44,506 research outputs found
Supplementarity is Necessary for Quantum Diagram Reasoning
The ZX-calculus is a powerful diagrammatic language for quantum mechanics and
quantum information processing. We prove that its \pi/4-fragment is not
complete, in other words the ZX-calculus is not complete for the so called
"Clifford+T quantum mechanics". The completeness of this fragment was one of
the main open problems in categorical quantum mechanics, a programme initiated
by Abramsky and Coecke. The ZX-calculus was known to be incomplete for quantum
mechanics. On the other hand, its \pi/2-fragment is known to be complete, i.e.
the ZX-calculus is complete for the so called "stabilizer quantum mechanics".
Deciding whether its \pi/4-fragment is complete is a crucial step in the
development of the ZX-calculus since this fragment is approximately universal
for quantum mechanics, contrary to the \pi/2-fragment. To establish our
incompleteness result, we consider a fairly simple property of quantum states
called supplementarity. We show that supplementarity can be derived in the
ZX-calculus if and only if the angles involved in this equation are multiples
of \pi/2. In particular, the impossibility to derive supplementarity for \pi/4
implies the incompleteness of the ZX-calculus for Clifford+T quantum mechanics.
As a consequence, we propose to add the supplementarity to the set of rules of
the ZX-calculus. We also show that if a ZX-diagram involves antiphase twins,
they can be merged when the ZX-calculus is augmented with the supplementarity
rule. Merging antiphase twins makes diagrammatic reasoning much easier and
provides a purely graphical meaning to the supplementarity rule.Comment: Generalised proof and graphical interpretation. 16 pages, submitte
Causal Dynamics of Discrete Surfaces
We formalize the intuitive idea of a labelled discrete surface which evolves
in time, subject to two natural constraints: the evolution does not propagate
information too fast; and it acts everywhere the same.Comment: In Proceedings DCM 2013, arXiv:1403.768
Towards a Minimal Stabilizer ZX-calculus
The stabilizer ZX-calculus is a rigorous graphical language for reasoning
about quantum mechanics. The language is sound and complete: one can transform
a stabilizer ZX-diagram into another one using the graphical rewrite rules if
and only if these two diagrams represent the same quantum evolution or quantum
state. We previously showed that the stabilizer ZX-calculus can be simplified
by reducing the number of rewrite rules, without losing the property of
completeness [Backens, Perdrix & Wang, EPTCS 236:1--20, 2017]. Here, we show
that most of the remaining rules of the language are indeed necessary. We do
however leave as an open question the necessity of two rules. These include,
surprisingly, the bialgebra rule, which is an axiomatisation of
complementarity, the cornerstone of the ZX-calculus. Furthermore, we show that
a weaker ambient category -- a braided autonomous category instead of the usual
compact closed category -- is sufficient to recover the meta rule 'only
connectivity matters', even without assuming any symmetries of the generators.Comment: 29 pages, minor updates for v
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Optimal coverage multi-path scheduling scheme with multiple mobile sinks for WSNs
Wireless Sensor Networks (WSNs) are usually formed with many tiny sensors which are randomly deployed within sensing field for target monitoring. These sensors can transmit their monitored data to the sink in a multi-hop communication manner. However, the ‘hot spots’ problem will be caused since nodes near sink will consume more energy during forwarding. Recently, mobile sink based technology provides an alternative solution for the long-distance communication and sensor nodes only need to use single hop communication to the mobile sink during data transmission. Even though it is difficult to consider many network metrics such as sensor position, residual energy and coverage rate etc., it is still very important to schedule a reasonable moving trajectory for the mobile sink. In this paper, a novel trajectory scheduling method based on coverage rate for multiple mobile sinks (TSCR-M) is presented especially for large-scale WSNs. An improved particle swarm optimization (PSO) combined with mutation operator is introduced to search the parking positions with optimal coverage rate. Then the genetic algorithm (GA) is adopted to schedule the moving trajectory for multiple mobile sinks. Extensive simulations are performed to validate the performance of our proposed method
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