2,592 research outputs found
On the complexity of solving linear congruences and computing nullspaces modulo a constant
We consider the problems of determining the feasibility of a linear
congruence, producing a solution to a linear congruence, and finding a spanning
set for the nullspace of an integer matrix, where each problem is considered
modulo an arbitrary constant k>1. These problems are known to be complete for
the logspace modular counting classes {Mod_k L} = {coMod_k L} in special case
that k is prime (Buntrock et al, 1992). By considering variants of standard
logspace function classes --- related to #L and functions computable by UL
machines, but which only characterize the number of accepting paths modulo k
--- we show that these problems of linear algebra are also complete for
{coMod_k L} for any constant k>1.
Our results are obtained by defining a class of functions FUL_k which are low
for {Mod_k L} and {coMod_k L} for k>1, using ideas similar to those used in the
case of k prime in (Buntrock et al, 1992) to show closure of Mod_k L under NC^1
reductions (including {Mod_k L} oracle reductions). In addition to the results
above, we briefly consider the relationship of the class FUL_k for arbitrary
moduli k to the class {F.coMod_k L} of functions whose output symbols are
verifiable by {coMod_k L} algorithms; and consider what consequences such a
comparison may have for oracle closure results of the form {Mod_k L}^{Mod_k L}
= {Mod_k L} for composite k.Comment: 17 pages, one Appendix; minor corrections and revisions to
presentation, new observations regarding the prospect of oracle closures.
Comments welcom
A linearized stabilizer formalism for systems of finite dimension
The stabilizer formalism is a scheme, generalizing well-known techniques
developed by Gottesman [quant-ph/9705052] in the case of qubits, to efficiently
simulate a class of transformations ("stabilizer circuits", which include the
quantum Fourier transform and highly entangling operations) on standard basis
states of d-dimensional qudits. To determine the state of a simulated system,
existing treatments involve the computation of cumulative phase factors which
involve quadratic dependencies. We present a simple formalism in which Pauli
operators are represented using displacement operators in discrete phase space,
expressing the evolution of the state via linear transformations modulo D <=
2d. We thus obtain a simple proof that simulating stabilizer circuits on n
qudits, involving any constant number of measurement rounds, is complete for
the complexity class coMod_{d}L and may be simulated by O(log(n)^2)-depth
boolean circuits for any constant d >= 2.Comment: 25 pages, 3 figures. Reorganized to collect complexity results; some
corrections and elaborations of technical results. Differs slightly from the
version to be published (fixed typos, changes of wording to accommodate page
breaks for a different article format). To appear as QIC vol 13 (2013),
pp.73--11
Quantum linear network coding as one-way quantum computation
Network coding is a technique to maximize communication rates within a
network, in communication protocols for simultaneous multi-party transmission
of information. Linear network codes are examples of such protocols in which
the local computations performed at the nodes in the network are limited to
linear transformations of their input data (represented as elements of a ring,
such as the integers modulo 2). The quantum linear network coding protocols of
Kobayashi et al [arXiv:0908.1457 and arXiv:1012.4583] coherently simulate
classical linear network codes, using supplemental classical communication. We
demonstrate that these protocols correspond in a natural way to
measurement-based quantum computations with graph states over over qudits
[arXiv:quant-ph/0301052, arXiv:quant-ph/0603226, and arXiv:0704.1263] having a
structure directly related to the network.Comment: 17 pages, 6 figures. Updated to correct an incorrect (albeit
hilarious) reference in the arXiv version of the abstrac
Sharp Quantum vs. Classical Query Complexity Separations
We obtain the strongest separation between quantum and classical query
complexity known to date -- specifically, we define a black-box problem that
requires exponentially many queries in the classical bounded-error case, but
can be solved exactly in the quantum case with a single query (and a polynomial
number of auxiliary operations). The problem is simple to define and the
quantum algorithm solving it is also simple when described in terms of certain
quantum Fourier transforms (QFTs) that have natural properties with respect to
the algebraic structures of finite fields. These QFTs may be of independent
interest, and we also investigate generalizations of them to noncommutative
finite rings.Comment: 13 pages, change in title, improvements in presentation, and minor
corrections. To appear in Algorithmic
One-qubit fingerprinting schemes
Fingerprinting is a technique in communication complexity in which two
parties (Alice and Bob) with large data sets send short messages to a third
party (a referee), who attempts to compute some function of the larger data
sets. For the equality function, the referee attempts to determine whether
Alice's data and Bob's data are the same. In this paper, we consider the
extreme scenario of performing fingerprinting where Alice and Bob both send
either one bit (classically) or one qubit (in the quantum regime) messages to
the referee for the equality problem. Restrictive bounds are demonstrated for
the error probability of one-bit fingerprinting schemes, and show that it is
easy to construct one-qubit fingerprinting schemes which can outperform any
one-bit fingerprinting scheme. The author hopes that this analysis will provide
results useful for performing physical experiments, which may help to advance
implementations for more general quantum communication protocols.Comment: 9 pages; Fixed some typos; changed order of bibliographical
reference
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