4,411 research outputs found
Generating binary partial Hadamard matrices
This paper deals with partial binary Hadamard matrices. Although there is a fast simple
way to generate about a half (which is the best asymptotic bound known so far, see de
Launey (2000) and de Launey and Gordon (2001)) of a full Hadamard matrix, it cannot
provide larger partial Hadamard matrices beyond this bound. In order to overcome such
a limitation, we introduce a particular subgraph Gt of Itoâs Hadamard Graph Î(4t) (Ito,
1985), and study some of its properties,which facilitates that a procedure may be designed
for constructing large partial Hadamard matrices. The key idea is translating the problem
of extending a given clique in Gt into a Constraint Satisfaction Problem, to be solved
by Minion (Gent et al., 2006). Actually, iteration of this process ends with large partial
Hadamard matrices, usually beyond the bound of half a full Hadamard matrix, at least as
our computation capabilities have led us thus far
Searching for partial Hadamard matrices
Three algorithms looking for pretty large partial Hadamard ma-
trices are described. Here âlargeâ means that hopefully about a third of a
Hadamard matrix (which is the best asymptotic result known so far, [8]) is
achieved. The first one performs some kind of local exhaustive search, and
consequently is expensive from the time consuming point of view. The second
one comes from the adaptation of the best genetic algorithm known so far
searching for cliques in a graph, due to Singh and Gupta [21]. The last one
consists in another heuristic search, which prioritizes the required processing
time better than the final size of the partial Hadamard matrix to be obtained. In
all cases, the key idea is characterizing the adjacency properties of vertices in a
particular subgraph Gt of Itoâs Hadamard Graph (4t) [18], since cliques of
order m in Gt can be seen as (m + 3) Ă 4t partial Hadamard matrices.Ministerio de Ciencia e InnovaciĂłn MTM2008-06578Junta de AndalucĂa FQM-016Junta de AndalucĂa P07-FQM-0298
Qdensity - a Mathematica Quantum Computer Simulation
This Mathematica 5.2 package~\footnote{QDENSITY is available at
http://www.pitt.edu/~tabakin/QDENSITY} is a simulation of a Quantum Computer.
The program provides a modular, instructive approach for generating the basic
elements that make up a quantum circuit. The main emphasis is on using the
density matrix, although an approach using state vectors is also implemented in
the package. The package commands are defined in {\it Qdensity.m} which
contains the tools needed in quantum circuits, e.g. multiqubit kets,
projectors, gates, etc. Selected examples of the basic commands are presented
here and a tutorial notebook, {\it Tutorial.nb} is provided with the package
(available on our website) that serves as a full guide to the package. Finally,
application is made to a variety of relevant cases, including Teleportation,
Quantum Fourier transform, Grover's search and Shor's algorithm, in separate
notebooks: {\it QFT.nb}, {\it Teleportation.nb}, {\it Grover.nb} and {\it
Shor.nb} where each algorithm is explained in detail. Finally, two examples of
the construction and manipulation of cluster states, which are part of ``one
way computing" ideas, are included as an additional tool in the notebook {\it
Cluster.nb}. A Mathematica palette containing most commands in QDENSITY is also
included: {\it QDENSpalette.nb} .Comment: The Mathematica 5+ package is available at:
http://www.pitt.edu/~tabakin/QDENSITY/QDENSITY.htm Minor corrections,
accepted in Computer Physics Communication
Improved Simulation of Stabilizer Circuits
The Gottesman-Knill theorem says that a stabilizer circuit -- that is, a
quantum circuit consisting solely of CNOT, Hadamard, and phase gates -- can be
simulated efficiently on a classical computer. This paper improves that theorem
in several directions. First, by removing the need for Gaussian elimination, we
make the simulation algorithm much faster at the cost of a factor-2 increase in
the number of bits needed to represent a state. We have implemented the
improved algorithm in a freely-available program called CHP
(CNOT-Hadamard-Phase), which can handle thousands of qubits easily. Second, we
show that the problem of simulating stabilizer circuits is complete for the
classical complexity class ParityL, which means that stabilizer circuits are
probably not even universal for classical computation. Third, we give efficient
algorithms for computing the inner product between two stabilizer states,
putting any n-qubit stabilizer circuit into a "canonical form" that requires at
most O(n^2/log n) gates, and other useful tasks. Fourth, we extend our
simulation algorithm to circuits acting on mixed states, circuits containing a
limited number of non-stabilizer gates, and circuits acting on general
tensor-product initial states but containing only a limited number of
measurements.Comment: 15 pages. Final version with some minor updates and corrections.
Software at http://www.scottaaronson.com/ch
On minors of maximal determinant matrices
By an old result of Cohn (1965), a Hadamard matrix of order n has no proper
Hadamard submatrices of order m > n/2. We generalise this result to maximal
determinant submatrices of Hadamard matrices, and show that an interval of
length asymptotically equal to n/2 is excluded from the allowable orders. We
make a conjecture regarding a lower bound for sums of squares of minors of
maximal determinant matrices, and give evidence in support of the conjecture.
We give tables of the values taken by the minors of all maximal determinant
matrices of orders up to and including 21 and make some observations on the
data. Finally, we describe the algorithms that were used to compute the tables.Comment: 35 pages, 43 tables, added reference to Cohn in v
Construction of a Large Class of Deterministic Sensing Matrices that Satisfy a Statistical Isometry Property
Compressed Sensing aims to capture attributes of -sparse signals using
very few measurements. In the standard Compressed Sensing paradigm, the
\m\times \n measurement matrix \A is required to act as a near isometry on
the set of all -sparse signals (Restricted Isometry Property or RIP).
Although it is known that certain probabilistic processes generate \m \times
\n matrices that satisfy RIP with high probability, there is no practical
algorithm for verifying whether a given sensing matrix \A has this property,
crucial for the feasibility of the standard recovery algorithms. In contrast
this paper provides simple criteria that guarantee that a deterministic sensing
matrix satisfying these criteria acts as a near isometry on an overwhelming
majority of -sparse signals; in particular, most such signals have a unique
representation in the measurement domain. Probability still plays a critical
role, but it enters the signal model rather than the construction of the
sensing matrix. We require the columns of the sensing matrix to form a group
under pointwise multiplication. The construction allows recovery methods for
which the expected performance is sub-linear in \n, and only quadratic in
\m; the focus on expected performance is more typical of mainstream signal
processing than the worst-case analysis that prevails in standard Compressed
Sensing. Our framework encompasses many families of deterministic sensing
matrices, including those formed from discrete chirps, Delsarte-Goethals codes,
and extended BCH codes.Comment: 16 Pages, 2 figures, to appear in IEEE Journal of Selected Topics in
Signal Processing, the special issue on Compressed Sensin
- âŠ