tout d'abord pour m'avoir fait découvrir ce domaine de recherche passionnant qu'est la Théorie de l'Information Quantique, mais aussi pour tous ses conseils, ses idées et son support sans lesquels cette thèse n'aurait pasété possible. Je le remercieégalement pour l'excellent environnement de travail qu'il a bâti au fil des années dans son service, rassemblant uneéquipe de chercheurs aussi sympathique que dynamique. Je remercieégalement Serge Massar pour ses idées souvent fructueuses, ce fut un réel plaisir de collaborer avec lui ainsi qu'avec Stefano Pironio sur les Inégalités de Bell. Je voudraiségalement remercier tous mes collègues,à commencer par Louis Lamoureux pour m'avoir changé les idées tous les jours avec son inégalable "humour canadien" et Sofyan Iblisdir avec qui j'espère encore avoir le plaisir de travailler, ne fut-ce que pour terminer cet article mis en veille depuis plus d'un an, sans oublier tous les autres, Gilles

ACM Transactions on Computation Theory, Dec 9, 2016

Does the information complexity of a function equal its communication complexity? We examine whet... more Does the information complexity of a function equal its communication complexity? We examine whether any currently known techniques might be used to show a separation between the two notions. Ganor et al. recently provided such a separation in the distributional case for a specific input distribution. We show that in the non-distributional setting, the relative discrepancy bound is smaller than the information complexity, hence it cannot separate information and communication complexity. In addition, in the distributional case, we provide a linear program formulation for relative discrepancy and relate it to variants of the partition bound, resolving also an open question regarding the relation of the partition bound and information complexity. Last, we prove the equivalence between the adaptive relative discrepancy and the public-coin partition, implying that the logarithm of the adaptive relative discrepancy bound is quadratically tight with respect to communication.

We show that quantum query complexity satisfies a strong direct product theorem. This means that ... more We show that quantum query complexity satisfies a strong direct product theorem. This means that computing k copies of a function with less than k times the quantum queries needed to compute one copy of the function implies that the overall success probability will be exponentially small in k. For a boolean function f we also show an XOR lemma-computing the parity of k copies of f with less than k times the queries needed for one copy implies that the advantage over random guessing will be exponentially small. We do this by showing that the multiplicative adversary method, which inherently satisfies a strong direct product theorem, is always at least as large as the additive adversary method, which is known to characterize quantum query complexity.

arXiv (Cornell University), Nov 29, 2019

Weak coin ipping (WCF) is a fundamental cryptographic primitive for two-party secure computation,... more Weak coin ipping (WCF) is a fundamental cryptographic primitive for two-party secure computation, where two distrustful parties need to remotely establish a shared random bit whilst having opposite preferred outcomes. It is the strongest known primitive with arbitrarily close to perfect security quantumly while classically, its security is completely compromised (unless one makes further assumptions, such as computational hardness). A WCF protocol is said to have bias ϵ if neither party can force their preferred outcome with probability greater than 1/2 + ϵ. Classical WCF protocols are shown to have bias 1/2, i.e., a cheating party can always force their preferred outcome. On the other hand, there exist quantum WCF protocols with arbitrarily small bias, as Mochon showed in his seminal work in 2007 [arXiv:0711.4114]. In particular, he proved the existence of a family of WCF protocols approaching bias ϵ(k) = 1/(4k + 2) for arbitrarily large k and proposed a protocol with bias 1/6. Last year, Arora, Roland and Weis presented a protocol with bias 1/10 and to go below this bias, they designed an algorithm that numerically constructs unitary matrices corresponding to WCF protocols with arbitrarily small bias [STOC'19, p.205-216]. In this work, we present new techniques which yield a fully analytical construction of WCF protocols with bias arbitrarily close to zero, thus achieving a solution that has been missing for more than a decade. Furthermore, our new techniques lead to a simpli ed proof of existence of WCF protocols by circumventing the non-constructive part of Mochon's proof. As an example, we illustrate the construction of a WCF protocol with bias 1/14.

arXiv (Cornell University), Mar 16, 2022

Certifying individual quantum devices with minimal assumptions is crucial for the development of ... more Certifying individual quantum devices with minimal assumptions is crucial for the development of quantum technologies. Here, we investigate how to leverage single-system contextuality to realize self-testing. We develop a robust self-testing protocol based on the simplest contextuality witness for the simplest contextual quantum system, the Klyachko-Can-Binicioğlu-Shumovsky (KCBS) inequality for the qutrit. We establish a lower bound on the fidelity of the state and the measurements (to an ideal configuration) as a function of the value of the witness under a pragmatic assumption on the measurements we call the KCBS orthogonality condition. We apply the method in an experiment with randomly chosen measurements on a single trapped 40 Ca + and near-perfect detection efficiency. The observed statistics allow us to self-test the system and provide the first experimental demonstration of quantum self-testing of a single system. Further, we quantify and report that deviations from our assumptions are minimal, an aspect previously overlooked by contextuality experiments.

Physical Review Letters, Oct 14, 2022

Continuous-time quantum walks provide a natural framework to tackle the fundamental problem of fi... more Continuous-time quantum walks provide a natural framework to tackle the fundamental problem of finding a node among a set of marked nodes in a graph, known as spatial search. Whether spatial search by continuous-time quantum walk provides a quadratic advantage over classical random walks has been an outstanding problem. Thus far, this advantage is obtained only for specific graphs or when a single node of the underlying graph is marked. In this article, we provide a new continuous-time quantum walk search algorithm that completely resolves this: our algorithm can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks. The overall algorithm is quite simple, requiring time evolution of the quantum walk Hamiltonian followed by a projective measurement. A key component of our algorithm is a purely analog procedure to perform operations on a state of the form e −tH 2 |ψ , for a given Hamiltonian H: it only requires evolving H for time scaling as √ t. This allows us to quadratically fast-forward the dynamics of a continuous-time classical random walk. The applications of our work thus go beyond the realm of quantum walks and can lead to new analog quantum algorithms for preparing ground states of Hamiltonians or solving optimization problems.

arXiv (Cornell University), Mar 21, 2018

We propose a new method for designing quantum search algorithms for finding a "marked" element in... more We propose a new method for designing quantum search algorithms for finding a "marked" element in the state space of a classical Markov chain. The algorithm is based on a quantum walkà la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.

arXiv (Cornell University), Feb 18, 2019

We analyze the quantum query complexity of sorting under partial information. In this problem, we... more We analyze the quantum query complexity of sorting under partial information. In this problem, we are given a partially ordered set P and are asked to identify a linear extension of P using pairwise comparisons. For the standard sorting problem, in which P is empty, it is known that the quantum query complexity is not asymptotically smaller than the classical information-theoretic lower bound. We prove that this holds for a wide class of partially ordered sets, thereby improving on a result from Yao (STOC'04).

Technical Report 2011-TR080info:eu-repo/semantics/publishe

Physical Review Letters

Quantum Information & Computation, Jul 1, 2011

We study a model of communication complexity that encompasses many well-studied problems, includi... more We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite measurements of shared quantum states, and XOR games. In this model, Alice gets an input x, Bob gets an input y, and their goal is to each produce an output a, b distributed according to some pre-specified joint distribution p(a, b|x, y). Our results apply to any non-signaling distribution, that is, those where Alice's marginal distribution does not depend on Bob's input, and vice versa, therefore our techniques apply to any communication problem that can be reduced to a non-signaling distribution, including quantum distributions, Boolean and non-Boolean functions, most relations, partial (promise) problems, in the two-player and multipartite settings. We give elementary proofs and very intuitive interpretations of the recent lower bounds of Linial and Shraibman, which we generalize to the problem of simulating any non-signaling distribution. The lower bounds we obtain are also expressed as linear programs (or SDPs for quantum communication). We show that the dual formulations have a striking interpretation, since they coincide with maximum violations of Bell and Tsirelson inequalities. The dual expressions are closely related to the winning probability of XOR games. We show that as in the case of Boolean functions, the gap between the quantum and classical lower bounds is at most linear in size of the support of the distribution, and does not depend on the size of the inputs. This tranlates into a bound on the gap between maximal Bell and Tsirelson inequalities, which was previously known only for the case of Boolean outcomes with uniform marginals. Finally, we give an exponential upper bound on quantum and classical communication complexity in the simultaneous messages model, for any non-signaling distribution. One consequence of this is a simple proof that any quantum distribution can be approximated with a constant number of bits of communication.

We show that almost all known lower bound methods for communication complexity are also lower bou... more We show that almost all known lower bound methods for communication complexity are also lower bounds for the information complexity. In particular, we define a relaxed version of the partition bound of Jain and Klauck [JK10] and prove that it lower bounds the information complexity of any function. Our relaxed partition bound subsumes all norm based methods (e.g. the γ 2 method) and rectangle-based methods (e.g. the rectangle/corruption bound, the smooth rectangle bound, and the discrepancy bound), except the partition bound. Our result uses a new connection between rectangles and zero-communication protocols where the players can either output a value or abort. We prove the following compression lemma: given a protocol for a function f with information complexity I, one can construct a zero-communication protocol that has non-abort probability at least 2 −O(I) and that computes f correctly with high probability conditioned on not aborting. Then, we show how such a zero-communication protocol relates to the relaxed partition bound. We use our main theorem to resolve three of the open questions raised by Braverman [Bra12]. First, we show that the information complexity of the Vector in Subspace Problem [KR11] is Ω(n 1/3), which, in turn, implies that there exists an exponential separation between quantum communication complexity and classical information complexity. Moreover, we provide an Ω(n) lower bound on the information complexity of the Gap Hamming Distance Problem.

Self-testing allows for characterising quantum systems under minimal assumptions. However, existi... more Self-testing allows for characterising quantum systems under minimal assumptions. However, existing schemes rely on quantum non-locality and cannot be applied to systems that are not entangled. Here, we introduce a robust method that achieves self-testing of individual systems by taking advantage of contextuality. The scheme is based on the simplest contextuality witness for the simplest contextual quantum system—the Klyachko-Can-Binicioglu-Shumovsky inequality for the qutrit. We establish a lower bound on the fidelity of the state and the measurements as a function of the value of the witness under a pragmatic assumption on the measurements. We apply the method in an experiment on a single trapped 40Ca+ and using randomly chosen measurements and perfect detection efficiency. Using the observed statistics, we obtain the first experimental demonstration of self-testing of a single quantum system with negligible deviations from the assumptions.

Rejection sampling is a well-known method to sample from a target distribution, given the ability... more Rejection sampling is a well-known method to sample from a target distribution, given the ability to sample from a given distribution. The method has been first formalized by von Neumann (1951) and has many applications in clas-sical computing. We define a quantum analogue of rejection sampling: given a black box producing a coherent superpo-sition of (possibly unknown) quantum states with some am-plitudes, the problem is to prepare a coherent superposition of the same states, albeit with different target amplitudes. The main result of this paper is a tight characterization of the query complexity of this quantum state generation prob-lem. We exhibit an algorithm, which we call quantum rejec-tion sampling, and analyze its cost using semidefinite pro-gramming. Our proof of a matching lower bound is based on the automorphism principle which allows to symmetrize any algorithm over the automorphism group of the prob-lem. Our main technical innovation is an extension of the automorphism ...

Physical Review A, 2020

Spatial search by a discrete-time quantum walk can find a marked node on any ergodic, reversible ... more Spatial search by a discrete-time quantum walk can find a marked node on any ergodic, reversible Markov chain P quadratically faster than its classical counterpart, i.e., in a time that is in the square root of the hitting time of P. However, in the framework of continuous-time quantum walks, it was previously unknown whether such general speedup is possible. In fact, in this framework, the widely used quantum algorithm by Childs and Goldstone fails to achieve such a speedup. Furthermore, it is not clear how to apply this algorithm for searching any Markov chain P. In this article we aim to reconcile the apparent differences between the running times of spatial search algorithms in these two frameworks. We first present a modified version of the Childs and Goldstone algorithm which can search for a marked element for any ergodic, reversible P by performing a quantum walk on its edges. Although this approach improves the algorithmic running time for several instances, it cannot provide a generic quadratic speedup for any P. Second, using the framework of interpolated Markov chains, we provide a spatial search algorithm by a continuous-time quantum walk which can find a marked node on any P in the square root of the classical hitting time. In the scenario where multiple nodes are marked, the algorithmic running time scales as the square root of a quantity known as the extended hitting time. Our results establish a connection between discrete-time and continuous-time quantum walks and can be used to develop a number of Markov chain-based quantum algorithms.

It is known that quantum correlations exhibited by a maximally entangled qubit pair can be simula... more It is known that quantum correlations exhibited by a maximally entangled qubit pair can be simulated with the help of shared randomness, supplemented with additional resources, such as communication, post-selection or non-local boxes. For instance, in the case of projective measurements, it is possible to solve this problem with protocols using one bit of communication or making one use of a non-local box. We show that this problem reduces to a distributed sampling problem. We give a new method to obtain samples from a biased distribution, starting with shared random variables following a uniform distribution, and use it to build distributed sampling protocols. This approach allows us to derive, in a simpler and unified way, many existing protocols for projective measurements, and extend them to positive operator value measurements. Moreover, this approach naturally leads to a local hidden variable model for Werner states.