38 research outputs found
Hidden Translation and Translating Coset in Quantum Computing
We give efficient quantum algorithms for the problems of Hidden Translation
and Hidden Subgroup in a large class of non-abelian solvable groups including
solvable groups of constant exponent and of constant length derived series. Our
algorithms are recursive. For the base case, we solve efficiently Hidden
Translation in , whenever is a fixed prime. For the induction
step, we introduce the problem Translating Coset generalizing both Hidden
Translation and Hidden Subgroup, and prove a powerful self-reducibility result:
Translating Coset in a finite solvable group is reducible to instances of
Translating Coset in and , for appropriate normal subgroups of
. Our self-reducibility framework combined with Kuperberg's subexponential
quantum algorithm for solving Hidden Translation in any abelian group, leads to
subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in
any solvable group.Comment: Journal version: change of title and several minor update
Finding central decompositions of p-groups
Polynomial-time algorithms are given to find a central decomposition of
maximum size for a finite p-group of class 2 and for a nilpotent Lie ring of
class 2. The algorithms use Las Vegas probabilistic routines to compute the
structure of finite *-rings and also the Las Vegas C-MeatAxe. When p is small,
the probabilistic methods can be replaced by deterministic polynomial-time
algorithms.
The methods introduce new group isomorphism invariants including new
characteristic subgroups.Comment: 28 page
Quantum Computing on Lattices using Global Two-Qubit Gate
We study the computation power of lattices composed of two dimensional
systems (qubits) on which translationally invariant global two-qubit gates can
be performed. We show that if a specific set of 6 global two qubit gates can be
performed, and if the initial state of the lattice can be suitably chosen, then
a quantum computer can be efficiently simulatedComment: 9 page
Pedagogical conditions of training future managers of foreign economic activity for cross-cultural communication
The study reveals the pedagogical conditions of training future managers of foreign economic activity for cross-cultural communication. The analysis of the scientific research in this field has shown that the essence of the pedagogical conditions has not been generalized or systemized yet. On the base of the invariable vectors of the pedagogical process, the pedagogical conditions of training future managers of foreign economic activity for cross-cultural communication are outlined. They are: creating multicultural surrounding in the process of training managers for gaining experience in cross-cultural communication (organization); adaptive gradual management of the educational activity with due regards for personal, professional, communicative qualities of future managers of foreign economic activity (management); subject and subject interaction, directed at the optimal management of cross-cultural conflicts (communication)
On the expressive power of read-once determinants
We introduce and study the notion of read- projections of the determinant:
a polynomial is called a {\it read-
projection of determinant} if , where entries of matrix are
either field elements or variables such that each variable appears at most
times in . A monomial set is said to be expressible as read-
projection of determinant if there is a read- projection of determinant
such that the monomial set of is equal to . We obtain basic results
relating read- determinantal projections to the well-studied notion of
determinantal complexity. We show that for sufficiently large , the permanent polynomial and the elementary symmetric
polynomials of degree on variables for are
not expressible as read-once projection of determinant, whereas
and are expressible as read-once projections of determinant. We
also give examples of monomial sets which are not expressible as read-once
projections of determinant
Symbolic determinant identity testing and non-commutative ranks of matrix Lie algebras
One approach to make progress on the symbolic determinant identity testing
(SDIT) problem is to study the structure of singular matrix spaces. After
settling the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson,
Found. Comput. Math. 2020; Ivanyos-Qiao-Subrahmanyam, Comput. Complex. 2018), a
natural next step is to understand singular matrix spaces whose non-commutative
rank is full. At present, examples of such matrix spaces are mostly sporadic,
so it is desirable to discover them in a more systematic way.
In this paper, we make a step towards this direction, by studying the family
of matrix spaces that are closed under the commutator operation, that is matrix
Lie algebras. On the one hand, we demonstrate that matrix Lie algebras over the
complex number field give rise to singular matrix spaces with full
non-commutative ranks. On the other hand, we show that SDIT of such spaces can
be decided in deterministic polynomial time. Moreover, we give a
characterization for the matrix Lie algebras to yield a matrix space possessing
singularity certificates as studied by Lov'asz (B. Braz. Math. Soc., 1989) and
Raz and Wigderson (Building Bridges II, 2019).Comment: 23 page
Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem
Schur duality decomposes many copies of a quantum state into subspaces
labeled by partitions, a decomposition with applications throughout quantum
information theory. Here we consider applying Schur duality to the problem of
distinguishing coset states in the standard approach to the hidden subgroup
problem. We observe that simply measuring the partition (a procedure we call
weak Schur sampling) provides very little information about the hidden
subgroup. Furthermore, we show that under quite general assumptions, even a
combination of weak Fourier sampling and weak Schur sampling fails to identify
the hidden subgroup. We also prove tight bounds on how many coset states are
required to solve the hidden subgroup problem by weak Schur sampling, and we
relate this question to a quantum version of the collision problem.Comment: 21 page