This thesis looks at two quantum motivated graphical calculi, the ZW calculus and ZX calculus. These two calculi are intended to perform qubit calculations using string diagrams. Both calculi are universal in the sense that they are able to represent all qubit maps, but they lack the completeness which means that they don't allow all qubit computation. The main goal of the thesis is to address the completeness issue. The ZW calculus was completed by Hadzihasanovic in his DPhil thesis. We build on his work to do a completion for a fragment of the ZW calculus which describes operations in the fermionic quantum circuits. In the process, we discovered an important family of maps, the even and odd projectors. The even projectors is particularly important for the completion process. Next, we further restrict the calculus to only "even" maps and complete this fragment too. The odd projectors are important for this completion process. We give an interesting story for the even fragment ZW calculus: it describes a world without "real" particles, and the fermionic particles are manifestation of the curvature of spacetime and self 'fermionic' intersection and interaction of spacetime. The ZX calculus was not complete and we solve this issue in this thesis. The technique used is a refinement of the one used in the paper on 'A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics' by Jeandel, Perdrix, and Vilmart. The basic idea of the technique is to translate the ZX calculus to the ZW calculus to analyse the problem, then translate it back. We also obtained many interesting generalising results which includes the completeness for the Clifford+T fragment and any fragments containing the Clifford+T fragment. Finally, we end off with some interesting possible continuation of the thesis. Many of them are low hanging fruits. One of them is the extension of the calculi to represent qudits maps. The other is to generalise the ZW calculus to take semi-ring parameters which may be suited for other areas of computer science.
This thesis looks at two quantum motivated graphical calculi, the ZW calculus and ZX calculus. These two calculi are intended to perform qubit calculations using string diagrams. Both calculi are universal in the sense that they are able to represent all qubit maps, but they lack the completeness which means that they don't allow all qubit computation. The main goal of the thesis is to address the completeness issue. The ZW calculus was completed by Hadzihasanovic in his DPhil thesis. We build on his work to do a completion for a fragment of the ZW calculus which describes operations in the fermionic quantum circuits. In the process, we discovered an important family of maps, the even and odd projectors. The even projectors is particularly important for the completion process. Next, we further restrict the calculus to only "even" maps and complete this fragment too. The odd projectors are important for this completion process. We give an interesting story for the even fragment ZW calculus: it describes a world without "real" particles, and the fermionic particles are manifestation of the curvature of spacetime and self 'fermionic' intersection and interaction of spacetime. The ZX calculus was not complete and we solve this issue in this thesis. The technique used is a refinement of the one used in the paper on 'A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics' by Jeandel, Perdrix, and Vilmart. The basic idea of the technique is to translate the ZX calculus to the ZW calculus to analyse the problem, then translate it back. We also obtained many interesting generalising results which includes the completeness for the Clifford+T fragment and any fragments containing the Clifford+T fragment. Finally, we end off with some interesting possible continuation of the thesis. Many of them are low hanging fruits. One of them is the extension of the calculi to represent qudits maps. The other is to generalise the ZW calculus to take semi-ring parameters which may be suited for other areas of computer science.
The ZX-calculus is a universal graphical language for qubit quantum computation, meaning that every linear map between qubits can be expressed in the ZX-calculus. Furthermore, it is a complete graphical rewrite system: any equation involving linear maps that is derivable in the Hilbert space formalism for quantum theory can also be derived in the calculus by rewriting. It has widespread usage within quantum industry and academia for a variety of tasks such as quantum circuit optimisation, error-correction, and education.The ZW-calculus is an alternative universal graphical language that is also complete for qubit quantum computing. In fact, its completeness was used to prove that the ZX-calculus is universally complete. This calculus has advanced how quantum circuits are compiled into photonic hardware architectures in the industry.Recently, by combining these two calculi, a new calculus has emerged for qubit quantum computation, the ZXW-calculus. Using this calculus, graphical-differentiation, -integration, and -exponentiation were made possible, thus enabling the development of novel techniques in the domains of quantum machine learning and quantum chemistry.Here, we generalise the ZXW-calculus to arbitrary finite dimensions, that is, to qudits. Moreover, we prove that this graphical rewrite system is complete for any finite dimension. This is the first completeness result for any universal graphical language beyond qubits.
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.
The ZX-calculus is a graphical language for reasoning about quantum computing and quantum information theory.As a complete graphical language, it incorporates a set of axioms rich enough to derive any equation of the underlying formalism.While completeness of the ZX-calculus has been established for qubits and the Clifford fragment of prime-dimensional qudits, universal completeness beyond two-level systems has remained unproven until now.In this paper, we present a proof establishing the completeness of finite-dimensional ZX-calculus, incorporating only the mixeddimensional Z-spider and the qudit X-spider as generators.Our approach builds on the completeness of another graphical language, the finite-dimensional ZW-calculus, with direct translations between these two calculi.By proving its completeness, we lay a solid foundation for the ZX-calculus as a versatile tool not only for quantum computation but also for various fields within finite-dimensional quantum theory.
ZW-calculus is a useful graphical language for pure qubit quantum computing. It is via the translation of the completeness of ZW-calculus that the first proof of completeness of ZX-calculus was obtained. A d-level generalisation of qubit ZW-calculus (anyonic qudit ZW-calculus) has been given in [Hadzihasanovic 2017] which is universal for pure qudit quantum computing. However, the interpretation of the W spider in this type of ZW-calculus has so-called q-binomial coefficients involved, thus makes computation quite complicated. In this paper, we give a new type of qudit ZW-calculus which has generators and rewriting rules similar to that of the qubit ZW-calculus. Especially, the Z spider is exactly the same as that of the qudit ZX-calculus as given in [Wang 2021], and the new W spider has much simpler interpretation as a linear map. Furthermore, we establish a translation between this qudit ZW-calculus and the qudit ZX-calculus which is universal as shown in [Wang 2021], therefore this qudit ZW-calculus is also universal for pure qudit quantum computing.
The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum mechanics and quantum information theory. An axiomatisation has recently been proven to be complete for an approximatively universal fragment of quantum mechanics, the so-called Clifford+T fragment. We focus here on the expressive power of this axiomatisation beyond Clifford+T Quantum mechanics. We consider the full pure qubit quantum mechanics, and mainly prove two results: (i) First, the axiomatisation for Clifford+T quantum mechanics is also complete for all equations involving some kind of linear diagrams. The linearity of the diagrams reflects the phase group structure, an essential feature of the ZX-calculus. In particular all the axioms of the ZX-calculus are involving linear diagrams. (ii) We also show that the axiomatisation for Clifford+T is not complete in general but can be completed by adding a single (non linear) axiom, providing a simpler axiomatisation of the ZX-calculus for pure quantum mechanics than the one recently introduced by Ng&Wang.
Different graphical calculi have been proposed to represent quantum computation. First the ZX-calculus [Coecke and Duncan, 2011], followed by the ZW-calculus [Hadzihasanovic, 2015] and then the ZH-calculus [Backens and Kissinger, 2018]. We can wonder if new ZX-like calculi will continue to be proposed forever. This article answers negatively. All those language share a common core structure we call Z^*-algebras. We classify Z^*-algebras up to isomorphism in two dimensional Hilbert spaces and show that they are all variations of the aforementioned calculi. We do the same for linear relations and show that the calculus of [Bonchi et al., 2017] is essentially the unique one.