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Abstract.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 gates together with V produces all n-qudit gates exactly. We show that (i) and (ii) are equivalent conditions on V , and they hold if and only if V is not a primitive gate. Here we say V is primitive if it transforms any decomposable tensor into a decomposable tensor. We discuss some applications and also relations with work of other authors. Statements of main resultsWe determine which 2-qudit gates V have the property that all 1-qudit gates together with V form a universal collection, in either the approximate sense or the exact sense. Here d is an arbitrary integer ≥ 2. Our results are new for the case of qubits, i.e., d = 2 (which for many is the case of primary interest). We treat the case d > 2 as well because it is of independent interest and requires no additional work.Since Deutsch [3] found a universal gate (on 3 qubits), universal gates for qubits have been extensively studied. We mention in particular the papers [1], [2] [4], [5] and [6] which will be further discussed in §2.First we set up some notations. A qudit is a (normalized) state in the Hilbert space C d . An n-qudit is a state in the tensor product Hilbert spaceThe computational basis of H is the orthonormal basis given by the d n classical n-quditswhere ||ψ|| 2 = |ψ i 1 i 2 ···in | 2 = 1. We say ψ is decomposable when it can be written as a tensor product |x 1 · · · x n = |x 1 ⊗ |x 2 ⊗ · · · ⊗ |x n of qudits.A quantum gate on n-qudits is a unitary operator L :⊗n . These gates form the unitary group U((. . , L k of gates constitutes a quantum circuit on n-qudits.

In [7], D. Kazhdan and G. Lusztig gave a conjecture on the multiplicity of simple modules which appear in a Jordan-H61der series of the Verma modules. This multiplicity is described in the terms of Coxeter groups and also by the geometry of Schubert cells in the flag manifold (see [8]). The purpose of this paper is to give the proof of their conjecture.The method employed here is to associate holonomic systems of linear differential equations with R.S. on the flag manifold with Verma modules and to use the correspondance of holonomic systems and constructible sheaves.Let G be a semi-simple Lie group defined over • and g its Lie algebra. We