269 research outputs found
A counterexample to the triangle conjecture
AbstractThe triangle conjecture sets a bound on the cardinality of a code formed by words of the form aibaj. A counterexample exceeding that bound is given. This also disproves a stronger conjecture that every code is commutatively equivalent to a prefix code
A lower bound for the length of a partial transversal in a latin square
AbstractIt is proved that every n Ă n Latin square has a partial transversal of length at least n â 5.53(log n)2
The Equivalence of Sampling and Searching
In a sampling problem, we are given an input x, and asked to sample
approximately from a probability distribution D_x. In a search problem, we are
given an input x, and asked to find a member of a nonempty set A_x with high
probability. (An example is finding a Nash equilibrium.) In this paper, we use
tools from Kolmogorov complexity and algorithmic information theory to show
that sampling and search problems are essentially equivalent. More precisely,
for any sampling problem S, there exists a search problem R_S such that, if C
is any "reasonable" complexity class, then R_S is in the search version of C if
and only if S is in the sampling version. As one application, we show that
SampP=SampBQP if and only if FBPP=FBQP: in other words, classical computers can
efficiently sample the output distribution of every quantum circuit, if and
only if they can efficiently solve every search problem that quantum computers
can solve. A second application is that, assuming a plausible conjecture, there
exists a search problem R that can be solved using a simple linear-optics
experiment, but that cannot be solved efficiently by a classical computer
unless the polynomial hierarchy collapses. That application will be described
in a forthcoming paper with Alex Arkhipov on the computational complexity of
linear optics.Comment: 16 page
Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision
The quantum query complexity of Boolean matrix multiplication is typically
studied as a function of the matrix dimension, n, as well as the number of 1s
in the output, \ell. We prove an upper bound of O (n\sqrt{\ell}) for all values
of \ell. This is an improvement over previous algorithms for all values of
\ell. On the other hand, we show that for any \eps < 1 and any \ell <= \eps
n^2, there is an \Omega(n\sqrt{\ell}) lower bound for this problem, showing
that our algorithm is essentially tight.
We first reduce Boolean matrix multiplication to several instances of graph
collision. We then provide an algorithm that takes advantage of the fact that
the underlying graph in all of our instances is very dense to find all graph
collisions efficiently
A family of sure-success quantum algorithms for solving a generalized Grover search problem
This work considers a generalization of Grover's search problem, viz., to
find any one element in a set of acceptable choices which constitute a fraction
f of the total number of choices in an unsorted data base. An infinite family
of sure-success quantum algorithms are introduced here to solve this problem,
each member for a different range of f. The nth member of this family involves
n queries of the data base, and so the lowest few members of this family should
be very convenient algorithms within their ranges of validity. The even member
{A}_{2n} of the family covers ever larger range of f for larger n, which is
expected to become the full range 0 infinity.Comment: 8 pages, including 4 figures in 4 page
Correcting the effects of spontaneous emission on cold trapped ions
We propose two quantum error correction schemes which increase the maximum
storage time for qubits in a system of cold trapped ions, using a minimal
number of ancillary qubits. Both schemes consider only the errors introduced by
the decoherence due to spontaneous emission from the upper levels of the ions.
Continuous monitoring of the ion fluorescence is used in conjunction with
selective coherent feedback to eliminate these errors immediately following
spontaneous emission events, and the conditional time evolution between quantum
jumps is removed by symmetrizing the quantum codewords.Comment: 19 pages; 2 figures; RevTex; The quantum codewords are extended to
achieve invariance under the conditional time evolution between jump
Decoherence of geometric phase gates
We consider the effects of certain forms of decoherence applied to both
adiabatic and non-adiabatic geometric phase quantum gates. For a single qubit
we illustrate path-dependent sensitivity to anisotropic noise and for two
qubits we quantify the loss of entanglement as a function of decoherence.Comment: 4 pages, 3 figure
Quantum Probabilistic Subroutines and Problems in Number Theory
We present a quantum version of the classical probabilistic algorithms
la Rabin. The quantum algorithm is based on the essential use of
Grover's operator for the quantum search of a database and of Shor's Fourier
transform for extracting the periodicity of a function, and their combined use
in the counting algorithm originally introduced by Brassard et al. One of the
main features of our quantum probabilistic algorithm is its full unitarity and
reversibility, which would make its use possible as part of larger and more
complicated networks in quantum computers. As an example of this we describe
polynomial time algorithms for studying some important problems in number
theory, such as the test of the primality of an integer, the so called 'prime
number theorem' and Hardy and Littlewood's conjecture about the asymptotic
number of representations of an even integer as a sum of two primes.Comment: 9 pages, RevTex, revised version, accepted for publication on PRA:
improvement in use of memory space for quantum primality test algorithm
further clarified and typos in the notation correcte
Partition-function zeros of spherical spin glasses and their relevance to chaos
We investigate partition-function zeros of the many-body interacting
spherical spin glass, the so-called -spin spherical model, with respect to
the complex temperature in the thermodynamic limit. We use the replica method
and extend the procedure of the replica symmetry breaking ansatz to be
applicable in the complex-parameter case. We derive the phase diagrams in the
complex-temperature plane and calculate the density of zeros in each phase.
Near the imaginary axis away from the origin, there is a replica symmetric
phase having a large density. On the other hand, we observe no density in the
spin-glass phases, irrespective of the replica symmetry breaking. We speculate
that this suggests the absence of the temperature chaos. To confirm this, we
investigate the multiple many-body interacting case which is known to exhibit
the chaos effect. The result shows that the density of zeros actually takes
finite values in the spin-glass phase, even on the real axis. These
observations indicate that the density of zeros is more closely connected to
the chaos effect than the replica symmetry breaking.Comment: 22 pages, 8 figure
No-relationship between impossibility of faster-than-light quantum communication and distinction of ensembles with the same density matrix
It has been claimed in the literature that impossibility of faster-than-light
quantum communication has an origin of indistinguishability of ensembles with
the same density matrix. We show that the two concepts are not related. We
argue that: 1) even with an ideal single-atom-precision measurement, it is
generally impossible to produce two ensembles with exactly the same density
matrix; or 2) to produce ensembles with the same density matrix, classical
communication is necessary. Hence the impossibility of faster-than-light
communication does not imply the indistinguishability of ensembles with the
same density matrix.Comment: 4 pages and 3 figure
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