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The Clifford hierarchy is a nested sequence of sets of quantum gates critical to achieving fault-tolerant quantum computation. Diagonal gates of the Clifford hierarchy and ‘nearly diagonal’ semi-Clifford gates are particularly important: they admit efficient gate teleportation protocols that implement these gates with fewer ancillary quantum resources such as magic states. Despite the practical importance of these sets of gates, many questions about their structure remain open; this is especially true in the higher-dimensional qudit setting. Our contribution is to leverage the discrete Stone–von Neumann theorem and the symplectic formalism of qudit stabilizer theory towards extending the results of Zeng et al. (2008) and Beigi & Shor (2010) to higher dimensions in a uniform manner. We further give a simple algorithm for recursively enumerating all gates of the Clifford hierarchy, a simple algorithm for recognizing and diagonalizing semi-Clifford gates, and a concise proof of the classification of the diagonal Clifford hierarchy gates due to Cui et al. (2016) for the single-qudit case. We generalize the efficient gate teleportation protocols of semi-Clifford gates to the qudit setting and prove that every third-level gate of one qudit (of any prime dimension) and of two qutrits can be implemented efficiently. Numerical evidence gathered via the aforementioned algorithms supports the conjecture that higher-level gates can be implemented efficiently.

We extend the work in a previous paper with David Li-Bland to construct the Wehrheim-Woodward category WW( $G\\mathbf{SLREL}$ ) of equivariant linear canonical relations between linear symplectic$G$ -spaces for a compact group$G$ . When$G$is the trivial group, this reduces to the previous result that the morphisms in WW( $\\mathbf{SLREL}$ ) may be identified with pairs$(L,k)$consisting of a linear canonical relation and a nonnegative integer.

A construction of Wehrheim and Woodward circumvents the problem that compositions of smooth canonical relations are not always smooth, building a category suitable for functorial quantization. To apply their construction to more examples, we introduce a notion of highly selective category, in which only certain morphisms and certain pairs of these morphisms are ''good''. We then apply this notion to the category [...] of linear canonical relations and the result [...] of our version of the WW construction, identifying the morphisms in the latter with pairs [...] consisting of a linear canonical relation and a nonnegative integer. We put a topology on this category of indexed linear canonical relations for which composition is continuous, unlike the composition in [...] itself. Subsequent papers will consider this category from the viewpoint of derived geometry and will concern quantum counterparts. [ProQuest: [...] denotes formulae omitted.]

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations.Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

A Maslov cycle is a singular variety in the lagrangian grassmannian Λ(V) of a symplectic vector space V consisting of all lagrangian subspaces having nonzero intersection with a fixed one. Givental has shown that a Maslov cycle is a Legendre singularity, i.e. the projection of a smooth conic lagrangian submanifold S in the cotangent bundle of Λ(V). We show here that S is the wavefront set of a Fourier integral distributionwhich is “evaluation at 0 of the quantizations”.

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

Let V be a vector space of dimension 2n, n even, over a field F, equipped with a nonsingular symplectic form. We define a new algebraic/combinatorial structure, a spread of nonsingular pairs, or nsp-spread, on V and show that nsp-spreads exist in considerable generality. We further examine in detail some particular cases.

We present a generalization of Bregman divergences in finite-dimensional symplectic vector spaces that we term symplectic Bregman divergences. Symplectic Bregman divergences are derived from a symplectic generalization of the Fenchel–Young inequality which relies on the notion of symplectic subdifferentials. The symplectic Fenchel–Young inequality is obtained using the symplectic Fenchel transform which is defined with respect to the symplectic form. Since symplectic forms can be built generically from pairings of dual systems, we obtain a generalization of Bregman divergences in dual systems obtained by equivalent symplectic Bregman divergences. In particular, when the symplectic form is derived from an inner product, we show that the corresponding symplectic Bregman divergences amount to ordinary Bregman divergences with respect to composite inner products. Some potential applications of symplectic divergences in geometric mechanics, information geometry, and learning dynamics in machine learning are touched upon.

We propose a generalization of Haiman’s conjecture on the diagonal coinvariant rings of real reflection groups to the context of irreducible quaternionic reflection groups (also known as symplectic reflection groups). For a reflection group W acting on a quaternionic vector space V , by regarding V as a complex vector space, we consider the scheme-theoretic fiber over zero of the quotient map V V/W . For W an irreducible reflection group of (quaternionic) rank at least 6 , we show that the ring of functions on this fiber admits a (g+1)^n -dimensional quotient arising from an irreducible representation of a symplectic reflection algebra, where g=2N/n , with N the number of reflections in W and n=dim_H(V) , and we conjecture that this holds in general. We observe that in fact the degree of the zero fiber is precisely g+1 for the rank one groups (corresponding to the Kleinian singularities). In an appendix, we give a proof that three variants of the Coxeter number, including g , are integers.

In the present self-contained paper, we want, first, to construct a fundamental diagram, called (S.C), in homage to Carl Siegel and I. Satake that connects the following groups: SU ( m , m ) , SO ∗ ( 2 m ) , Sp ( 2 m , R ) , Sp ( 4 m , R ) , SO ∗ ( 4 m ) . Then, we define and study three Clifford algebras related to that diagram. First, we consider the morphism from Sp ( 2 m , R ) into SU ( m , m ) , shown in the construction of the diagram (S.C.). Then, we define a Clifford algebra C l m , m , naturally associated with the group U ( m , m ) . Let ( E ,  b ) be an m -dimensional skew-hermitian space over H . For any x , y ∈ E , write b ( x , y ) = h ( x , y ) + j a ( x , y ) . It is well known that h is a skew-hermitian complex form on E 2 m , the complex 2 m -dimensional vector space underlying E ,  and a is a symmetric bilinear complex form on E 2 m . We proved previously in [ 4 ] that the special unitary group SU ( E , b ) of a skew-hermitian H -right vector space ( E ,  b ),  m -dimensional over H , can be identified with the group SO ∗ ( 2 m ) defined by E. Cartan. We define a real Clifford algebra, namely C l R ∗ ( 2 m ) , whose complexified algebra is C 2 m + ( E 2 m , a ) , the even complex Clifford algebra associated with a . Both algebras are associated with the geometry of the skew-hermitian H -space ( E ,  b ). Let V = ( R 2 m , Sp ( 2 m , R ) ) be the standard model of a real symplectic space. We present some connections between the geometry of V and the algebras C l m , m , C 2 m + ( E 2 m , a ) , C l R ∗ ( 2 m ) . The last section wants to give a sketch of the prospects offered by these algebras for the study of the real conformal symplectic geometry. An appendix gives some indispensable recalls and some complements.