ZX-calculus publicationshttp://zxcalculus.com/publications.rssAn up to date list of the newest publications related to the ZX-calculusen-USSat, 17 Jul 2021 13:19:37 GMTrfeed v1.0.0https://github.com/svpino/rfeed/blob/master/README.mdQuantum Multiple-Valued Decision Diagrams in Graphical Calculihttp://arxiv.org/abs/2107.01186Graphical calculi such as the ZH-calculus are powerful tools in the study and analysis of quantum processes, with links to other models of quantum computation such as quantum circuits, measurement-based computing, etc. A somewhat compact but systematic way to describe a quantum process is through the use of quantum multiple-valued decision diagrams (QMDDs), which have already been used for the synthesis of quantum circuits as well as for verification. We show in this paper how to turn a QMDD into an equivalent ZH-diagram, and vice-versa, and show how reducing a QMDD translates in the ZH-Calculus, hence allowing tools from one formalism to be used into the other.Renaud VilmartFri, 02 Jul 2021 00:00:00 GMThttp://arxiv.org/abs/2107.01186Quantum double aspects of surface code modelshttp://arxiv.org/abs/2107.04411We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry, where $G$ is a finite group. We provide projection operators for its quasiparticles content as irreducible representations of $D(G)$ and combine this with $D(G)$-bimodule properties of open ribbon excitation spaces $L(s_0,s_1)$ to show how open ribbons can be used to teleport information between their endpoints $s_0,s_1$. We give a self-contained account that builds on earlier work but emphasises applications to quantum computing as surface code theory, including gates on $D(S_3)$. We show how the theory reduces to a simpler theory for toric codes in the case of $D( \Bbb Z_n)≅ \Bbb C\Bbb Z_n^2$, including toric ribbon operators and their braiding. In the other direction, we show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$, including site actions of $D(H)$ and partial results on ribbon equivariance even when the Hopf algebra is not semisimple.Alexander Cowtan and Shahn MajidFri, 25 Jun 2021 00:00:00 GMThttp://arxiv.org/abs/2107.04411[Video] Relating measurement patterns to circuits via Pauli flow and Pauli Dependency DAGs 🎥https://www.youtube.com/watch?v=XcrD_Hc4zUYThe one-way model of Measurement-Based Quantum Computing and the gate-based circuit model give two different presentations of how quantum computation can be performed. There are known methods for converting any gate-based quantum circuit into a one-way computation, whereas the reverse is only efficient given some constraints on the structure of the measurement pattern. Causal flow and generalised flow have already been shown as sufficient, with efficient algorithms for identifying these properties and performing the circuit extraction. Pauli flow is a weaker set of conditions that extends generalised flow to use the knowledge that some vertices are measured in a Pauli basis. In this paper, we show that Pauli flow can similarly be identified efficiently and that any measurement pattern whose underlying graph admits a Pauli flow can be efficiently transformed into a gate-based circuit without using ancilla qubits. We then use this relationship to derive simulation results for the effects of graph theoretic rewrites in the ZX-calculus using a more circuit-like data structure - the Pauli Dependency DAG.Will SimonsWed, 23 Jun 2021 00:00:00 GMThttps://www.youtube.com/watch?v=XcrD_Hc4zUYThe role of compositionality in constructing complementary classical structures within qubit systemshttps://arxiv.org/abs/2105.11966Observables in a quantum system, represented by a Hilbert space, are given by the orthogonal bases of the aforementioned Hilbert space. Categorical Quantum Mechanics provides further abstraction of such observables, allowing for a diagrammatic representation of measurements that extends to quantum processes. Our research studies this abstraction of observables, which has been dubbed as classical structures, in a subtheory of quantum mechanics which focuses on qubit systems (or 2-dimensional quantum system and its composites). We have constructed a procedure that takes the complementary classical structures of a single qubit system and compose them separably via the Kronecker product or 'entangle' them via Bell states to obtain complementary classical structures in n-qubit systems. In this present work, we apply our procedure to two qubit and three qubit systems as examples. Then, using rewriting rules of ZX-calculus and tools in graph theory, we searched for maximal complete sets of mutually complementary classical structures (the categorical counterpart of mutually unbiased bases) among our constructed composite classical structures. For two qubits, we found 13 maximal complete sets of mutually complementary classical structures, and for three qubits, we found 32,448 maximal complete sets of mutually complementary classical structures.Siti Aqilah Binti Muhamad RasatMon, 24 May 2021 00:00:00 GMThttps://arxiv.org/abs/2105.11966A Graphical Calculus for Lagrangian Relationshttp://arxiv.org/abs/2105.06244Symplectic vector spaces are the phase space of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constraints on the evolution -- of various physical systems. We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled' category of linear relations. More precisely, we show that it arises as a variation of Selinger's CPM construction applied to linear relations, where the covariant orthogonal complement functor plays of the role of conjugation. Furthermore, for linear relations over prime fields, this corresponds exactly to the CPM construction for a suitable choice of dagger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation, we prove the equivalence of the prop of affine Lagrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several disparate process theories, including electrical circuits, Spekkens' toy theory, and odd-prime-dimensional stabilizer quantum circuits.Cole Comfort and Aleks KissingerThu, 13 May 2021 00:00:00 GMThttp://arxiv.org/abs/2105.06244[Video] Vanishing 2-qubit gates with non-simplification ZX-rules 🎥https://www.youtube.com/watch?v=tUIcqXKEFhkTraditional quantum circuit optimization is performed directly at the circuit level. Alternatively, a quantum circuit can be translated to a ZX-diagram which can be simplified using the rules of the ZX-calculus, after which a simplified circuit can be extracted. However, the best-known extraction procedures can drastically increase the number of 2-qubit gates. In this work, we take advantage of the fact that local changes in a ZX-diagram can drastically affect the complexity of the extracted circuit. We use a pair of congruences (i.e., non-simplification rewrite rules) based on the graph-theoretic notions of local complementation and pivoting to generate local variants of a simplified ZX-diagram. We explore the space of equivalent ZX-diagrams generated by these congruences using simulated annealing and genetic algorithms to obtain a simplified circuit with fewer 2-qubit gates.
Our method can reliably outperform state-of-the-art optimization techniques for low-qubit, less than 10 qubit, circuits and serves as a proof-of-concept for a new circuit optimization strategy in the ZX-calculus.
Ryan KruegerFri, 23 Apr 2021 00:00:00 GMThttps://www.youtube.com/watch?v=tUIcqXKEFhkQufinite ZX-calculus: a unified framework of qudit ZX-calculihttp://arxiv.org/abs/2104.06429ZX-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. We would expect a reconstruction of finite quantum theory within the framework of qufinite ZX-calculus which focuses on compositionality without resorting to any probability theory or sum structures.Quanlong WangTue, 13 Apr 2021 00:00:00 GMThttp://arxiv.org/abs/2104.06429Quantum Algorithms and Oracles with the Scalable ZX-calculushttp://arxiv.org/abs/2104.01043The ZX-calculus was introduced as a graphical language able to represent specific quantum primitives in an intuitive way. The recent completeness results have shown the theoretical possibility of a purely graphical description of quantum processes. However, in practice, such approaches are limited by the intrinsic low level nature of ZX calculus. The scalable notations have been proposed as an attempt to recover an higher level point of view while maintaining the topological rewriting rules of a graphical language. We demonstrate that the scalable ZX-calculus provides a formal, intuitive, and compact framework to describe and prove quantum algorithms. As a proof of concept, we consider the standard oracle-based quantum algorithms: Deutsch-Jozsa, Bernstein-Vazirani, Simon, and Grover algorithms, and we show they can be described and proved graphically.Titouan Carette, Yohann D'Anello and Simon PerdrixFri, 02 Apr 2021 00:00:00 GMThttp://arxiv.org/abs/2104.01043ZX-Calculus and Extended Wolfram Model Systems II: Fast Diagrammatic Reasoning with an Application to Quantum Circuit Simplificationhttp://arxiv.org/abs/2103.15820This article presents a novel algorithmic methodology for performing automated diagrammatic deductions over combinatorial structures, using a combination of modified equational theorem-proving techniques and the extended Wolfram model hypergraph rewriting formalism developed by the authors in previous work. We focus especially upon the application of this new algorithm to the problem of automated circuit simplification in quantum information theory, using Wolfram model multiway operator systems combined with the ZX-calculus formalism for enacting fast diagrammatic reasoning over linear transformations between qubits. We show how to construct a generalization of the deductive inference rules for Knuth-Bendix completion in which equation matches are selected on the basis of causal edge density in the associated multiway system, before proceeding to demonstrate how to embed the higher-order logic of the ZX-calculus rules within this first-order equational framework. After showing explicitly how the (hyper)graph rewritings of both Wolfram model systems and the ZX-calculus can be effectively realized within this formalism, we proceed to exhibit comparisons of time complexity vs. proof complexity for this new algorithmic approach when simplifying randomly-generated Clifford circuits down to pseudo-normal form, as well as when reducing the number of T-gates in randomly-generated non-Clifford circuits, with circuit sizes ranging up to 3000 gates, illustrating that the method performs favorably in comparison with existing circuit simplification frameworks, and also exhibiting the approximately quadratic speedup obtained by employing the causal edge density optimization. Finally, we present a worked example of an automated proof of correctness for a simple quantum teleportation protocol, in order to demonstrate more clearly the internal operations of the theorem-proving procedure.Jonathan Gorard, Manojna Namuduri and Xerxes D. ArsiwallaMon, 29 Mar 2021 00:00:00 GMThttp://arxiv.org/abs/2103.15820Diagrammatic Differentiation for Quantum Machine Learninghttp://arxiv.org/abs/2103.07960We introduce diagrammatic differentiation for tensor calculus by generalising the dual number construction from rigs to monoidal categories. Applying this to ZX diagrams, we show how to calculate diagrammatically the gradient of a linear map with respect to a phase parameter. For diagrams of parametrised quantum circuits, we get the well-known parameter-shift rule at the basis of many variational quantum algorithms. We then extend our method to the automatic differentation of hybrid classical-quantum circuits, using diagrams with bubbles to encode arbitrary non-linear operators. Moreover, diagrammatic differentiation comes with an open-source implementation in DisCoPy, the Python library for monoidal categories. Diagrammatic gradients of classical-quantum circuits can then be simplified using the PyZX library and executed on quantum hardware via the tket compiler. This opens the door to many practical applications harnessing both the structure of string diagrams and the computational power of quantum machine learning.Alexis Toumi, Richie Yeung and Giovanni de FeliceSun, 14 Mar 2021 00:00:00 GMThttp://arxiv.org/abs/2103.07960