ZX-calculus publicationshttp://zxcalculus.com/publications.rssAn up to date list of the newest publications related to the ZX-calculusen-USMon, 27 Nov 2023 11:04:03 GMTrfeed v1.0.0https://github.com/svpino/rfeed/blob/master/README.mdVyZX: Formal Verification of a Graphical Quantum Languagehttp://arxiv.org/abs/2311.11571Mathematical representations of graphs often resemble adjacency matrices or lists, representations that facilitate whiteboard reasoning and algorithm design. In the realm of proof assistants, inductive representations effectively define semantics for formal reasoning. This highlights a gap where algorithm design and proof assistants require a fundamentally different structure of graphs, particularly for process theories which represent programs using graphs. To address this gap, we present VyZX, a verified library for reasoning about inductively defined graphical languages. These inductive constructs arise naturally from category theory definitions. A key goal for VyZX is to Verify the ZX-calculus, a graphical language for reasoning about quantum computation. The ZX-calculus comes with a collection of diagrammatic rewrite rules that preserve the graph's semantic interpretation. We show how inductive graphs in VyZX are used to prove the correctness of the ZX-calculus rewrite rules and apply them in practice using standard proof assistant techniques. VyZX integrates easily with the proof engineer's workflow through visualization and automation.Adrian Lehmann, Ben Caldwell, Bhakti Shah and Robert RandMon, 20 Nov 2023 00:00:00 GMThttp://arxiv.org/abs/2311.11571Minimal Equational Theories for Quantum Circuitshttp://arxiv.org/abs/2311.07476We introduce the first minimal and complete equational theory for quantum circuits. Hence, we show that any true equation on quantum circuits can be derived from simple rules, all of them being standard except a novel but intuitive one which states that a multi-control $2π$ rotation is nothing but the identity. Our work improves on the recent complete equational theories for quantum circuits, by getting rid of several rules including a fairly unpractical one. One of our main contributions is to prove the minimality of the equational theory, i.e. none of the rules can be derived from the other ones. More generally, we demonstrate that any complete equational theory on quantum circuits (when all gates are unitary) requires rules acting on an unbounded number of qubits. Finally, we also simplify the complete equational theories for quantum circuits with ancillary qubits and/or qubit discarding.Alexandre Clément, Noé Delorme and Simon PerdrixMon, 13 Nov 2023 00:00:00 GMThttp://arxiv.org/abs/2311.07476Enriching Diagrams with Algebraic Operationshttp://arxiv.org/abs/2310.11288In this paper, we extend diagrammatic reasoning in monoidal categories with algebraic operations and equations. We achieve this by considering monoidal categories that are enriched in the category of Eilenberg-Moore algebras for a monad. Under the condition that this monad is monoidal and affine, we construct an adjunction between symmetric monoidal categories and symmetric monoidal categories enriched over algebras for the monad. This allows us to devise an extension, and its semantics, of the ZX-calculus with probabilistic choices by freely enriching over convex algebras, which are the algebras of the finite distribution monad. We show how this construction can be used for diagrammatic reasoning of noise in quantum systems.Alejandro Villoria, Henning Basold and Alfons LaarmanTue, 17 Oct 2023 00:00:00 GMThttp://arxiv.org/abs/2310.11288Flexible entangled state generation in linear opticshttp://arxiv.org/abs/2310.06832Fault-tolerant quantum computation can be achieved by creating constant-sized, entangled resource states and performing entangling measurements on subsets of their qubits. Linear optical quantum computers can be designed based on this approach, even though entangling operations at the qubit level are non-deterministic in this platform. Probabilistic generation and measurement of entangled states must be pushed beyond the required threshold by some combination of scheme optimisation, introduction of redundancy and auxiliary state assistance. We report progress in each of these areas. We explore multi-qubit fusion measurements on dual-rail photonic qubits and their role in measurement-based resource state generation, showing that it is possible to boost the success probability of photonic GHZ state analysers with single photon auxiliary states. By incorporating generators of basic entangled "seed" states, we provide a method that simplifies the process of designing and optimising generators of complex, encoded resource states by establishing links to ZX diagrams.Brendan Pankovich, Alex Neville, Angus Kan, Srikrishna Omkar, Kwok Ho Wan and Kamil BrádlerTue, 10 Oct 2023 00:00:00 GMThttp://arxiv.org/abs/2310.06832Completeness of qufinite ZXW calculus, a graphical language for mixed-dimensional quantum computinghttp://arxiv.org/abs/2309.13014Finite-dimensional quantum theory serves as the theoretical foundation for quantum information and computation based on 2-dimensional qubits, d-dimensional qudits, and their interactions. The qufinite ZX calculus has been used as a framework for mixed-dimensional quantum computing; however, it lacked the crucial property of completeness, which ensures that the calculus incorporates a set of rules rich enough to prove any equation. The ZXW calculus is a complete language for qudit quantum computing with applications previously unreachable solely with the ZX or ZW calculus. In this paper, we introduce the qufinite ZXW calculus, a unification of all qudit ZXW calculi in a single framework for mixed-dimensional quantum computing. We provide a set of rewrite rules and a unique normal form that make the calculus complete for finite-dimensional quantum theory. This work paves the way for the optimization of mixed dimensional circuits and tensor networks appearing in different areas of quantum computing including quantum chemistry, compilation, and quantum many-body systems.Quanlong Wang and Boldizsár PoórFri, 22 Sep 2023 00:00:00 GMThttp://arxiv.org/abs/2309.13014Reinforcement Learning based Circuit Compilation via ZX-calculushttps://diposit.ub.edu/dspace/bitstream/2445/202911/1/Memoria_TFM-JanNogue.pdfZX-calculus is a formalism that can be used for quantum circuit compilation and optimization. We developed a Reinforcement Learning approach for enhanced circuit optimization via the ZX-diagram graph representation of the quantum circuit. The agent is trained using the well-established Proximal Policy Optimization (PPO) algorithm, and it uses Conditional Action Trees to perform Invalid Action Masking to reduce the space of actions available to the agent and speed up its training. Additionally, we also design and implement a Genetic Algorithm for the same task. Both the genetic algorithm and the most widely used ZX-calculus-based library for circuit optimization, the PyZX library, are used to benchmark our RL approach. We find our RL algorithm to be competitive against both approaches, but further exploration is required.Jan Nogué GómezMon, 28 Aug 2023 00:00:00 GMThttps://diposit.ub.edu/dspace/bitstream/2445/202911/1/Memoria_TFM-JanNogue.pdfFault-tolerant complexeshttp://arxiv.org/abs/2308.07844Fault-tolerant complexes describe surface-code fault-tolerant protocols from a single geometric object. We first introduce fusion complexes that define a general family of fusion-based quantum computing (FBQC) fault-tolerant quantum protocols based on surface codes. We show that any 3-dimensional cell complex where each edge has four incident faces gives a valid fusion complex. This construction enables an automated search for fault tolerance schemes, allowing us to identify 627 examples within a moderate search time. We implement this using the open-source software tool Gavrog and present threshold results for a variety of schemes, finding fusion networks with higher erasure and Pauli thresholds than those existing in the literature. We then define more general structures we call fault-tolerant complexes that provide a homological description of fault tolerance from a large family of low-level error models, which include circuit-based computation, floquet-based computation, and FBQC with multi-qubit measurements. This extends the applicability of homological descriptions of fault tolerance, and enables the generation of many new schemes which have not been previously identified. We also define families of fault-tolerant complexes for color codes and 3d single-shot subsystem codes, which enables similar constructive methods, and we present several new examples of each.Hector Bombin, Chris Dawson, Terry Farrelly, Yehua Liu, Naomi Nickerson, Mihir Pant, Fernando Pastawski and Sam RobertsTue, 15 Aug 2023 00:00:00 GMThttp://arxiv.org/abs/2308.07844Rewriting and Completeness of Sum-Over-Paths in Dyadic Fragments of Quantum Computinghttp://arxiv.org/abs/2307.14223The "Sum-Over-Paths" formalism is a way to symbolically manipulate linear maps that describe quantum systems, and is a tool that is used in formal verification of such systems. We give here a new set of rewrite rules for the formalism, and show that it is complete for "Toffoli-Hadamard", the simplest approximately universal fragment of quantum mechanics. We show that the rewriting is terminating, but not confluent (which is expected from the universality of the fragment). We do so using the connection between Sum-over-Paths and graphical language ZH-calculus, and also show how the axiomatisation translates into the latter. We provide generalisations of the presented rewrite rules, that can prove useful when trying to reduce terms in practice, and we show how to graphically make sense of these new rules. We show how to enrich the rewrite system to reach completeness for the dyadic fragments of quantum computation, used in particular in the Quantum Fourier Transform, and obtained by adding phase gates with dyadic multiples of $π$ to the Toffoli-Hadamard gate-set. Finally, we show how to perform sums and concatenation of arbitrary terms, something which is not native in a system designed for analysing gate-based quantum computation, but necessary when considering Hamiltonian-based quantum computation.Renaud VilmartWed, 26 Jul 2023 00:00:00 GMThttp://arxiv.org/abs/2307.14223Floquetifying the Colour Codehttps://dx.doi.org/10.4204/EPTCS.384.14Floquet codes are a recently discovered type of quantum error correction code. They can be thought of as generalising stabilizer codes and subsystem codes, by allowing the logical Pauli operators of the code to vary dynamically over time. In this work, we use the ZX-calculus to create new Floquet codes that are in a definable sense equivalent to known stabilizer codes. In particular, we find a Floquet code that is equivalent to the colour code, but has the advantage that all measurements required to implement it are of weight one or two. Notably, the qubits can even be laid out on a square lattice. This circumvents current difficulties with implementing the colour code fault-tolerantly, while preserving its advantages over other well-studied codes, and could furthermore allow one to benefit from extra features exclusive to Floquet codes. On a higher level, as in arXiv:2303.08829, this work shines a light on the relationship between 'static' stabilizer and subsystem codes and 'dynamic' Floquet codes; at first glance the latter seems a significant generalisation of the former, but in the case of the codes that we find here, the difference is essentially just a few basic ZX-diagram deformations.Alex Townsend-Teague, Julio Magdalena de la Fuente and Markus KesselringThu, 20 Jul 2023 00:00:00 GMThttp://arxiv.org/abs/2307.11136The Qudit ZH-Calculus: Generalised Toffoli+Hadamard and Universalityhttps://dx.doi.org/10.4204/EPTCS.384.9We introduce the qudit ZH-calculus and show how to generalise the phase-free qubit rules to qudits. We prove that for prime dimensions $d$, the phase-free qudit ZH-calculus is universal for matrices over the ring $ℤ[e^2π i/d]$. For qubits, there is a strong connection between phase-free ZH-diagrams and Toffoli+Hadamard circuits, a computationally universal fragment of quantum circuits. We generalise this connection to qudits, by finding that the two-qudit $|0ångle$-controlled $X$ gate can be used to construct all classical reversible qudit logic circuits in any odd qudit dimension, which for qubits requires the three-qubit Toffoli gate. We prove that our construction is asymptotically optimal up to a logarithmic term. Twenty years after the celebrated result by Shi proving universality of Toffoli+Hadamard for qubits, we prove that circuits of $|0\n̊gle$-controlled $X$ and Hadamard gates are approximately universal for qudit quantum computing for any odd prime $d$, and moreover that phase-free ZH-diagrams correspond precisely to such circuits allowing postselections.Patrick Roy, John van de Wetering and Lia YehWed, 19 Jul 2023 00:00:00 GMThttp://arxiv.org/abs/2307.10095