%We will exploit further the recent completeness results to give representations for mixed state qubit quantum theory. We will
%We will exploit further the recent completeness results to give representations for mixed state qubit quantum theory. We will
%extend the \textsc{zx} tensor formalism from the qubit domain to higher dimensions.
%extend the \textsc{zx} tensor formalism from the qubit domain to higher dimensions.
We will extend the completeness results of the \textsc{zx}-calculus from the qubit domain to higher dimensions, to have complete qudit \textsc{zx}-calculus. Furthermore, we will combine all the qudit \textsc{zx}-calculus into a single framework so that we can deal with the whole finite-dimensional quantum theory in a \textsc{zx} style. In addition, we will exploit techniques from the \textsc{zw}-calculus to understand the deep structure of W-type tensors.
We will extend the completeness results of the \textsc{zx}-calculus from the qubit domain to higher dimensions, to have complete qudit \textsc{zx}-calculus. Furthermore, we will combine all the qudit \textsc{zx}-calculus into a single framework so that we can deal with the whole finite-dimensional quantum theory in a \textsc{zx} style. In addition, we will exploit techniques from the \textsc{zw}-calculus to understand the deep structure of W-type tensors.
% and exploit the translation from \textsc{zx}- to \textsc{zw}-calculus.
% and exploit the translation from \textsc{zx}- to \textsc{zw}-calculus.
% Support simple control flow at the level of \azx, making it a more suitable target for compiling from a high-level language. In particular, add support for repetition and recursive definitions of diagrams, e.g. for expressing and transforming regular families of circuits.
% Support simple control flow at the level of \azx, making it a more suitable target for compiling from a high-level language. In particular, add support for repetition and recursive definitions of diagrams, e.g. for expressing and transforming regular families of circuits.
We will use parametric \zx terms to support simple control flow at the level of the \dzxc system, making it a more suitable target for compiling from a high-level language. In particular, we will add support for repetition and recursive definitions of diagrams, e.g. for expressing and transforming regular families of circuits.
We will use parametric \zx terms to support simple control flow at the level of the \dzxc system, making it a more suitable target for compiling from a high-level language. In particular, we will add support for repetition and recursive definitions of diagrams, e.g. for expressing and transforming regular families of circuits.
}
}
\WPtask[\label{task:resources}]{Resources and axioms
\WPtask[\label{task:resources}]{Resources and axioms
We will exploit the three axiom sets for Clifford, Clifford+T, and universal qubit QM,
We will exploit the three axiom sets for Clifford, Clifford+T, and universal qubit QM,
to identify and distill specific resources that are necessary to quantum speed-up. In particular, to focus on finding multiple resource elements (rather than simply magic states), and to characterise post-classical composition as a resource.
to identify and distill specific resources that are necessary to quantum speed-up. In particular, to focus on finding multiple resource elements (rather than simply magic states), and to characterise post-classical composition as a resource.
This includes developing \zx representations of contextuality, as a possible post-classical resource.
This includes developing \zx representations of contextuality, as a possible post-classical resource.
We will use the existing graph re-writing and automated theorem proving tools of Quantomatic and PyZX to determine parts of the re-writing process that are difficult to compute classically. This will then be used to extract candidate subroutines for sources of quantum speed-up. Along with the previous task, these will be used to develop procedures for characterising if a \zx-represented algorithm demonstrates speed-up or not.
We will use the existing graph re-writing and automated theorem proving tools of Quantomatic and PyZX to determine parts of the re-writing process that are difficult to compute classically. This will then be used to extract candidate subroutines for sources of quantum speed-up. Along with the previous task, these will be used to develop procedures for characterising if a \zx-represented algorithm demonstrates speed-up or not.
}
}
\end{WPtasks}
\end{WPtasks}
\begin{WPdeliverables}
\begin{WPdeliverables}
\WPdeliverable{M12}{Preliminary assessment of the comparative study of the axiomatizations of paradigms of quantum computation}
\WPdeliverable{M9}{Preliminary assessment of the comparative study of the axiomatizations of paradigms of quantum computation}
\WPdeliverable{M15}{\zx representation and explanation of the result that promotes magic states as a resource of quantum computation in the state injection paradigm}
\WPdeliverable{M14}{Completeness of qudit \zx calculus}
\WPdeliverable{M18}{Preliminary assessment of nonclassicality of re-writing processes}
\WPdeliverable{M18}{\zx formalism for recursion and control}
\WPdeliverable{M24}{\zx formulation of contextuality (Kochen--Specker and/or generalised Spekken's type)}
\WPdeliverable{M20}{Preliminary assessment of nonclassicality of re-writing processes}
\WPdeliverable{M24}{\zx representation and explanation of the result that promotes magic states as a resource of quantum computation}
\WPdeliverable{M30}{\zx formulation of contextuality (Kochen--Specker and/or Spekken's)}
\WPdeliverable{M36}{Characterisation of set of generic non-classical resources for quantum speed-up}
