arXiv subjects

Luke Robitaille

Publications and source records attributed to Luke Robitaille.

4 recordsLinked to original sources

arXiv2025

Characterization of permutation gates in the third level of the Clifford hierarchy

The Clifford hierarchy is a fundamental structure in quantum computation whose mathematical properties are not fully understood. In this work, we characterize permutation gates -- unitaries which permute the $2^n$ basis states -- in the third level of the hierarchy. We prove that any permutation gate in the third level must be a product of Toffoli gates in what we define as \emph{staircase form}, up to left and right multiplications by Clifford permutations. We then present necessary and sufficient conditions for a staircase form permutation gate to be in the third level of the Clifford hierarchy. As a corollary, we construct a family of non-semi-Clifford permutation gates $\{U_k\}_{k\geq 3}$ in staircase form such that each $U_k$ is in the third level but its inverse is not in the $k$-th level.

quant-ph↗

arXiv2024

Permutation gates in the third level of the Clifford hierarchy

The Clifford hierarchy is a fundamental structure in quantum computation, classifying unitary operators based on their commutation relations with the Pauli group. Despite its significance, the mathematical structure of the hierarchy is not well understood at the third level and higher. In this work, we study permutations in the hierarchy: gates which permute the $2^n$ basis states. We fully characterize all the semi-Clifford permutation gates. Moreover, we prove that any permutation gate in the third level, not necessarily semi-Clifford, must be a product of Toffoli gates in what we define as staircase form, up to left and right multiplications of Clifford permutations. Finally, we show that the smallest number of qubits for which there exists a non-semi-Clifford permutation in the third level is $7$.

quant-ph↗

arXiv2023

Simple Braids Tend toward Positive Entropy

A simple braid is a positive braid that can be drawn so that any two strands cross at most once. We prove that as $n \to \infty$, the proportion of simple braids on $n$ strands that have positive topological entropy tends toward $100\%$. Notably, such braids are either pseudo-Anosov or reducible with a pseudo-Anosov component. Our proof involves a method of reduction from simple braids to non-simple $3$-strand braids that may be of independent interest.

math.GT↗

arXiv2021

Mixed tensor products and Capelli-type determinants

In this paper we study properties of a homomorphism $\rho$ from the universal enveloping algebra $U=U(\mathfrak{gl}(n+1))$ to a tensor product of an algebra $\mathcal D'(n)$ of differential operators and $U(\mathfrak{gl}(n))$. We find a formula for the image of the Capelli determinant of $\mathfrak{gl}(n+1)$ under $\rho$, and, in particular, of the images under $\rho$ of the Gelfand generators of the center $Z(\mathfrak{gl}(n+1))$ of $U$. This formula is proven by relating $\rho$ to the corresponding Harish-Chandra isomorphisms, and, alternatively, by using a purely computational approach. Furthermore, we define a homomorphism from $\mathcal D'(n) \otimes U(\mathfrak{gl}(n))$ to an algebra containing $U$ as a subalgebra, so that $\sigma (\rho (u)) - u \in G_1 U$, for all $u \in U$, where $G_1 = \sum_{i=0}^{n} E_{ii}$.

math.RT↗