555 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
The ZX-calculus is complete for the single-qubit Clifford+T group
The ZX-calculus is a graphical calculus for reasoning about pure state qubit
quantum mechanics. It is complete for pure qubit stabilizer quantum mechanics,
meaning any equality involving only stabilizer operations that can be derived
using matrices can also be derived pictorially. Stabilizer operations include
the unitary Clifford group, as well as preparation of qubits in the state |0>,
and measurements in the computational basis. For general pure state qubit
quantum mechanics, the ZX-calculus is incomplete: there exist equalities
involving non-stabilizer unitary operations on single qubits which cannot be
derived from the current rule set for the ZX-calculus. Here, we show that the
ZX-calculus for single qubits remains complete upon adding the operator T to
the single-qubit stabilizer operations. This is particularly interesting as the
resulting single-qubit Clifford+T group is approximately universal, i.e. any
unitary single-qubit operator can be approximated to arbitrary accuracy using
only Clifford operators and T.Comment: In Proceedings QPL 2014, arXiv:1412.810
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
ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T quantum mechanics
The ZX-Calculus is a powerful graphical language for quantum mechanics and
quantum information processing. The completeness of the language -- i.e. the
ability to derive any true equation -- is a crucial question. In the quest of a
complete ZX-calculus, supplementarity has been recently proved to be necessary
for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity
consists in merging two subdiagrams when they are parameterized by antipodal
angles. We introduce a generalised supplementarity -- called cyclotomic
supplementarity -- which consists in merging n subdiagrams at once, when the n
angles divide the circle into equal parts. We show that when n is an odd prime
number, the cyclotomic supplementarity cannot be derived, leading to a
countable family of new axioms for diagrammatic quantum reasoning.We exhibit
another new simple axiom that cannot be derived from the existing rules of the
ZX-Calculus, implying in particular the incompleteness of the language for the
so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of
an extended ZX-Calculus, including an axiom schema for the cyclotomic
supplementarity.Comment: Mathematical Foundations of Computer Science, Aug 2017, Aalborg,
Denmar
A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics
We introduce the first complete and approximatively universal diagrammatic
language for quantum mechanics. We make the ZX-Calculus, a diagrammatic
language introduced by Coecke and Duncan, complete for the so-called Clifford+T
quantum mechanics by adding four new axioms to the language. The completeness
of the ZX-Calculus for Clifford+T quantum mechanics was one of the main open
questions in categorical quantum mechanics. We prove the completeness of the
Clifford+T fragment of the ZX-Calculus using the recently studied ZW-Calculus,
a calculus dealing with integer matrices. We also prove that the Clifford+T
fragment of the ZX-Calculus represents exactly all the matrices over some
finite dimensional extension of the ring of dyadic rationals
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
Completeness of the ZX-Calculus
The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum
mechanics and quantum information theory. It comes equipped with an equational
presentation. We focus here on a very important property of the language:
completeness, which roughly ensures the equational theory captures all of
quantum mechanics. We first improve on the known-to-be-complete presentation
for the so-called Clifford fragment of the language - a restriction that is not
universal - by adding some axioms. Thanks to a system of back-and-forth
translation between the ZX-Calculus and a third-party complete graphical
language, we prove that the provided axiomatisation is complete for the first
approximately universal fragment of the language, namely Clifford+T.
We then prove that the expressive power of this presentation, though aimed at
achieving completeness for the aforementioned restriction, extends beyond
Clifford+T, to a class of diagrams that we call linear with Clifford+T
constants. We use another version of the third-party language - and an adapted
system of back-and-forth translation - to complete the language for the
ZX-Calculus as a whole, that is, with no restriction. We briefly discuss the
added axioms, and finally, we provide a complete axiomatisation for an altered
version of the language which involves an additional generator, making the
presentation simpler
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