arXiv subjects

Shai Evra

Publications and source records attributed to Shai Evra.

13 recordsLinked to original sources

arXiv2025

Multi-Qubit Golden Gates

Our goal in this paper is to construct optimal topological generators for compact unitary Lie groups, extending the work of a letter of Sarnak and arXiv:1704.02106 on golden and super-golden gates to higher dimensions. To do so we consider a variant of the Sarnak--Xue Density Hypotheses in the weight aspect for definite projective unitary groups and prove it using the endoscopic classification of automorphic representations. Our main motivation is to construct efficient multi-qubit universal gate sets for quantum computers. For example, we find a set of universal gates that, for a given accuracy, can heuristically approximate arbitrary unitary operations on 2 qubits with $\approx$10 times fewer ``expensive'' $T$-type gates than the standard Clifford+$T$ set. Our framework also covers the 2-qubit Clifford+CS gate set, well-known for being particularly friendly to fault-tolerant implementation. We thereby prove tight upper bounds on the required CS count for approximations (specifically, $4.8$x fewer non-Clifford gates than Clifford+$T$).

math.NT↗

arXiv2025

The Cohomological Sarnak-Xue Density Hypothesis for $SO_5$

We prove the cohomological version of the Sarnak--Xue Density Hypothesis for $SO_{5}$ over a totally real field and for inner forms split at all finite places. The proof relies on recent lines of work in the Langlands program: (i) Arthur's Endoscopic Classification of Representations of classical groups, extended to inner forms by Taïbi and its explicit description for $SO_{5}$ by Schmidt, and (ii) the Generalized Ramanujan--Petersson Theorem, proved for cohomological self-dual cuspidal representations of general linear groups. We give applications to the growth of cohomology of arithmetic manifolds, density-Ramanujan complexes, cutoff phenomena and optimal strong approximation.

math.NT↗

arXiv2024

Polynomial Bogolyubov for special linear groups via tensor rank

We prove a polynomial Bogolyubov type lemma for the special linear group over finite fields. Specifically, we show that there exists an absolute constant $C>0,$ such that if $A$ is a density $α$ subset of the special linear group, then the set $AA^{-1}AA^{-1}$ contains a subgroup $H$ of density $α^C$. Moreover, this subgroup is isomorphic to a special linear group of a smaller rank. We also show that if $A$ is an approximate subgroups then it can be covered by the union of few cosets of $H$. Our proof makes use of the Gurevich--Howe notion of tensor rank, and of a strengthened Bonami type Lemma for global functions on the bilinear scheme. We also present applications to spectral bounds for global convolution operators, global product free sets, and covering numbers corresponding to global sets.

math.CO↗

arXiv2024

Arithmeticity, thinness and efficiency of qutrit Clifford+T gates

The Clifford+T gate set is a topological generating set for PU(2), which has been well-studied from the perspective of quantum computation on a single qubit. The discovery that it generates a full S-arithmetic subgroup of PU(2) has led to a fruitful interaction between quantum computation and number theory, resulting in a proof that words in these gates cover PU(2) in an almost-optimal manner. In this paper we study the analogue gate set for PU(3). We show that in PU(3) the group generated by the Clifford+T gates is not arithmetic - in fact, it is a thin matrix group, namely a Zariski-dense group of infinite index in its ambient S-arithmetic group. On the other hand, we study a recently proposed extension of the Clifford+T gates, called Clifford+D, and show that these do generate a full S-arithmetic subgroup of PU(3), and satisfy a slightly weaker almost-optimal covering property than that of Clifford+T in PU(2). The proofs are different from those for PU(2): while both gate sets act naturally on a (Bruhat-Tits) tree, in PU(2) the generated group acts transitively on the vertices of the tree, and this is a main ingredient in proving both arithmeticity and efficiency. In the PU(3) Clifford+D case the action on the tree is far from being transitive. This makes the proof of arithmeticity considerably harder, and the study of efficiency by automorphic representation theory becomes more involved, and results in a covering rate which differs from the optimal one by a factor of $log_3(105)\approx 4.236$.

quant-ph↗

arXiv2023

Ramanujan Bigraphs

In their seminal paper, Lubotzky, Phillips and Sarnak (LPS) defined the notion of regular Ramanujan graphs and gave an explicit construction of infinite families of $(p+1)$-regular Ramanujan Cayley graphs, for infinitely many primes $p$. In this paper we extend the work of LPS and its successors to bigraphs (biregular bipartite graphs), in several aspects: we investigate the combinatorial properties of various generalizations of the notion of Ramanujan graphs, define a notion of Cayley bigraphs, and give explicit constructions of infinite families of $(p^3+1,p+1)$-regular Ramanujan Cayley bigraphs, for infinitely many $p$. Both the LPS graphs and our ones are arithmetic, arising as quotients of Bruhat-Tits trees by congruence subgroups of arithmetic lattices in a $p$-adic group, $PGL_2(\mathbb{Q}_p)$ for LPS and $PU_3(\mathbb{Q}_p)$ for us. In both cases the Ramanujan property relates to the Ramanujan Conjecture (RC) on the respective groups. But while for $PGL_2$ the RC holds unconditionally, this is not so in the case of $PU_3$. We find explicit cases where the RC does and does not hold, and use this to construct arithmetic non-Ramanujan Cayley bigraphs as well, and prove that nevertheless they satisfy the Sarnak-Xue density hypothesis. On the combinatorial side, we present a pseudorandomness characterization of Ramanujan bigraphs, and a more general notion of biexpanders. We show that our Ramanujan bigraphs exhibit the cutoff phenomenon with bounded window size for non-backtracking random walks, either as a consequence of the Ramanujan property, or of the Sarnak-Xue density hypothesis. Finally, we present some other applications of our work: golden gates for $PU_3$, Ramanujan and non-Ramanujan complexes of type $\widetilde{A}_2$, optimal strong approximation for $p$-arithmetic subgroups of $PSU_3$ and vanishing of first Betti numbers of Picard modular surfaces.

math.NT↗

arXiv2022

Good Locally Testable Codes

An explicit construction of locally testable codes of constant rate, constant distance and constant number of queries is given. Hence answering affirmatively the $c^3$-problem.

cs.IT↗

arXiv2022

Ramanujan complexes and Golden Gates in PU(3)

In a seminal series of papers from the 80's, Lubotzky, Phillips and Sarnak applied the Ramanujan-Petersson Conjecture for $GL_{2}$ (Deligne's theorem), to a special family of arithmetic lattices, which act simply-transitively on the Bruhat-Tits trees associated with $SL_{2}(\mathbb{Q}_{p})$. As a result, they obtained explicit Ramanujan Cayley graphs from $PSL_{2}\left(\mathbb{F}_{p}\right)$, as well as optimal topological generators ("Golden Gates") for the compact Lie group $PU(2)$. In higher dimension, the naive generalization of the Ramanujan Conjecture fails, due to the phenomenon of endoscopic lifts. In this paper we overcome this problem for $PU_{3}$ by constructing a family of arithmetic lattices which act simply-transitively on the Bruhat-Tits buildings associated with $SL_{3}(\mathbb{Q}_{p})$ and $SU_{3}(\mathbb{Q}_{p})$, while at the same time do not admit any representation which violates the Ramanujan Conjecture. This gives us Ramanujan complexes from $PSL_{3}(\mathbb{F}_{p})$ and $PSU_{3}(\mathbb{F}_{p})$, as well as golden gates for $PU(3)$.

math.NT↗

arXiv2021

Locally Testable Codes with constant rate, distance, and locality

A locally testable code (LTC) is an error-correcting code that has a property-tester. The tester reads $q$ bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter $q$ is called the locality of the tester. LTCs were initially studied as important components of PCPs, and since then the topic has evolved on its own. High rate LTCs could be useful in practice: before attempting to decode a received word, one can save time by first quickly testing if it is close to the code. An outstanding open question has been whether there exist "$c^3$-LTCs", namely LTCs with *c*onstant rate, *c*onstant distance, and *c*onstant locality. In this work we construct such codes based on a new two-dimensional complex which we call a left-right Cayley complex. This is essentially a graph which, in addition to vertices and edges, also has squares. Our codes can be viewed as a two-dimensional version of (the one-dimensional) expander codes, where the codewords are functions on the squares rather than on the edges.

cs.IT↗

arXiv2020

Decodable quantum LDPC codes beyond the $\sqrt{n}$ distance barrier using high dimensional expanders

Constructing quantum LDPC codes with a minimum distance that grows faster than a square root of the length has been a major challenge of the field. With this challenge in mind, we investigate constructions that come from high-dimensional expanders, in particular Ramanujan complexes. These naturally give rise to very unbalanced quantum error correcting codes that have a large $X$-distance but a much smaller $Z$-distance. However, together with a classical expander LDPC code and a tensoring method that generalises a construction of Hastings and also the Tillich-Zemor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 3-dimensional Ramanujan complex, we show that its 2-systole behaves like a square of the log of the complex size, which results in an overall quantum code of minimum distance $n^{1/2}\log n$, and sets a new record for quantum LDPC codes. When we use a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance $n^{1/2}\log^{1/2}n$. We then exploit the expansion properties of the complex to devise the first polynomial time algorithm that decodes above the square root barrier for quantum LDPC codes.

quant-ph↗

arXiv2017

Good cyclic codes and the uncertainty principle

A long standing problem in the area of error correcting codes asks whether there exist good cyclic codes. Most of the known results point in the direction of a negative answer. The uncertainty principle is a classical result of harmonic analysis asserting that given a non-zero function $f$ on some abelian group, either $f$ or its Fourier transform $\hat{f}$ has large support. In this note, we observe a connection between these two subjects. We point out that even a weak version of the uncertainty principle for fields of positive characteristic would imply that good cyclic codes do exist. We also provide some heuristic arguments supporting that this is indeed the case.

cs.IT↗

arXiv2017

Bounded Degree Cosystolic Expanders of Every Dimension

In this work we present a new local to global criterion for proving a form of high dimensional expansion, which we term cosystolic expansion. Applying this criterion on Ramanujan complexes, yields for every dimension, an infinite family of bounded degree complexes with the topological overlapping property. This answer affirmatively an open question raised by Gromov.

math.CO↗

arXiv2015

Finite quotients of Bruhat-Tits buildings as geometric expanders

In \cite{FGLNP}, Fox, Gromov, Lafforgue, Naor and Pach, in a respond to a question of Gromov \cite{G}, constructed bounded degree geometric expanders, namely, simplical complexes having the affine overlapping property. Their explicit constructions are finite quotients of $\tilde{A_d}$-buildings, for $d\geq 2$, over local fields. In this paper, this result is extended to general high rank Bruhat-Tits buildings.

math.CO↗

arXiv2014

Mixing properties and the chromatic number of Ramanujan complexes

Ramanujan complexes are high dimensional simplical complexes generalizing Ramanujan graphs. A result of Oh on quantitative property (T) for Lie groups over local fields is used to deduce a Mixing Lemma for such complexes. As an application we prove that non-partite Ramanujan complexes have 'high girth' and high chromatic number, generalizing a well known result about Ramanujan graphs.

math.CO↗