Algebrahttps://hdl.handle.net/11441/108032024-03-28T19:49:42Z2024-03-28T19:49:42ZA generalisation of the Phase Kick-Backhttps://hdl.handle.net/11441/1553702024-02-20T12:10:42Z2023-03-13T00:00:00ZA generalisation of the Phase Kick-Back
In this paper, we present a generalisation of the Phase Kick-Back technique, which is
central to some of the classical algorithms in quantum computing. We will begin by
recalling the Phase Kick-Back technique to then introduce the new generalised version
for f : {0, 1}n → {0, 1}m functions using the eigenvalues of the oracle function U f .
After that, we will present a new generalised version of the Deutsch–Jozsa problem
and how it can be solved using the previously defined technique. We will also deal
with a generalised version of the Bernstein–Vazirani problem and solve it using the
generalised Phase Kick-Back. Finally, we show how we can use this technique to
obtain an algorithm for Simon’s problem that improves the classical one.
2023-03-13T00:00:00ZAll linear symmetries of the SU(3) tensor multiplicitieshttps://hdl.handle.net/11441/1548222024-02-07T12:09:34Z2023-12-07T00:00:00ZAll linear symmetries of the SU(3) tensor multiplicities
The SU(3) tensor multiplicities are piecewise polynomial of degree 1 in their labels. The pieces are the chambers of a complex of cones. We describe in detail this chamber complex and determine the group of all linear symmetries (of order 144) for these tensor multiplicities. We represent the cells by diagrams showing clearly the inclusions as well as the actions of the group of symmetries and of its remarkable subgroups.
2023-12-07T00:00:00ZVector partition functions and Kronecker coefficientshttps://hdl.handle.net/11441/1548152024-02-07T11:52:40Z2021-04-01T00:00:00ZVector partition functions and Kronecker coefficients
The Kronecker coefficients are the structure constants for the restriction of irreducible representations of the general linear group GL(nm) into irreducibles for
the subgroup GL(n) × GL(m). In this work we study the quasipolynomial nature
of the Kronecker function using elementary tools from polyhedral geometry.We
write the Kronecker function in terms of coefficients of a vector partition function. This allows us to define a new family of coefficients, the atomic Kronecker
coefficients. Our derivation is explicit and self-contained, and gives a new exact
formula and an upper bound for the Kronecker coefficients in the first nontrivial
case.
2021-04-01T00:00:00ZPartial symmetries of iterated plethysmshttps://hdl.handle.net/11441/1547132024-02-06T13:32:43Z2023-05-03T00:00:00ZPartial symmetries of iterated plethysms
This work highlights the existence of partial symmetries in large families of iterated plethystic coefficients. The plethystic coefficients involved come from the expansion in the Schur basis of iterated plethysms of Schur functions indexed by one-row partitions.The partial symmetries are described in terms of an involution on partitions, the flip involution, that generalizes the ubiquitous � involution. Schur-positive symmetric functions possessing this partial symmetry are termed flip-symmetric. The operation of taking plethysm with �� preserves flip-symmetry, provided that � is a partition of two. Explicit formulas for the iterated plethysms �2∘��∘�� and ��∘�2∘��, with a, b, and c ≥ 2 allow us to show that these two families of iterated plethysms are flip-symmetric. The article concludes with some observations, remarks, and open questions on the unimodality and asymptotic normality of certain flip-symmetric sequences of iterated plethystic coefficients.
2023-05-03T00:00:00Z