arXiv subjects

Shelly Garion

Publications and source records attributed to Shelly Garion.

At least 19 recordsLinked to original sources

arXiv2024

Benchmarking the performance of quantum computing software

We present Benchpress, a benchmarking suite for evaluating the performance and range of functionality of multiple quantum computing software development kits. This suite consists of a collection of over $1000$ tests measuring key performance metrics for a wide variety of operations on quantum circuits comprised of up to $930$ qubits and $\mathcal{O}(10^{6})$ two-qubit gates, as well as an execution framework for running the tests over multiple quantum software packages in a unified manner. We give a detailed overview of the benchmark suite, its methodology, and generate representative results over seven different quantum software packages. The flexibility of the Benchpress framework allows for benchmarking that not only keeps pace with quantum hardware improvements but can preemptively gauge the quantum circuit processing costs of future device architectures. Being open-source, Benchpress ensures the transparency and verification of performance claims.

quant-ph↗

arXiv2024

Linear Circuit Synthesis using Weighted Steiner Trees

CNOT circuits are a common building block of general quantum circuits. The problem of synthesizing and optimizing such circuits has received a lot of attention in the quantum computing literature. This problem is especially challenging for quantum devices with restricted connectivity, where two-qubit gates can only be placed between adjacent qubits. The state-of-the-art algorithms for optimizing the number of CNOT gates are heuristic algorithms that are based on Gaussian elimination and that use Steiner trees to connect between different subsets of qubits. In this article, we suggest considering weighted Steiner trees, and we present a simple low-cost heuristic to compute weights. The simulated evaluation shows that the suggested heuristic is almost always beneficial and reduces the number of CNOT gates by up to 10%.

quant-ph↗

arXiv2024

Diagonalization of large many-body Hamiltonians on a quantum processor

The estimation of low energies of many-body systems is a cornerstone of computational quantum sciences. Variational quantum algorithms can be used to prepare ground states on pre-fault-tolerant quantum processors, but their lack of convergence guarantees and impractical number of cost function estimations prevent systematic scaling of experiments to large systems. Alternatives to variational approaches are needed for large-scale experiments on pre-fault-tolerant devices. Here, we use a superconducting quantum processor to compute eigenenergies of quantum many-body systems on two-dimensional lattices of up to 56 sites, using the Krylov quantum diagonalization algorithm, an analog of the well-known classical diagonalization technique. We construct subspaces of the many-body Hilbert space using Trotterized unitary evolutions executed on the quantum processor, and classically diagonalize many-body interacting Hamiltonians within those subspaces. These experiments show that quantum diagonalization algorithms are poised to complement their classical counterpart at the foundation of computational methods for quantum systems.

quant-ph↗

arXiv2021

Experimental implementation of non-Clifford interleaved randomized benchmarking with a controlled-S gate

Hardware efficient transpilation of quantum circuits to a quantum devices native gateset is essential for the execution of quantum algorithms on noisy quantum computers. Typical quantum devices utilize a gateset with a single two-qubit Clifford entangling gate per pair of coupled qubits, however, in some applications access to a non-Clifford two-qubit gate can result in more optimal circuit decompositions and also allows more flexibility in optimizing over noise. We demonstrate calibration of a low error non-Clifford Controlled-$\fracπ{2}$ phase (CS) gate on a cloud based IBM Quantum computing using the Qiskit Pulse framework. To measure the gate error of the calibrated CS gate we perform non-Clifford CNOT-Dihedral interleaved randomized benchmarking. We are able to obtain a gate error of $5.9(7) \times 10^{-3}$ at a gate length 263 ns, which is close to the coherence limit of the associated qubits, and lower error than the backends standard calibrated CNOT gate.

quant-ph↗

arXiv2020

Synthesis of CNOT-Dihedral circuits with optimal number of two qubit gates

In this note we present explicit canonical forms for all the elements in the two-qubit CNOT-Dihedral group, with minimal numbers of controlled-S (CS) and controlled-X (CX) gates, using the generating set of quantum gates [X, T, CX, CS]. We provide an algorithm to successively construct the n-qubit CNOT-Dihedral group, asserting an optimal number of controlled-X (CX) gates. These results are needed to estimate gate errors via non-Clifford randomized benchmarking and may have further applications to circuit optimization over fault-tolerant gate sets.

quant-ph↗

arXiv2020

AIOps for a Cloud Object Storage Service

With the growing reliance on the ubiquitous availability of IT systems and services, these systems become more global, scaled, and complex to operate. To maintain business viability, IT service providers must put in place reliable and cost efficient operations support. Artificial Intelligence for IT Operations (AIOps) is a promising technology for alleviating operational complexity of IT systems and services. AIOps platforms utilize big data, machine learning and other advanced analytics technologies to enhance IT operations with proactive actionable dynamic insight. In this paper we share our experience applying the AIOps approach to a production cloud object storage service to get actionable insights into system's behavior and health. We describe a real-life production cloud scale service and its operational data, present the AIOps platform we have created, and show how it has helped us resolving operational pain points.

cs.DC↗

arXiv2013

Beauville surfaces and probabilistic group theory

A Beauville surface is a complex algebraic surface that can be presented as a quotient of a product of two curves by a suitable action of a finite group. Bauer, Catanese and Grunewald have been able to intrinsically characterize the groups appearing in minimal presentations of Beauville surfaces in terms of the existence of a so-called "Beauville structure". They conjectured that all finite simple groups, except A5, admit such a structure. This conjecture has recently been proved by Guralnick-Malle and Fairbairn-Magaard-Parker. In this survey we demonstrate another approach towards the proof of this conjecture, based on probabilistic group theoretical methods, by describing the following three works. The first is the work of Garion, Larsen and Lubotzky, showing that the above conjecture holds for almost all finite simple groups of Lie type. The second is the work of Garion and Penegini on Beauville structures of alternating groups, based on results of Liebeck and Shalev, and the third is the case of the group PSL(2,p^e), in which we give bounds on the probability of generating a Beauville structure. We also discuss other related problems regarding finite simple quotients of hyperbolic triangle groups and present some open questions and conjectures.

math.GR↗

arXiv2013

On Beauville Structures for PSL(2,q)

We characterize Beauville surfaces of unmixed type with group either PSL(2,p^e) or PGL(2,p^e), thus extending previous results of Bauer, Catanese and Grunewald, Fuertes and Jones, and Penegini and the author.

math.GR↗

arXiv2013

Expansion of conjugacy classes in PSL(2,q)

For any conjugacy class C in G=PSL(2,q) we compute C^2 and discuss whether C contains a triple of elements whose product is 1 which generate G. Moreover, we determine which elements in G can be written as a product of two conjugate elements that generate G.

math.GR↗

arXiv2013

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods, group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.

math.AG↗

arXiv2012

New Beauville surfaces and finite simple groups

In this paper we construct new Beauville surfaces with group either $\PSL(2,p^e)$, or belonging to some other families of finite simple groups of Lie type of low Lie rank, or an alternating group, or a symmetric group, proving a conjecture of Bauer, Catanese and Grunewald. The proofs rely on probabilistic group theoretical results of Liebeck and Shalev, on classical results of Macbeath and on recent results of Marion.

math.GR↗

arXiv2011

Highly Transitive Actions of Out(Fn)

An action of a group on a set is called k-transitive if it is transitive on ordered k-tuples and highly transitive if it is k-transitive for every k. We show that for n>3 the group Out(Fn) = Aut(Fn)/Inn(Fn) admits a faithful highly transitive action on a countable set.

math.GR↗

arXiv2011

Surjectivity and equidistribution of the word x^ay^b on PSL(2,q) and SL(2,q)

We determine the positive integers a,b and the prime powers q for which the word map w(x,y)=x^ay^b is surjective on the group PSL(2,q) (and SL(2,q)). We moreover show that this map is almost equidistributed for the family of groups PSL(2,q) (and SL(2,q)). Our proof is based on the investigation of the trace map of positive words.

math.GR↗

arXiv2011

Beauville surfaces, moduli spaces and finite groups

In this paper we give the asymptotic growth of the number of connected components of the moduli space of surfaces of general type corresponding to certain families of Beauville surfaces with group either $\PSL(2,p)$, or an alternating group, or a symmetric group or an abelian group. We moreover extend these results to regular surfaces isogenous to a higher product of curves.

math.AG↗

arXiv2011

On the Surjectivity of Engel Words on PSL(2,q)

We investigate the surjectivity of the word map defined by the n-th Engel word on the groups PSL(2,q) and SL(2,q). For SL(2,q), we show that this map is surjective onto the subset SL(2,q)\{-id} provided that q>Q(n) is sufficiently large. Moreover, we give an estimate for Q(n). We also present examples demonstrating that this does not hold for all q. We conclude that the n-th Engel word map is surjective for the groups PSL(2,q) when q>Q(n). By using the computer, we sharpen this result and show that for any n<5, the corresponding map is surjective for all the groups PSL(2,q). This provides evidence for a conjecture of Shalev regarding Engel words in finite simple groups. In addition, we show that the n-th Engel word map is almost measure preserving for the family of groups PSL(2,q), with q odd, answering another question of Shalev. Our techniques are based on the method developed by Bandman, Grunewald and Kunyavskii for verbal dynamical systems in the group SL(2,q).

math.GR↗

arXiv2010

Beauville surfaces and finite simple groups

A Beauville surface is a rigid complex surface of the form (C1 x C2)/G, where C1 and C2 are non-singular, projective, higher genus curves, and G is a finite group acting freely on the product. Bauer, Catanese, and Grunewald conjectured that every finite simple group G, with the exception of A5, gives rise to such a surface. We prove that this is so for almost all finite simple groups (i.e., with at most finitely many exceptions). The proof makes use of the structure theory of finite simple groups, probability theory, and character estimates.

math.GR↗

arXiv2008

Connectivity of the Product Replacement Graph of Simple Groups of Bounded Lie Rank

The Product Replacement Algorithm is a practical algorithm for generating random elements of a finite group. The algorithm can be described as a random walk on a graph whose vertices are the generating k-tuples of the group (for a fixed integer k). We show that there is a function c(r) such that for any finite simple group of Lie type, with Lie rank r, the product replacement graph of the generating k-tuples is connected for any k > c(r). The proof uses results of Larsen and Pink and does not rely on the classification of finite simple groups.

math.GR↗

arXiv2007

Connectivity of the Product Replacement Algorithm Graph of PSL(2,q)

The product replacement algorithm is a practical algorithm to construct random elements of a finite group G. It can be described as a random walk on a graph whose vertices are the generating k-tuples of G (for a fixed k). We show that if G is PSL(2,q) or PGL(2,q), where q is a prime power, then this graph is connected for any k>=4. This generalizes former results obtained by Gilman and Evans.

math.GR↗