38 research outputs found
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
Fibred Coalgebraic Logic and Quantum Protocols
Motivated by applications in modelling quantum systems using coalgebraic
techniques, we introduce a fibred coalgebraic logic. Our approach extends the
conventional predicate lifting semantics with additional modalities relating
conditions on different fibres. As this fibred setting will typically involve
multiple signature functors, the logic incorporates a calculus of modalities
enabling the construction of new modalities using various composition
operations. We extend the semantics of coalgebraic logic to this setting, and
prove that this extension respects behavioural equivalence.
We show how properties of the semantics of modalities are preserved under
composition operations, and then apply the calculational aspect of our logic to
produce an expressive set of modalities for reasoning about quantum systems,
building these modalities up from simpler components. We then demonstrate how
these modalities can describe some standard quantum protocols. The novel
features of our logic are shown to allow for a uniform description of unitary
evolution, and support local reasoning such as "Alice's qubit satisfies
condition" as is common when discussing quantum protocols.Comment: In Proceedings QPL 2013, arXiv:1412.791
Certainty and Uncertainty in Quantum Information Processing
This survey, aimed at information processing researchers, highlights
intriguing but lesser known results, corrects misconceptions, and suggests
research areas. Themes include: certainty in quantum algorithms; the "fewer
worlds" theory of quantum mechanics; quantum learning; probability theory
versus quantum mechanics.Comment: Invited paper accompanying invited talk to AAAI Spring Symposium
2007. Comments, corrections, and suggestions would be most welcom
Weakly complete axiomatization of exogenous quantum propositional logic
A weakly complete finitary axiomatization for EQPL (exogenous quantum
propositional logic) is presented. The proof is carried out using a non trivial
extension of the Fagin-Halpern-Megiddo technique together with three Henkin
style completions.Comment: 28 page