15,425 research outputs found
Interpolation Methods for Binary and Multivalued Logical Quantum Gate Synthesis
A method for synthesizing quantum gates is presented based on interpolation
methods applied to operators in Hilbert space. Starting from the diagonal forms
of specific generating seed operators with non-degenerate eigenvalue spectrum
one obtains for arity-one a complete family of logical operators corresponding
to all the one-argument logical connectives. Scaling-up to n-arity gates is
obtained by using the Kronecker product and unitary transformations. The
quantum version of the Fourier transform of Boolean functions is presented and
a Reed-Muller decomposition for quantum logical gates is derived. The common
control gates can be easily obtained by considering the logical correspondence
between the control logic operator and the binary propositional logic operator.
A new polynomial and exponential formulation of the Toffoli gate is presented.
The method has parallels to quantum gate-T optimization methods using powers of
multilinear operator polynomials. The method is then applied naturally to
alphabets greater than two for multi-valued logical gates used for quantum
Fourier transform, min-max decision circuits and multivalued adders
From p-adic to real Grassmannians via the quantum
Let F be a local field. The action of GL(n,F) on the Grassmann variety
Gr(m,n,F) induces a continuous representation of the maximal compact subgroup
of GL(n,F) on the space of L^2-functions on Gr(m,n,F). The irreducible
constituents of this representation are parameterized by the same underlying
set both for Archimedean and non-Archimedean fields.
This paper connects the Archimedean and non-Archimedean theories using the
quantum Grassmannian. In particular, idempotents in the Hecke algebra
associated to this representation are the image of the quantum zonal spherical
functions after taking appropriate limits. Consequently, a correspondence is
established between some irreducible representations with Archimedean and
non-Archimedean origin.Comment: 24 pages, final version, to appear in Advances in Mathematic
Interpolation of SUSY quantum mechanics
Interpolation of two adjacent Hamiltonians in SUSY quantum mechanics
, is discussed together
with related operators. For a wide variety of shape-invariant degree one
quantum mechanics and their `discrete' counterparts, the interpolation
Hamiltonian is also shape-invariant, that is it takes the same form as the
original Hamiltonian with shifted coupling constant(s).Comment: 18 page
The Capelli eigenvalue problem for quantum groups
We introduce and study quantum Capelli operators inside newly constructed
quantum Weyl algebras associated to three families of symmetric pairs. Both the
center of a particular quantized enveloping algebra and the Capelli operators
act semisimply on the polynomial part of these quantum Weyl algebras. We show
how to transfer well-known properties of the center arising from the theory of
quantum symmetric pairs to the Capelli operators. Using this information, we
provide a natural realization of Knop-Sahi interpolation polynomials as
functions that produce eigenvalues for quantum Capelli operators
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