arXiv subjects

Elia Gorokhovsky

Publications and source records attributed to Elia Gorokhovsky.

5 recordsLinked to original sources

arXiv2026

4-rank distribution of Picard groups of hyperelliptic curves via $C$-symmetric matrices

We determine the large-genus limiting distribution of the 4-rank of the Picard group of hyperelliptic curves over a fixed finite field $\mathbb F_q$ of odd characteristic. This is a function field analogue of a result of Fouvry and Klüners. Our computation agrees with (the Picard group analogue of) the Cohen--Lenstra--Gerth heuristics in the case $q \equiv 3\pmod{4}$, i.e., in the absence of roots of unity in the base field. When roots of unity are present, the result is of the same form as conjectured distribution for class groups of quadratic extensions of number fields containing roots of unity. The limiting distribution does not change when imposing finitely many conditions on the ramification behavior of the curves. In the process, we determine the rank distribution of a certain class of random matrix ensembles over finite fields determined by symmetry conditions.

math.NT↗

arXiv2026

Good Stabilizer Codes from Shallow Clifford Circuits with Random Matchings

Encoding quantum information with low circuit overhead is a fundamental challenge in fault-tolerant quantum computation. Random circuits provide a natural mechanism for rapidly spreading logical information through simple gates applied in parallel. Brown and Fawzi showed that random Clifford circuits on two-qubit Clifford gates provide such encoders that achieve the quantum Gilbert-Varshamov rate-distance tradeoff with depth $O(\log^3 n)$. We show that the same asymptotic tradeoff is attained in optimal $O(\log n)$ depth under a gate distribution with a more restricted support. For every fixed $δ>0$ and sufficiently large $n$, if $\frac kn < 1 - H(\frac{d}{n}) - \frac{d}{n}\log_2 3 - δ$, we can construct random circuits of depth $O(\log n)$ which define, with high probability, an $[n,k]$ stabilizer code of distance at least $d+1$, which matches the $Ω(\log n)$ light-cone lower bound for linear distance encoders. Our ensemble employs a random matching circuit architecture consisting of $T$ independent permutation-invariant layers. In each layer, the qubits are paired up by a uniformly random perfect matching, and a random independent two-qubit Clifford gate is applied to each pair. The gate distribution need not be uniform over, or even have full support on, the two-qubit Clifford group; rather, we allow for very general distributions on Clifford gates satisfying three regularity conditions. In particular, the construction can be implemented using $n/2$ CNOT gates on randomly matched pairs in each layer, with parallel one-qubit Clifford twirls. These regularity conditions allow us to reduce the second-moment dynamics of our random circuits to a reversible Markov chain on binary support strings. We establish logarithmic hitting-time bounds for this Markov chain and comparisons of its stationary distribution to prove the coding properties of the circuits.

quant-ph↗

arXiv2026

Time-inhomogeneous random walks on finite groups and cokernels of random integer block matrices

We study time-inhomogeneous random walks on finite groups in the case where each random walk step need not be supported on a generating set of the group. When the supports of the random walk steps satisfy a natural condition involving normal subgroups of quotients of the group, we show that the random walk converges to the uniform distribution on the group and give bounds for the convergence rate using spectral properties of the random walk steps. As an application, we use the moment method of Wood to prove a universality theorem for cokernels of random integer matrices allowing some dependence between entries.

math.PR↗

arXiv2026

Nearly-symmetric matrices in the Cohen-Lenstra universality class

In this paper, we study cokernels of random $n\times n$ matrices over $\mathbb Z$ with symmetry conditions determined by fixed alternating bilinear forms on $\mathbb Z^n$. These include perturbations of random symmetric matrices at a very small (but unbounded with $n$) number of entries. We show that, subject to fairly weak conditions on the distributions of the entries, the distribution of these cokernels converges weakly to the Cohen-Lenstra distribution, which is the limiting distribution of cokernels of random matrices with no symmetry constraints. This result demonstrates that the cokernel distributions of symmetric matrices are quite sensitive to small perturbations of the symmetry conditions.

math.PR↗

arXiv2022

A quantitative Neumann lemma for finitely generated groups

We study the coset covering function $\mathfrak{C}(r)$ of a finitely generated group: the number of cosets of infinite index subgroups needed to cover the ball of radius $r$. We show that $\mathfrak{C}(r)$ is of order at least $\sqrt{r}$ for all groups. Moreover, we show that $\mathfrak{C}(r)$ is linear for a class of amenable groups including virtually nilpotent and polycyclic groups, and that it is exponential for property (T) groups.

math.GR↗