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arXiv:2302.12135 · ACM/IEEE Symposium on Logic in Computer Science (LICS)

Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus

Boldizsár Poór, Quanlong Wang, Razin Abdulrauf Shaikh, Lia Yeh, Richie Wai Kit Yeung, Bob Coecke

Published Jun 26, 2023• 13 citations
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Abstract

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.

DOI
10.1109/lics56636.2023.10175672

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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.

•Jan 1, 2018•2 citations
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 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.
Finite-dimensional quantum theory serves as the theoretical foundation for quantum information and computation. Mathematically, it is formalized in the category FHilb, comprising all finite-dimensional Hilbert spaces and linear maps between them. However, there has not been a graphical language for FHilb which is both universal and complete and thus incorporates a set of rules rich enough to derive any equality of the underlying formalism solely by rewriting. In this paper, we introduce the qufinite ZXW calculus - a graphical language for reasoning about finite-dimensional quantum theory. We set up a unique normal form to represent an arbitrary tensor and prove the completeness of this calculus by demonstrating that any qufinite ZXW diagram can be rewritten into its normal form. This result implies the equivalence of the qufinite ZXW calculus and the category FHilb, leading to a purely diagrammatic framework for finite-dimensional quantum theory with the same reasoning power. In addition, we identify several domains where the application of the qufinite ZXW calculus holds promise. These domains include spin networks, interacting mixed-dimensional systems in quantum chemistry, quantum programming, high-level description of quantum algorithms, and mixed-dimensional quantum computing. Our work paves the way for a comprehensive diagrammatic description of quantum physics, opening the doors of this area to the wider public.
ZX-calculus is graphical language for quantum computing which usually focuses on qubits. In this paper, we generalise qubit ZX-calculus to qudit ZX-calculus in any finite dimension by introducing suitable generators, especially a carefully chosen triangle node. As a consequence we obtain a set of rewriting rules which can be seen as a direct generalisation of qubit rules, and a normal form for any qudit vectors. Based on the qudit ZX-calculi, we propose a graphical formalism called qufinite ZX-calculus as a unified framework for all qudit ZX-calculi, which is universal for finite quantum theory due to a normal form for matrix of any finite size. As a result, it would be interesting to give a fine-grained version of the diagrammatic reconstruction of finite quantum theory [Selby2021reconstructing] within the framework of qufinite ZX-calculus.
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 ZW-calculus is a graphical language capable of representing 2-dimensional quantum systems (qubit) through its diagrams, and manipulating them through its equational theory. We extend the formalism to accommodate finite dimensional Hilbert spaces beyond qubit systems. First we define a qudddit version of the language, where all systems have the same arbitrary finite dimension ddd, and show that the provided equational theory is both complete -- i.e. semantical equivalence is entirely captured by the equations -- and minimal -- i.e. none of the equations are consequences of the others. We then extend the graphical language further to allow for mixed-dimensional systems. We again show the completeness and minimality of the provided equational theory.
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.