11,112 research outputs found
Tight Upper Bounds for Streett and Parity Complementation
Complementation of finite automata on infinite words is not only a
fundamental problem in automata theory, but also serves as a cornerstone for
solving numerous decision problems in mathematical logic, model-checking,
program analysis and verification. For Streett complementation, a significant
gap exists between the current lower bound and upper
bound , where is the state size, is the number of
Streett pairs, and can be as large as . Determining the complexity
of Streett complementation has been an open question since the late '80s. In
this paper show a complementation construction with upper bound for and for ,
which matches well the lower bound obtained in \cite{CZ11a}. We also obtain a
tight upper bound for parity complementation.Comment: Corrected typos. 23 pages, 3 figures. To appear in the 20th
Conference on Computer Science Logic (CSL 2011
Pivoting makes the ZX-calculus complete for real stabilizers
We show that pivoting property of graph states cannot be derived from the
axioms of the ZX-calculus, and that pivoting does not imply local
complementation of graph states. Therefore the ZX-calculus augmented with
pivoting is strictly weaker than the calculus augmented with the Euler
decomposition of the Hadamard gate. We derive an angle-free version of the
ZX-calculus and show that it is complete for real stabilizer quantum mechanics.Comment: In Proceedings QPL 2013, arXiv:1412.791
Graph States, Pivot Minor, and Universality of (X,Z)-measurements
The graph state formalism offers strong connections between quantum
information processing and graph theory. Exploring these connections, first we
show that any graph is a pivot-minor of a planar graph, and even a pivot minor
of a triangular grid. Then, we prove that the application of measurements in
the (X,Z)-plane over graph states represented by triangular grids is a
universal measurement-based model of quantum computation. These two results are
in fact two sides of the same coin, the proof of which is a combination of
graph theoretical and quantum information techniques
Complexity of Graph State Preparation
The graph state formalism is a useful abstraction of entanglement. It is used
in some multipartite purification schemes and it adequately represents
universal resources for measurement-only quantum computation. We focus in this
paper on the complexity of graph state preparation. We consider the number of
ancillary qubits, the size of the primitive operators, and the duration of
preparation. For each lexicographic order over these parameters we give upper
and lower bounds for the complexity of graph state preparation. The first part
motivates our work and introduces basic notions and notations for the study of
graph states. Then we study some graph properties of graph states,
characterizing their minimal degree by local unitary transformations, we
propose an algorithm to reduce the degree of a graph state, and show the
relationship with Sutner sigma-game.
These properties are used in the last part, where algorithms and lower bounds
for each lexicographic order over the considered parameters are presented.Comment: 17 page
State of B\"uchi Complementation
Complementation of B\"uchi automata has been studied for over five decades
since the formalism was introduced in 1960. Known complementation constructions
can be classified into Ramsey-based, determinization-based, rank-based, and
slice-based approaches. Regarding the performance of these approaches, there
have been several complexity analyses but very few experimental results. What
especially lacks is a comparative experiment on all of the four approaches to
see how they perform in practice. In this paper, we review the four approaches,
propose several optimization heuristics, and perform comparative
experimentation on four representative constructions that are considered the
most efficient in each approach. The experimental results show that (1) the
determinization-based Safra-Piterman construction outperforms the other three
in producing smaller complements and finishing more tasks in the allocated time
and (2) the proposed heuristics substantially improve the Safra-Piterman and
the slice-based constructions.Comment: 28 pages, 4 figures, a preliminary version of this paper appeared in
the Proceedings of the 15th International Conference on Implementation and
Application of Automata (CIAA
Expressing an observer in preferred coordinates by transforming an injective immersion into a surjective diffeomorphism
When designing observers for nonlinear systems, the dynamics of the given
system and of the designed observer are usually not expressed in the same
coordinates or even have states evolving in different spaces. In general, the
function, denoted (or its inverse, denoted ) giving one state in
terms of the other is not explicitly known and this creates implementation
issues. We propose to round this problem by expressing the observer dynamics in
the the same coordinates as the given system. But this may impose to add extra
coordinates, problem that we call augmentation. This may also impose to modify
the domain or the range of the augmented" or , problem that we
call extension. We show that the augmentation problem can be solved partly by a
continuous completion of a free family of vectors and that the extension
problem can be solved by a function extension making the image of the extended
function the whole space. We also show how augmentation and extension can be
done without modifying the observer dynamics and therefore with maintaining
convergence.Several examples illustrate our results.Comment: Submitted for publication in SIAM Journal of Control and Optimizatio
The ZX-calculus is complete for stabilizer quantum mechanics
The ZX-calculus is a graphical calculus for reasoning about quantum systems
and processes. It is known to be universal for pure state qubit quantum
mechanics, meaning any pure state, unitary operation and post-selected pure
projective measurement can be expressed in the ZX-calculus. The calculus is
also sound, i.e. any equality that can be derived graphically can also be
derived using matrix mechanics. Here, we show that the ZX-calculus is complete
for pure qubit stabilizer quantum mechanics, meaning any equality that can be
derived using matrices can also be derived pictorially. The proof relies on
bringing diagrams into a normal form based on graph states and local Clifford
operations.Comment: 26 page
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