Claude Gravel

Andriy Miranskyy, Adam Sorrenti, Jasmine Thind et al.

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.

Claude Gravel, Daniel Panario, Bastien Rigault

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.

Claude Gravel, Daniel Panario

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.

Claude Gravel, Daniel Panario, David Thomson

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 $σ$.