On the spectral gap for finitely-generated subgroups of SU(2)

Bourgain, Jean; Gamburd, Alex

doi:10.1007/s00222-007-0072-z

Cited by 133 publications

(186 citation statements)

References 22 publications

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“…These developments should include verification of the spectral gap conjecture. Currently it is known to be true under the additional assumption that gates have algebraic entries [6,7].In this paper we present an approach that allows to decide universality of S by checking the spectra of the gates and solving some linear equations whose coefficients are polynomial in the entries of the gates and their complex conjugates. Moreover, for non-universal S, our method indicates what type of gates can be added to make S universal.…”

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confidence: 99%

Criteria for universality of quantum gates

Sawicki

Karnas

2017

Phys. Rev. A

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We consider the problem of deciding if a set of quantum one-qudit gates S = {U1, . . . , Un} is universal. We provide the compact form criteria leading to a simple algorithm that allows deciding universality of any given set of gates in a finite number of steps. Moreover, for a non-universal S our criteria indicate what type of gates can be added to S to turn it into a universal set.Universal quantum gates play an important role in quantum computing and quantum optics [15,24,30]. The ability to effectively manufacture gates operating on many modes, using for example optical networks that couple modes of light [9,29], is a natural motivation to consider the universality problems not only for qubits but also for higher dimensional systems, i.e. qudits (see also [27,28] for fermionic linear optics and quantum metrology). For quantum computing with qudits, a universal set of gates consists of all one-qudit gates together with an additional two-qudit gate that does not map separable states onto separable states [10] (see [11,[35][36][37] for recent results in the context of universal Hamiltonians). The set of all one-qudit gates can be, however, generated using a finite number of gates [23]. We say that one-qudit gates S = {U 1 , . . . , U n } ⊂ SU (d) are universal if any gate from SU (d) can be built, with an arbitrary precision, using gates from S. It is known that almost all sets of qudit gates are universal, i.e. non-universal sets S of the given cardinality are of measure zero and can be characterised by vanishing of a finite number of polynomials in the gates entries and their conjugates [17,23]. Surprisingly, however, these polynomials are not known and it is hard to find operationally simple criteria that decide one-qudit gates universality. Some special cases of optical 3-mode gates have been recently studied in [5,32] and the approach providing an algorithm for deciding universality of a given set of quantum gates that can be implemented on a quantum automata has been proposed [13] (see also [1,2,14] for algorithms deciding if a finitely generated group is infinite). The main obstruction in the problems considered in [5,32] is the lack of classification of finite subgroups of SU (d) for d > 4. Nevertheless, as we show in this paper one can still provide some reasonable conditions for universality of one-qudit gates without this knowledge.The efficiency of universal sets is typically measured by the number of gates that are needed to approximate other gates with a given precision ǫ. The Solovay-Kitaev theorem states that all universal sets are roughly the same efficient. More precisely, the number of gates needed to approximate any gate U ∈ SU (d) is bounded by O(log c (1/ǫ)) [26], where c may depend only on d and c ≥ 1. Recently there has been a bit of flurry in the * E-mail: a.sawicki@cft.edu.pl, karnas@cft.edu.pl area of single qubit gates [3,21,22,34] showing that using some number theoretic results and conjectures one can construct universal sets with c = 1. The approach presented in these contribution...

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confidence: 99%

Criteria for universality of quantum gates

Sawicki

Karnas

2017

Phys. Rev. A

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“…Let G = SL n (R) and K = SO n (R), for n 3. Let Γ < G be a countable subgroup which contains a matrix g 0 ∈ G \ K as well as matrices g 1 , ..., g l ∈ K that have algebraic entries and generate a dense subgroup of K. If n = 3, then the work of J. Bourgain and A. Gamburd [BG06] shows that the action Γ ∩ K (K, m K ) has spectral gap. Moreover, the very recent work [BdS14] implies that this statement holds for any n 3.…”

Section: Then the Actionsmentioning

confidence: 99%

Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions

Ioana

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We study equivalence relations that arise from translation actions Γ G which are associated to dense embeddings Γ < G of countable groups into second countable locally compact groups. Assuming that G is simply connected and the action Γ G is strongly ergodic, we prove that Γ G is orbit equivalent to another such translation action Λ H if and only if there exists an isomorphism δ : G → H such that δ(Γ) = Λ. If G is moreover a real algebraic group, then we establish analogous rigidity results for the translation actions of Γ on homogeneous spaces of the form G/Σ, where Σ < G is either a discrete or an algebraic subgroup. We also prove that if G is simply connected and the action Γ G has property (T), then any cocycle w : Γ × G → Λ with values into a countable group Λ is cohomologous to a homomorphism δ : Γ → Λ. As a consequence, we deduce that the action Γ G is orbit equivalent superrigid: any free nonsingular action Λ Y which is orbit equivalent to Γ G, is necessarily conjugate to an induction of Γ G.

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“…Inspired by work of Gamburd-Jakobson-Sarnak [GJS99] and BourgainGamburd [BG08] on the spectral gap problem for finitely generated subgroups of compact Lie groups, we defined in a previous article [ABRdS14a] the notion of diophantine subgroup of an arbitrary Lie group G. The definition is as follows. Any finite symmetric subset S := {1,…”

Section: Diophantine Approximation On Lie Groupsmentioning

confidence: 99%

On metric Diophantine approximation in matrices and Lie groups

Aka

Breuillard

Rosenzweig

et al. 2015

Comptes Rendus. Mathématique

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Abstract. We study the diophantine exponent of analytic submanifolds of m × n real matrices, answering questions of Beresnevich, Kleinbock and Margulis. We identify a family of algebraic obstructions to the extremality of such a submanifold, and give a formula for the exponent when the submanifold is algebraic and defined over Q. We then apply these results to the determination of the diophantine exponent of rational nilpotent Lie groups.

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On the spectral gap for finitely-generated subgroups of SU(2)

Cited by 133 publications

References 22 publications

Criteria for universality of quantum gates

Criteria for universality of quantum gates

Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions

On metric Diophantine approximation in matrices and Lie groups

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