6,025 research outputs found
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
The ZX-calculus is incomplete for quantum mechanics
We prove that the ZX-calculus is incomplete for quantum mechanics. We suggest
the addition of a new 'color-swap' rule, of which currently no analytical
formulation is known and which we suspect may be necessary, but not sufficient
to make the ZX-calculus complete.Comment: In Proceedings QPL 2014, arXiv:1412.810
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
Qutrit Dichromatic Calculus and Its Universality
We introduce a dichromatic calculus (RG) for qutrit systems. We show that the
decomposition of the qutrit Hadamard gate is non-unique and not derivable from
the dichromatic calculus. As an application of the dichromatic calculus, we
depict a quantum algorithm with a single qutrit. Since it is not easy to
decompose an arbitrary d by d unitary matrix into Z and X phase gates when d >
2, the proof of the universality of qudit ZX calculus for quantum mechanics is
far from trivial. We construct a counterexample to Ranchin's universality
proof, and give another proof by Lie theory that the qudit ZX calculus contains
all single qudit unitary transformations, which implies that qudit ZX calculus,
with qutrit dichromatic calculus as a special case, is universal for quantum
mechanics.Comment: In Proceedings QPL 2014, arXiv:1412.810
A complete graphical calculus for Spekkens' toy bit theory
While quantum theory cannot be described by a local hidden variable model, it
is nevertheless possible to construct such models that exhibit features
commonly associated with quantum mechanics. These models are also used to
explore the question of {\psi}-ontic versus {\psi}-epistemic theories for
quantum mechanics. Spekkens' toy theory is one such model. It arises from
classical probabilistic mechanics via a limit on the knowledge an observer may
have about the state of a system. The toy theory for the simplest possible
underlying system closely resembles stabilizer quantum mechanics, a fragment of
quantum theory which is efficiently classically simulable but also non-local.
Further analysis of the similarities and differences between those two theories
can thus yield new insights into what distinguishes quantum theory from
classical theories, and {\psi}-ontic from {\psi}-epistemic theories.
In this paper, we develop a graphical language for Spekkens' toy theory.
Graphical languages offer intuitive and rigorous formalisms for the analysis of
quantum mechanics and similar theories. To compare quantum mechanics and a toy
model, it is useful to have similar formalisms for both. We show that our
language fully describes Spekkens' toy theory and in particular, that it is
complete: meaning any equality that can be derived using other formalisms can
also be derived entirely graphically. Our language is inspired by a similar
graphical language for quantum mechanics called the ZX-calculus. Thus Spekkens'
toy bit theory and stabilizer quantum mechanics can be analysed and compared
using analogous graphical formalisms.Comment: Major revisions for v2. 22+7 page
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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