5 research outputs found
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
Thinking about Trust: People, Process, and Place
This brief paper is about trust. It explores the phenomenon from various angles, with the implicit assumptions that trust can be measured in some ways, that trust can be compared and rated, and that trust is of worth when we consider entities from data, through artificial intelligences, to humans, with side trips along the way to animals. It explores trust systems and trust empowerment as opposed to trust enforcement, the creation of trust models, applications of trust, and the reasons why trust is of worth. [Abstract copyright: © 2020 The Authors.
There and back again:a circuit extraction tale
Translations between the quantum circuit model and the measurement-based
one-way model are useful for verification and optimisation of quantum
computations. They make crucial use of a property known as gflow. While gflow
is defined for one-way computations allowing measurements in three different
planes of the Bloch sphere, most research so far has focused on computations
containing only measurements in the XY-plane. Here, we give the first
circuit-extraction algorithm to work for one-way computations containing
measurements in all three planes and having gflow. The algorithm is efficient
and the resulting circuits do not contain ancillae. One-way computations are
represented using the ZX-calculus, hence the algorithm also represents the most
general known procedure for extracting circuits from ZX-diagrams. In developing
this algorithm, we generalise several concepts and results previously known for
computations containing only XY-plane measurements. We bring together several
known rewrite rules for measurement patterns and formalise them in a unified
notation using the ZX-calculus. These rules are used to simplify measurement
patterns by reducing the number of qubits while preserving both the semantics
and the existence of gflow. The results can be applied to circuit optimisation
by translating circuits to patterns and back again.Comment: 47 pages in body, 15 pages in appendice