7 research outputs found
A Near-Minimal Axiomatisation of ZX-Calculus for Pure Qubit Quantum Mechanics
International audienceRecent developments in the ZX-Calculus have resulted in complete axiomatisations first for an approximately universal restriction of the language, and then for the whole language. The main drawbacks were that the axioms that were added to achieve completeness were numerous, tedious to manipulate and lacked a physical interpretation. We present in this paper two complete axiomatisations for the general ZX-Calculus, that we believe are optimal, in that all their equations are necessary and moreover have a nice physical interpretation
Graphical Calculi and their Conjecture Synthesis
Categorical Quantum Mechanics, and graphical calculi in particular, has
proven to be an intuitive and powerful way to reason about quantum computing.
This work continues the exploration of graphical calculi, inside and outside of
the quantum computing setting, by investigating the algebraic structures with
which we label diagrams. The initial aim for this was Conjecture Synthesis; the
algorithmic process of creating theorems. To this process we introduce a
generalisation step, which itself requires the ability to infer and then verify
parameterised families of theorems. This thesis introduces such inference and
verification frameworks, in doing so forging novel links between graphical
calculi and fields such as Algebraic Geometry and Galois Theory. These
frameworks inspired further research into the design of graphical calculi, and
we introduce two important new calculi here. First is the calculus RING, which
is initial among ring-based qubit graphical calculi, and in turn inspired the
introduction and classification of phase homomorphism pairs also presented
here. The second is the calculus ZQ, an edge-decorated calculus which naturally
expresses arbitrary qubit rotations, eliminating the need for non-linear rules
such as (EU) of ZX. It is expected that these results will be of use to those
creating optimisation schemes and intermediate representations for quantum
computing, to those creating new graphical calculi, and for those performing
conjecture synthesis.Comment: DPhil Thesis, University of Oxford. 222 pages, inline diagram
Completeness of the ZH-calculus
There are various gate sets used for describing quantum computation. A
particularly popular one consists of Clifford gates and arbitrary single-qubit
phase gates. Computations in this gate set can be elegantly described by the
ZX-calculus, a graphical language for a class of string diagrams describing
linear maps between qubits. The ZX-calculus has proven useful in a variety of
areas of quantum information, but is less suitable for reasoning about
operations outside its natural gate set such as multi-linear Boolean operations
like the Toffoli gate. In this paper we study the ZH-calculus, an alternative
graphical language of string diagrams that does allow straightforward encoding
of Toffoli gates and other more complicated Boolean logic circuits. We find a
set of simple rewrite rules for this calculus and show it is complete with
respect to matrices over , which correspond to the
approximately universal Toffoli+Hadamard gateset. Furthermore, we construct an
extended version of the ZH-calculus that is complete with respect to matrices
over any ring where is not a zero-divisor.Comment: 64 pages, many many diagram