arXiv subjects

Claude Gravel

Publications and source records attributed to Claude Gravel.

15 recordsLinked to original sources

arXiv2026

Oblivious Self-Distance Symmetric Rendezvous on the Integer Line

Symmetric rendezvous on the line is a search problem in which two agents, initially placed at distance $2d$, must follow the same randomized strategy to meet as quickly as possible. In the standard model, agents may condition their actions on the entire execution history, and both the known- and unknown-distance variants admit expected rendezvous time $Θ(d)$. We study the role of memory by introducing oblivious self-distance strategies, in which an agent's decision depends only on her position relative to her own starting location. For an initial separation of $2d$, let $R_d$ denote the optimal oblivious expected rendezvous time in the known-distance setting. We develop two finite-state frameworks based on absorbing Markov chains. Truncated chains give computable upper bounds through finite-support strategies, while weak-peek chains give lower bounds through a revealed-information relaxation. Together, they provide a mechanism for certifying optimality. Using that mechanism, we determine $R_1$ exactly and prove that it is attained by a finite-support strategy. For $d=2,\ldots,6$, numerical optimization gives the same truncation structure and objective values, yielding rigorous upper bounds below $7.83d^2$. We do not prove that the computed weak-peek minimizers are global, but the stability of the computations leads us to conjecture that they are, in which case the corresponding truncated strategies are optimal. We also prove that $R_d=Θ(d^2)$. In the unknown-distance setting, we construct a universal strategy, independent of $d$, with expected rendezvous time $O(d^{2+η})$ for every fixed $η>0$. Thus, under the memory restriction, the known-distance rendezvous time becomes quadratic, while near-quadratic performance remains possible even without knowing $d$. The asymptotic analysis uses birth-death Markov chains and their electrical-network interpretation.

cs.DM↗

arXiv2026

Algorithms, Complexity, and Entropy of the Bernard-Letac Fair-Sampling Construction

Bernard and Letac (1971) introduced a method for uniform random sampling among m outcomes from an unknown biased source of independent and identically distributed symbols. The process terminates when the multinomial coefficient of the cumulative symbol counts equals zero modulo m. This study extends the computational and information-theoretic analysis of their construction by presenting five algorithms with formal correctness guarantees and comprehensive complexity analyses. For prime m = p, the Bernard-Letac framework is analyzed in greater detail. The Rényi entropies of the source yield an exact product formula for the expected number of draws. A first-order approximation consistently overestimates this value, and the entropy lower bound is never attained. As p approaches 1, the expected cost converges to a constant greater than 1, determined by the entire source distribution. Furthermore, a seven-state automaton computes the mod-2 first-passage kernel of the binary walk, reducing the fair assignment cost from quadratic to nearly linear.

cs.IT↗

arXiv2026

Improving Zero-Noise Extrapolation via Physically Bounded Models

Zero-noise extrapolation (ZNE) mitigates errors in near-term quantum devices by extrapolating measurements obtained at amplified noise levels to estimate noise-free expectation values. In practice, commonly used extrapolation models are fitted without enforcing physical constraints, which can yield predictions outside the valid range of quantum observables. In this work, we introduce physically bounded variants of polynomial, exponential, and polynomial--exponential extrapolation models by explicitly parameterizing the zero-noise estimate and constraining it during optimization. We evaluate the approach using a large synthetic benchmark comprising 180,000 circuits and approximately 3.6 million ZNE experiments generated under realistic device noise models derived from IBM quantum backends. We also perform preliminary validation on real quantum hardware using GHZ and W-state circuits. Across the synthetic benchmark, bounded extrapolation substantially reduces unphysical predictions and improves the stability of exponential- and polynomial--exponential-family models, whereas polynomial models show little difference between bounded and unbounded variants. Hardware experiments show similar qualitative behaviour: bounded models generally avoid pathological extrapolations and often provide a more reliable balance between accuracy and usable coverage. At the same time, the results highlight practical limitations of current devices, including stronger-than-expected noise effects and variability not fully captured by simulation models. These results suggest that enforcing physical constraints during extrapolation improves the reliability of ZNE and that this approach can be incorporated into existing workflows with minimal modification.

quant-ph↗

arXiv2026

Software Testing in the Quantum World

Quantum computing offers significant speedups for simulating physical, chemical, and biological systems, and for optimization and machine learning. As quantum software grows in complexity, the classical simulation of quantum computers, which has long been essential for quality assurance, becomes infeasible. This shift requires new quality-assurance methods that operate directly on real quantum computers. This paper presents the key challenges in testing large-scale quantum software and offers software engineering perspectives for addressing them.

cs.SE↗

arXiv2023

Evaluating the generalized Buchshtab function and revisiting the variance of the distribution of the smallest components of combinatorial objects

Let $n\geq 1$ and $X_{n}$ be the random variable representing the size of the smallest component of a random combinatorial object made of $n$ elements. A combinatorial object could be a permutation, a monic polynomial over a finite field, a surjective map, a graph, and so on. By a random combinatorial object, we mean a combinatorial object that is chosen uniformly at random among all possible combinatorial objects of size $n$. It is understood that a component of a permutation is a cycle, an irreducible factor for a monic polynomial, a connected component for a graph, etc. Combinatorial objects are categorized into parametric classes. In this article, we focus on the exp-log class with parameter $K=1$ (permutations, derangements, polynomials over finite field, etc.) and $K=1/2$ (surjective maps, $2$-regular graphs, etc.) The generalized Buchstab function $Ω_{K}$ plays an important role in evaluating probabilistic and statistical quantities. For $K=1$, Theorem $5$ from \cite{PanRic_2001_small_explog} stipulates that $\mathrm{Var}(X_{n})=C(n+O(n^{-ε}))$ for some $ε>0$ and sufficiently large $n$. We revisit the evaluation of $C=1.3070\ldots$ using different methods: analytic estimation using tools from complex analysis, numerical integration using Taylor expansions, and computation of the exact distributions for $n\leq 4000$ using the recursive nature of the counting problem. In general for any $K$, Theorem $1.1$ from \cite{BenMasPanRic_2003} connects the quantity $1/Ω_{K}(x)$ for $x\geq 1$ with the asymptotic proportion of $n$-objects with large smallest components. We show how the coefficients of the Taylor expansion of $Ω_{K}(x)$ for $\lfloor x\rfloor \leq x < \lfloor x\rfloor+1$ depends on those for $\lfloor x\rfloor-1 \leq x-1 < \lfloor x\rfloor$. We use this family of coefficients to evaluate $Ω_{K}(x)$.

math.CO↗

arXiv2021

Feedback linearly extended discrete functions

We study a new flexible method to extend linearly the graph of a non-linear, and usually not bijective, function so that the resulting extension is a bijection. Our motivation comes from cryptography. Examples from symmetric cryptography are given as how the extension was used implicitly in the construction of some well-known block ciphers. The method heavily relies on ideas brought from linear coding theory and secret sharing. We are interested in the behaviour of the composition of many extensions, and especially the space of parameters that defines a family of equations based on finite differences or linear forms. For any linear extension, we characterize entirely the space of parameters for which such equations are solvable in terms of the space of parameters that render those equations for the corresponding non-linear extended functions solvable. Conditions are derived to assess the solvability of those kind of equations in terms of the number of compositions or iterations. We prove a relation between the number of compositions and the dimensions of vector spaces that appear in our results. The proofs of those properties rely mostly on tools from linear algebra.

cs.CR↗

arXiv2021

A generalization of the Von Neumann extractor

An iterative randomness extraction algorithm which generalized the Von Neumann's extraction algorithm is detailed, analyzed and implemented in standard C++. Given a sequence of independently and identically distributed biased Bernoulli random variables, to extract randomness from the aforementioned sequence pertains to produce a new sequence of independently and identically distributed unbiased Bernoulli random variables. The iterative construction here is inspired from the work of Stout and Warren 1984 who modified appropriately the tree of probabilities produced by recursively repeating the Von Neumann's extraction algorithm. The correctness of the iterative algorithm is proven. The number of biased Bernoulli random variables needed to produce one unbiased instance is the complexity of interest. The complexity depends on the bias of the source. The expected complexity converges toward 3.10220648... when the bias tends to 0 and diverges when the bias tends to 1/2. In addition to the expected complexity, some other results that concern the limiting asymptotic construction, and that seem unnoticed in the literature so far, are proven.

cs.IT↗

arXiv2020

Random variate generation using only finitely many unbiased, independently and identically distributed random bits

For any discrete probability distributions with bounded entropy, we can generate exactly a random variate using only a finite expected number of perfect coin flips. A perfect coin flip is the outcome of an unbiased Bernoulli random variable. Coin flips are unbiased, independently and identically distributed in all our work. We survey well-known algorithms for the discrete case such as the one from Knuth and Yao as well as the one from Han and Hoshi. We also discuss briefly about a practical implementation for the algorithm proposed by Knuth and Yao. For the continuous case, only approximations can be hoped for. The freedom to choose the accuracy for the approximations matters, and, for that, we propose to measure accuracy in terms of the Wasserstein $L_\infty$-metric. We derive a universal lower bound for the expected number of perfect coin flips required to reach a desired accuracy. We also provide several algorithms for absolutely continuous distributions that come within our universal lower bound.

cs.IT↗

arXiv2020

Resource reduction for distributed quantum information processing using quantum multiplexed photons

Distributed quantum information processing is based on the transmission of quantum data over lossy channels between quantum processing nodes. These nodes may be separated by a few microns or on planetary scale distances, but transmission losses due to absorption/scattering in the channel are the major source of error for most distributed quantum information tasks. Of course quantum error detection (QED) /correction (QEC) techniques can be used to mitigate such effects but error detection approaches have severe performance limitations due to the signaling constraints between nodes and so error correction approaches are preferable -assuming one has sufficient high quality local operations. Typically, performance comparisons between loss-mitigating codes assume one encoded qubit per photon. However single photons can carry more than one qubit of information and so our focus in this work is to explore whether loss-based QEC codes utilizing quantum multiplexed photons are viable and advantageous, especially as photon loss results in more than one qubit of information being lost. We show that quantum multiplexing enables significant resource reduction: in terms of the number of single photon sources while at the same time maintaining (or even lowering) the number of two-qubit gates required. Further, our multiplexing approach requires only conventional optical gates already necessary for the implementation of these codes.

quant-ph↗

arXiv2020

Finding linearly generated subsequences

We develop a new algorithm to compute determinants of all possible Hankel matrices made up from a given finite length sequence over a finite field. Our algorithm fits within the dynamic programming paradigm by exploiting new recursive relations on the determinants of Hankel matrices together with new observations concerning the distribution of zero determinants among the possible matrix sizes allowed by the length of the original sequence. The algorithm can be used to isolate \emph{very} efficiently linear shift feedback registers hidden in strings with random prefix and random postfix for instance and, therefore, recovering the shortest generating vector. Our new mathematical identities can be used also in any other situations involving determinants of Hankel matrices. We also implement a parallel version of our algorithm. We compare our results empirically with the trivial algorithm which consists of computing determinants for each possible Hankel matrices made up from a given finite length sequence. Our new accelerated approach on a single processor is faster than the trivial algorithm on 160 processors for input sequences of length 16384 for instance.

cs.CR↗

arXiv2019

Unicyclic Strong Permutations

In this paper, we study some properties of a certain kind of permutation $σ$ over $\mathbb{F}_{2}^{n}$, where $n$ is a positive integer. The desired properties for $σ$ are: (1) the algebraic degree of each component function is $n-1$; (2) the permutation is unicyclic; (3) the number of terms of the algebraic normal form of each component is at least $2^{n-1}$. We call permutations that satisfy these three properties simultaneously unicyclic strong permutations. We prove that our permutations $σ$ always have high algebraic degree and that the average number of terms of each component function tends to $2^{n-1}$. We also give a condition on the cycle structure of $σ$. We observe empirically that for $n$ even, our construction does not provide unicylic permutations. For $n$ odd, $n \leq 11$, we conduct an exhaustive search of all $σ$ given our construction for specific examples of unicylic strong permutations. We also present some empirical results on the difference tables and linear approximation tables of $σ$.

cs.CR↗

arXiv2018

Remote Sampling with Applications to General Entanglement Simulation

We show how to sample exactly discrete probability distributions whose defining parameters are distributed among remote parties. For this purpose, von Neumann's rejection algorithm is turned into a distributed sampling communication protocol. We study the expected number of bits communicated among the parties and also exhibit a trade-off between the number of rounds of the rejection algorithm and the number of bits transmitted in the initial phase. Finally, we apply remote sampling to the simulation of quantum entanglement in its most general form possible, when an arbitrary number of parties share systems of arbitrary dimensions on which they apply arbitrary measurements (not restricted to being projective measurements). In case the dimension of the systems and the number of possible outcomes per party is bounded by a constant, it suffices to communicate an expected O(m^2) bits in order to simulate exactly the outcomes that these measurements would have produced on those systems, where m is the number of participants.

quant-ph↗

arXiv2016

The expected bit complexity of the von Neumann rejection algorithm

In 1952, von Neumann introduced the rejection method for random variate generation. We revisit this algorithm when we have a source of perfect bits at our disposal. In this random bit model, there are universal lower bounds for generating a random variate with a given density to within an accuracy $ε$ derived by Knuth and Yao, and refined by the authors. In general, von Neumann's method fails in this model. We propose a modification that insures proper behavior for all Riemann-integrable densities on compact sets, and show that the expected number of random bits needed behaves optimally with respect to universal lower bounds. In particular, we introduce the notion of an oracle that evaluates the supremum and infimum of a function on any rectangle of $\mathbb{R}^{d}$, and develop a quadtree-style extension of the classical rejection method.

cs.IT↗

arXiv2015

Exact simulation of the GHZ distribution

John Bell has shown that the correlations entailed by quantum mechanics cannot be reproduced by a classical process involving non-communicating parties. But can they be simulated with the help of bounded communication? This problem has been studied for more than two decades and it is now well understood in the case of bipartite entanglement. However, the issue was still widely open for multipartite entanglement, even for the simplest case, which is the tripartite Greenberger-Horne-Zeilinger (GHZ) state. We give an exact simulation of arbitrary independent von Neumann measurements on general n-partite GHZ states. Our protocol requires O(n^2) bits of expected communication between the parties, and O(n log n) expected time is sufficient to carry it out in parallel. Furthermore, we need only an expectation of O(n) independent unbiased random bits, with no need for the generation of continuous real random variables nor prior shared random variables. In the case of equatorial measurements, we improve on the prior art with a protocol that needs only O(n log n) bits of communication and O(log^2 n) parallel time. At the cost of a slight increase in the number of bits communicated, these tasks can be accomplished with a constant expected number of rounds.

cs.IT↗

arXiv2012

Spectral geometry over the disk : Weyl's law and nodal sets

In this M.Sc. thesis (Université de Montréal, 2007), we consider problems arising in the study of the spectrum of the Dirichlet Laplacian on a disk as well as on a circular sector. The first part of the thesis is concerned with the location of the nodal line of the second eigenfunction of a sector. In the second part of the thesis we develop an efficient algorithm for ordering the eigenvalues of a disk, and study numerically the growth of the error term in Weyl's law. We also give a detailed proof of a theorem due to Kuznetsov and Fedosov (1965), who obtained a van der Corput type estimate on the remainder. The result of Kuznetsov and Fedosov was rediscovered in 2011 by Y. Colin de Verdière using similar techniques, see http://arxiv.org/pdf/1104.2233v2.pdf .

math.SP↗