16,765 research outputs found
An Algorithm for Constructing Polynomial Systems Whose Solution Space Characterizes Quantum Circuits
An algorithm and its first implementation in C# are presented for assembling
arbitrary quantum circuits on the base of Hadamard and Toffoli gates and for
constructing multivariate polynomial systems over the finite field Z_2 arising
when applying the Feynman's sum-over-paths approach to quantum circuits. The
matrix elements determined by a circuit can be computed by counting the number
of common roots in Z_2 for the polynomial system associated with the circuit.
To determine the number of solutions in Z_2 for the output polynomial system,
one can use the Groebner bases method and the relevant algorithms for computing
Groebner bases.Comment: 10 pages, 9 Postscript figures, report presented on QI 200
A Software Package to Construct Polynomial Sets over Z_2 for Determining the Output of Quantum Computations
A C# package is presented that allows a user for an input quantum circuit to
generate a set of multivariate polynomials over the finite field Z_2 whose
total number of solutions in Z_2 determines the output of the quantum
computation defined by the circuit. The generated polynomial system can further
be converted to the canonical Groebner basis form which provides a universal
algorithmic tool for counting the number of common roots of the polynomials.Comment: 5 pages, 4 Postscript figures, report presented on ACAT 200
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
On solving systems of random linear disequations
An important subcase of the hidden subgroup problem is equivalent to the
shift problem over abelian groups. An efficient solution to the latter problem
would serve as a building block of quantum hidden subgroup algorithms over
solvable groups. The main idea of a promising approach to the shift problem is
reduction to solving systems of certain random disequations in finite abelian
groups. The random disequations are actually generalizations of linear
functions distributed nearly uniformly over those not containing a specific
group element in the kernel. In this paper we give an algorithm which finds the
solutions of a system of N random linear disequations in an abelian p-group A
in time polynomial in N, where N=(log|A|)^{O(q)}, and q is the exponent of A.Comment: 13 page
Finite-size corrections to the rotating string and the winding state
We compute higher order finite size corrections to the energies of the
circular rotating string on AdS_5 x S^5, of its orbifolded generalization on
AdS_5 x S^5/Z_M and of the winding state which is obtained as the limit of the
orbifolded circular string solution when J -> infinity and J/M^2 is kept fixed.
We solve, at the first order in lambda'=lambda/J^2, where lambda is the 't
Hooft coupling, the Bethe equations that describe the anomalous dimensions of
the corresponding gauge dual operators in an expansion in m/K, where m is the
winding number and K is the "magnon number", and to all orders in the angular
momentum J. The solution for the circular rotating string and for the winding
state can be matched to the energy computed from an effective quantum
Landau-Lifshitz model beyond the first order correction in 1/J. For the leading
1/J corrections to the circular rotating string in m^2 and m^4 and for the
subleading 1/J^2 corrections to the m^2 term, we find agreement. For the
winding state we match the energy completely up to, and including, the order
1/J^2 finite-size corrections. The solution of the Bethe equations
corresponding to the spinning closed string is also provided in an expansion in
m/K and to all orders in J.Comment: v2: 33 pages, misprints corrected, references added, version accepted
for publication in JHE
Higher order terms for the quantum evolution of a Wick observable within the Hepp method
The Hepp method is the coherent state approach to the mean field dynamics for
bosons or to the semiclassical propagation. A key point is the asymptotic
evolution of Wick observables under the evolution given by a time-dependent
quadratic Hamiltonian. This article provides a complete expansion with respect
to the small parameter \epsilon > 0 which makes sense within the
infinite-dimensional setting and fits with finite-dimensional formulae
From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups
We approach the hidden subgroup problem by performing the so-called pretty
good measurement on hidden subgroup states. For various groups that can be
expressed as the semidirect product of an abelian group and a cyclic group, we
show that the pretty good measurement is optimal and that its probability of
success and unitary implementation are closely related to an average-case
algebraic problem. By solving this problem, we find efficient quantum
algorithms for a number of nonabelian hidden subgroup problems, including some
for which no efficient algorithm was previously known: certain metacyclic
groups as well as all groups of the form (Z_p)^r X| Z_p for fixed r (including
the Heisenberg group, r=2). In particular, our results show that entangled
measurements across multiple copies of hidden subgroup states can be useful for
efficiently solving the nonabelian HSP.Comment: 18 pages; v2: updated references on optimal measuremen
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