6,659 research outputs found
Classical simulations of Abelian-group normalizer circuits with intermediate measurements
Quantum normalizer circuits were recently introduced as generalizations of
Clifford circuits [arXiv:1201.4867]: a normalizer circuit over a finite Abelian
group is composed of the quantum Fourier transform (QFT) over G, together
with gates which compute quadratic functions and automorphisms. In
[arXiv:1201.4867] it was shown that every normalizer circuit can be simulated
efficiently classically. This result provides a nontrivial example of a family
of quantum circuits that cannot yield exponential speed-ups in spite of usage
of the QFT, the latter being a central quantum algorithmic primitive. Here we
extend the aforementioned result in several ways. Most importantly, we show
that normalizer circuits supplemented with intermediate measurements can also
be simulated efficiently classically, even when the computation proceeds
adaptively. This yields a generalization of the Gottesman-Knill theorem (valid
for n-qubit Clifford operations [quant-ph/9705052, quant-ph/9807006] to quantum
circuits described by arbitrary finite Abelian groups. Moreover, our
simulations are twofold: we present efficient classical algorithms to sample
the measurement probability distribution of any adaptive-normalizer
computation, as well as to compute the amplitudes of the state vector in every
step of it. Finally we develop a generalization of the stabilizer formalism
[quant-ph/9705052, quant-ph/9807006] relative to arbitrary finite Abelian
groups: for example we characterize how to update stabilizers under generalized
Pauli measurements and provide a normal form of the amplitudes of generalized
stabilizer states using quadratic functions and subgroup cosets.Comment: 26 pages+appendices. Title has changed in this second version. To
appear in Quantum Information and Computation, Vol.14 No.3&4, 201
Efficient quantum processing of ideals in finite rings
Suppose we are given black-box access to a finite ring R, and a list of
generators for an ideal I in R. We show how to find an additive basis
representation for I in poly(log |R|) time. This generalizes a recent quantum
algorithm of Arvind et al. which finds a basis representation for R itself. We
then show that our algorithm is a useful primitive allowing quantum computers
to rapidly solve a wide variety of problems regarding finite rings. In
particular we show how to test whether two ideals are identical, find their
intersection, find their quotient, prove whether a given ring element belongs
to a given ideal, prove whether a given element is a unit, and if so find its
inverse, find the additive and multiplicative identities, compute the order of
an ideal, solve linear equations over rings, decide whether an ideal is
maximal, find annihilators, and test the injectivity and surjectivity of ring
homomorphisms. These problems appear to be hard classically.Comment: 5 page
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
An Efficient Quantum Algorithm for some Instances of the Group Isomorphism Problem
In this paper we consider the problem of testing whether two finite groups
are isomorphic. Whereas the case where both groups are abelian is well
understood and can be solved efficiently, very little is known about the
complexity of isomorphism testing for nonabelian groups. Le Gall has
constructed an efficient classical algorithm for a class of groups
corresponding to one of the most natural ways of constructing nonabelian groups
from abelian groups: the groups that are extensions of an abelian group by
a cyclic group with the order of coprime with . More precisely,
the running time of that algorithm is almost linear in the order of the input
groups. In this paper we present a quantum algorithm solving the same problem
in time polynomial in the logarithm of the order of the input groups. This
algorithm works in the black-box setting and is the first quantum algorithm
solving instances of the nonabelian group isomorphism problem exponentially
faster than the best known classical algorithms.Comment: 20 pages; this is the full version of a paper that will appear in the
Proceedings of the 27th International Symposium on Theoretical Aspects of
Computer Science (STACS 2010
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