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In this paper we study universality for quantum gates acting on qudits. Qudits are states in a Hilbert space of dimension d where d can be any integer ≥ 2. We determine which 2-qudit gates V have the properties (i) the collection of all 1-qudit gates together with V produces all n-qudit gates up to arbitrary precision, or (ii) the collection of all 1-qudit… (More)

- RANEE BRYLINSKI
- 2000

Lecomte and Ovsienko constructed SL n+1 (R)-equivariant quantization maps Q λ for symbols of differential operators on λ-densities on RP n. We derive some formulas for the associated graded equivariant star products ⋆ λ on the symbol algebra Pol(T * RP n). These give some measure of the failure of locality. Our main result expresses (for n odd) the… (More)

Let M be a G-covering of a nilpotent orbit in g where G is a complex semisimple Lie group and g = Lie(G). We prove that under Poisson bracket the space R[2] of homogeneous functions on M of degree 2 is the unique maximal semisimple Lie subalgebra of R = R(M) containing g. The action of g ′ ≃ R[2] exponentiates to an action of the corresponding Lie group G ′… (More)

- R Brylinski, B Kostant
- Proceedings of the National Academy of Sciences…
- 1994

In the framework of geometric quantization we explicitly construct, in a uniform fashion, a unitary minimal representation pio of every simply-connected real Lie group Go such that the maximal compact subgroup of Go has finite center and Go admits some minimal representation. We obtain algebraic and analytic results about pio. We give several results on the… (More)

- Ranee Brylinski
- 1998

In this paper, we begin a quantization program for nilpotent orbits O R of a real semisimple Lie group G R. These orbits arise naturally as the coadjoint orbits of G R which are stable under scaling, and thus they have a canonical symplectic structure ω where the G R-action is Hamiltonian. These orbits and their covers generalize the oscillator phase space… (More)

- R Brylinski, B Kostant
- Proceedings of the National Academy of Sciences…
- 1994

We explicitly construct, in a uniform fashion, the (unique) minimal and spherical representation pi0 of the split real Lie group of exceptional type E6, E7, or E8. We obtain several algebraic and analytic results about pi0.

- ALEXANDER ASTASHKEVICH, RANEE BRYLINSKI
- 2000

We construct a unique G-equivariant graded star product on the algebra S (g)/I of polynomial functions on the minimal nilpotent coadjoint orbit O min of G where G is a complex simple Lie group and g = sl(2, C). This strengthens the result of Arnal, Benamor and Cahen. Our main result is to compute, for G classical, the star product of a momentum function µ x… (More)

- RANEE BRYLINSKI
- 2001

Let X R be a (generalized) flag manifold of a non-compact real semisimple Lie group G R , where X R and G R have complexifications X and G. We investigate the problem of constructing a graded star product on Pol(T * X R) which corresponds to a G R-equivariant quantization of symbols into smooth differential operators acting on half-densities on X R. We show… (More)

- Alexander Astashkevich, Ranee Brylinski
- 1998

Let O be the minimal nilpotent adjoint orbit in a classical complex semisim-ple Lie algebra g. O is a smooth quasi-affine variety stable under the Eu-ler dilation action C * on g. The algebra of differential operators on O is D(O) = D(Cl(O)) where the closure Cl(O) is a singular cone in g. See [J] and [B-K1] for some results on the geometry and quantization… (More)

- Jean-Luc Brylinski, Ranee Brylinski
- ArXiv
- 2003

Immanants are polynomial functions of n by n matrices attached to irre-ducible characters of the symmetric group Sn, or equivalently to Young diagrams of size n. Immanants include determinants and permanents as extreme cases. Valiant proved that computation of permanents is a complete problem in his algebraic model of NP theory , i.e., it is VNP-complete.… (More)