45 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
Difficult instances of the counting problem for 2-quantum-SAT are very atypical
The problem 2-quantum-satisfiability (2-QSAT) is the generalisation of the
2-CNF-SAT problem to quantum bits, and is equivalent to determining whether or
not a spin-1/2 Hamiltonian with two-body terms is frustration-free. Similarly
to the classical problem 2-SAT, the counting problem #2-QSAT of determining the
size (i.e. the dimension) of the set of satisfying states is #P-complete.
However, if we consider random instances of #2-QSAT in which constraints are
sampled from the Haar measure, intractible instances have measure zero. An
apparent reason for this is that almost all two-qubit constraints are
entangled, which more readily give rise to long-range constraints.
We investigate under which conditions product constraints also give rise to
efficiently solvable families of #2-QSAT instances. We consider #2-QSAT
involving only discrete distributions over tensor product operators, which
interpolates between classical #2-SAT and #2-QSAT involving arbitrary product
constraints. We find that such instances of #2-QSAT, defined on Erdos--Renyi
graphs or bond-percolated lattices, are asymptotically almost surely
efficiently solvable except to the extent that they are biased to resemble
monotone instances of #2-SAT.Comment: 25 pages, 2 figures. Fixed errata concerning frustrated figure eights
(relating to the junction probability), and the threshold for a decoupled
regime on bond-percolated 3D cubic lattice