The ZX Seminar

Speaker: Haytham McDowall-Rose (University of Oxford)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: Mapping fermionic systems to qubits on a quantum computer is often the first step for algorithms in quantum chemistry and condensed matter physics. However, it is difficult to reconcile the many different approaches that have been proposed, such as those based on binary matrices, ternary trees, and stabilizer codes. This challenge is further exacerbated by the many ways to describe them—transformation of Majorana operators, action on Fock states, encoder circuits, and stabilizers of local encodings—making it challenging to know when the mappings are equivalent. In this work, we present a graphical framework for fermion-to-qubit mappings that streamlines and unifies various representations through the ZX-calculus.

To start, we present the correspondence between linear encodings of the Fock basis and phase-free ZX-diagrams. Using the commutation rules of scalable ZX-calculus, we can derive fermionic operators under any linear encoding. Next, we give a translation from ternary tree mappings to scalable ZX-diagrams, which not only directly represents the encoder map as a CNOT circuit, but also retains the same structure as the tree. Consequently, we graphically prove that ternary tree transformations are linear encodings, a recent result by Chiew et. al. The scalable ZX representation moreover enables us to construct an algorithm to directly compute the binary matrix for any ternary tree mapping. Lastly, we present the graphical representation of local fermion-to-qubit encodings. Its encoder ZX-diagram has the same connectivity as the interaction graph of the fermionic Hamiltonian and also allows us to easily identify stabilizers of the encoding.

Joint-Work With: Razin A. Shaikh and Lia Yeh

Speaker: Marco Schumann (Institute for Quantum Computing Analytics (PGI-12) - Jülich Research Center, Saarland University)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Germany/Netherlands

Abstract: Quantum circuit cutting refers to a series of techniques that allow one to partition a quantum computation on a large quantum computer into several quantum computations on smaller devices. This usually comes at the price of a sampling overhead, that is quantified by the 1-norm of the associated decomposition. The applicability of these techniques relies on the possibility of finding decompositions of the ideal, global unitaries into quantum operations that can be simulated onto each sub-register, which should ideally minimize the 1-norm. In this work, we show how these decompositions can be obtained diagrammatically using ZX-calculus expanding on the work of Ref. [1]. The central idea of our work is that since in ZX-calculus only connectivity matters, it should be possible to cut wires in ZX-diagrams by inserting known decompositions of the identity in standard quantum circuits. We show how, using this basic idea, many of the gate decompositions known in the literature can be re-interpreted as an instance of wire cuts in ZX-diagrams. Furthermore, we obtain improved decompositions for multi-qubit controlled-Z (MCZ) gates with 1-norm equal to 3 for any number of qubits and any partition, which we argue to be optimal. Our work gives new ways of thinking about circuit cutting that can be particularly valuable for finding decompositions of large unitary gates. Besides, it sheds light on the question of why exploiting classical communication decreases the 1-norm of a wire cut but does not do so for certain gate decompositions. In particular, using wire cuts with classical communication, we obtain gate decompositions that do not require classical communication.

[1] Ufrecht, C., Periyasamy, M., Rietsch, S., Scherer, D. D., Plinge, A., & Mutschler, C. (2023). Cutting Multi-Control Quantum Gates with ZX Calculus. Quantum, 7, 1147.

Joint-Work With: Tobias Stollenwerk and Alessandro Ciani

Speaker: Benjamin Rodatz (University of Oxford)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: TBA

Speaker: Lia Yeh (University of Oxford)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: TBA

Joint-Work With: Jiaxin Huang, Aleks Kissinger, Sarah Meng Li, and John van de Wetering

Speaker: Yichen Xu (Cornell University) and Arpit Dua (Virgina Tech)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Germany/Netherlands

Abstract: We introduce a framework called spacetime concatenation for fault-tolerant compilation of syndrome extraction circuits of stabilizer codes. Spacetime concatenation enables efficient compilation of syndrome extraction circuits into dynamical codes through structured gadget layouts and encoding matrices, facilitating low-weight measurements while preserving logical information. Our framework uses conditions that are sufficient for fault-tolerance of the dynamical code, including not measuring logical operators and preserving the spacetime distance. We construct explicit examples of dynamical codes using this framework, including the dynamical bivariate bicycle code and a dynamical Haah code, while illustrating their fault-tolerant properties. Furthermore, we analyze the classification and resource trade-offs of dynamical codes, demonstrating their adaptability to hardware constraints, including fabrication defects and qubit dropout scenarios.

Speaker: Boldizsár Poór (University of Oxford)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: Stabiliser codes with large weight measurements can be challenging to implement fault-tolerantly. To overcome this, we propose a Floquetification procedure which, given a stabiliser code, synthesises a novel Floquet code that only uses single- and two-qubit operations. Moreover, this procedure preserves the distance and number of logicals of the original code. The new Floquet code requires additional physical qubits. This overhead is linear in the weight of the largest measurement of the original code.

Our method is based on the ZX calculus, a graphical language for representing and rewriting quantum circuits. However, a problem arises with the use of ZX in the context of rewriting error-correcting codes: ZX rewrites generally do not preserve code distance. Tackling this issue, we define the notion of distance-preserving rewrite that enables the transformation of error-correcting codes without changing their distance.

These distance-preserving rewrites are used to decompose arbitrary weight stabiliser measurements into quantum circuits with single- and two-qubit operations. As we only use distance-preserving rewrites, we are guaranteed that a single error in the resulting circuit creates at most a single error on the data qubits. These decompositions enable us to generalise the Floquetification procedure of Townsend-Teague et al. (2023) to arbitrary stabiliser codes, provably preserving the distance and number of logicals of the original code.

Joint-Work With: Benjamin Rodatz and Aleks Kissinger

Recording: https://youtu.be/KrPsjIFx_Y8

Speaker: Neil Dowling (University of Cologne)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Germany/Netherlands

Abstract: Magic quantifies the non-Clifford operations required for preparing a state on quantum processors and sets bounds on the classical computational complexity of simulating quantum dynamics. We study a non-stabilizerness resource theory for operators, which is dual to that describing states. We identify that the stabilizer Rényi entropy analog in operator space is a good magic monotone satisfying the usual conditions, while inheriting efficient computability properties and providing a tight lower-bound to the minimum number of non-Clifford gates in a circuit. It is operationally well-defined as quantifying how well one can approximate an operator with one that has only few Pauli strings; analogous to the relation between entanglement entropy and tensor-network truncation. An immediate advantage is that the operator stabilizer entropies exhibit inherent locality through a Lieb-Robinson bound, making them particularly suited for studying local dynamical magic generation in many-body systems. We compute this quantity analytically in two distinct regimes. First, we show that random evolution typically has approximately maximal magic in the Heisenberg picture for all Rényi indices, and evaluate the Page correction. Second, harnessing both dual unitarity and ZX graphical calculus, we compute the operator stabilizer entropy evolution for an interacting integrable XXZ circuit. In this case, magic quickly saturates to a constant. This monotone reveals structural properties of many-body magic generation, and can inspire Clifford-assisted tensor network methods.

Joint-Work With: Pavel Kos and Xhek Turkeshi

Recording: https://youtu.be/gh50B8LKEpE

Speaker: Matthew Sutcliffe (University of Oxford)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: We introduce a novel method for strong classical simulation of quantum circuits based on optimally kpartitioning ZX-diagrams, reducing each part individually, and then efficiently cross-referencing their results to conclude the overall probability amplitude of the original circuit. We then analyse how this method fares against the alternatives for circuits of various size, shape, and interconnectedness and demonstrate how it is often liable to outperform those alternatives in speed by orders of magnitude.

Recording: https://www.youtube.com/watch?v=kQoT2PXct-Y

Speaker: Giovanni de Felice (Quantinuum)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: Photonics offers a promising platform for implementations of measurement-based quantum computing. Recently proposed fusion-based architectures aim to achieve universality and fault-tolerance. In these approaches, computation is carried out by performing fusion and single-qubit measurements on a resource graph state. The verification of these architectures requires linear algebraic, probabilistic, and control flow structures to be combined in a unified formal language. This paper develops a framework for photonic quantum computing by bringing together linear optics, ZX calculus, and dataflow programming. We characterize fusion measurements that induce Pauli errors and show that they are correctable using a novel flow structure for fusion networks. We prove the correctness of new repeat-until-success protocols for the realization of arbitrary fusions and provide a graph-theoretic proof of universality for linear optics with entangled photon sources. The proposed framework paves the way for the development of compilation algorithms for photonic quantum computing.

Joint-Work With: Boldizsár Poór, Lia Yeh, and William Cashman

Recording: https://youtu.be/riAQLbto8Ho

No ZX Seminar

Speaker: Tom Ewen (Fraunhofer ITWM)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Germany/Netherlands

Abstract: This paper presents a novel approach to quantum architecture search by integrating the techniques of ZX-calculus with Genetic Programming (GP) to optimize the structure of parameterized quantum circuits employed in Quantum Machine Learning (QML). Recognizing the challenges in designing efficient quantum circuits for QML, we propose a GP framework that utilizes mutations defined via ZX-calculus, a graphical language that can simplify visualizing and working with quantum circuits. Our methodology focuses on evolving quantum circuits with the aim of enhancing their capability to approximate functions relevant in various machine learning tasks. We introduce several mutation operators inspired by the transformation rules of ZX-calculus and investigate their impact on the learning efficiency and accuracy of quantum circuits. The empirical analysis involves a comparative study where these mutations are applied to a diverse set of quantum regression problems, measuring performance metrics such as the percentage of valid circuits after the mutation, improvement of the objective, as well as circuit depth and width. Our results indicate that certain ZX-calculus-based mutations perform significantly better than others for Quantum Architecture Search (QAS) in all metrics considered. They suggest that ZX-diagram based QAS results in shallower circuits and more uniformly allocated gates than crude genetic optimization based on the circuit model.

Joint-Work With: Ivica Turkalj, Patrick Holzer, and Mark-Oliver Wolf

Recording: https://youtu.be/7rVUGd6W7kY

Speaker: Hironari Nagayoshi (The University of Tokyo)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands = 11 pm - 12 am Japan

Abstract: Continuous-variable (CV) quantum information processing is a promising candidate for large-scale fault-tolerant quantum computation. However, analysis of CV quantum process relies mostly on direct computation of the evolution of operators in the Heisenberg picture, and the features of CV space has yet to be thoroughly investigated in an intuitive manner. One key ingredient for further exploration of CV quantum computing is the construction of a computational model that brings visual intuition and new tools for analysis. In this paper, we delve into a graphical computational model, inspired by a similar model for qubit-based systems called the ZX calculus, that enables the representation of arbitrary CV quantum process as a simple directed graph. We demonstrate the utility of our model as a graphical tool to comprehend CV processes intuitively by showing how equivalences between two distinct quantum processes can be proven as a sequence of diagrammatic transformations in certain cases. We also examine possible applications of our model, such as measurement-based quantum computing, characterization of Gaussian and non-Gaussian processes, and circuit optimization.

Joint-Work With: Warit Asavanant, Ryuhoh Ide, Kosuke Fukui, Atsushi Sakaguchi, Jun-ichi Yoshikawa, Nicolas C. Menicucci, and Akira Furusawa

Speaker: Joe Bacchus George (University of Cambridge)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: It was shown in [7] that universal resources can still be extracted from faulty cluster states in measurement-based quantum computing. This computational model lends itself naturally to the diagrammatic language of ZX-Calculus, a perspective not exploited in the original discussions. Independently, percolation methods have been used to decipher phase transitions in the computational efficiencies of resources. Whilst both percolation theory and the ZX language carry evident relations to graph theory, their symbiosis is yet to be exploited in simultaneity. We hope to promote a bridging of these two perspectives, demonstrating a natural duality for tackling faulty measurement-based systems.

In doing so, the techniques are used to build from the original work of [7], where we propose alternative insights regarding the extraction process of universal clusters. We eventually argue that for certain lattices, the supercritical regime is sufficient for universality in the thermodynamic limit. The results hold innately under both site and bond percolation, as well as differing lattice shapes obeying certain symmetries. Subsequently, we estimate a resource overhead, necessary for real-life cluster state preparations. Finally, our tools are adapted to more recently proposed fusion-based models [3], using network renormalisation techniques. A potential avenue for hypergraph treatments is briefly motivated and concludes our work.

Recording: https://youtu.be/g-_h0nBQ7rQ [Slides]

Speaker: Andrey Boris Khesin (Massachusetts Institute of Technology)

Time: 10 - 11 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: We introduce a representation of \([[n,k]]\) stabilizer codes as semi-bipartite graphs wherein k "input" nodes map to n "output" nodes, such that output nodes may connect to each other but input nodes may not. We prove that this graph representation is in bijection with tableaus and give an efficient compilation algorithm that transforms tableaus into graphs. We then show that this map is efficiently invertible, which gives a new universal recipe for code construction by way of finding graphs with sufficiently nice properties.

The graph representation gives insight into both code construction and algorithms. To the former, we argue that graphs provide a flexible platform for building codes particularly at smaller (non-asymptotic) scales. We construct as examples constant-size codes, e.g. a \([[54, 6, 5]]\) code and families of \([[n, \Theta(\frac{n}{log n}), \Theta(log n)]]\) and \([[n, \Omega(n^{3/5}), \Theta(n^{1/5})]]\) codes. We also leverage graphs in a probabilistic analysis to extend the quantum Gilbert-Varshamov bound into a three-way distance-rate-weight trade-off. To the latter, we show that key coding algorithms—distance approximation, weight reduction, and decoding—are unified as instances of a single optimization game on a graph. Moreover, key code properties such as distance, weight, and encoding circuit depth, are all controlled by the graph degree. We give efficient algorithms for producing simple encoding circuits whose depths scale as twice the degree and for implementing logical diagonal and certain Clifford gates with reduced depth. Finally, we construct a simple efficient decoding algorithm and prove a performance guarantee for a certain class of graphs, including the above two families. This class contains all graphs of girth at least 9, building a bridge between stabilizer codes and extremal graph theory. These results give evidence that graphs are generically useful for the study of stabilizer codes and their practical implementations.

Joint-Work With: Jonathan Z. Lu and Peter W. Shor

Recording: https://youtu.be/H4OeqkILalg

Speaker: Razin A. Shaikh (University of Oxford & Quantinuum)

Time: 10 - 11 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: 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 ZXcalculus, incorporating only the mixed-dimensional Z-spider and the qudit X-spider as generators. Our approach builds on the completeness of another graphical language, the finitedimensional 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.

Joint-Work With: Boldizsár Poór and Quanlong Wang

Recording: https://youtu.be/-35o41JfwFU

Speaker: Pranay Gorantla (University of Chicago)

Time: 9 - 10 am Chicago = 10 - 11 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: Tensor networks provide a natural language for non-invertible symmetries in general Hamiltonian lattice models. We use ZX-diagrams, which are tensor network presentations of quantum circuits, to define a non-invertible operator implementing the Wegner duality in \(3+1d\) lattice \(\mathbb{Z}_2\) gauge theory. The non-invertible algebra, which mixes with lattice translations, can be efficiently computed using ZX-calculus. We further deform the \(\mathbb{Z}_2\) gauge theory while preserving the duality and find a model with nine exactly degenerate ground states on a torus, consistent with the Lieb-Schultz-Mattis-type constraint imposed by the symmetry. Finally, we provide a ZX-diagram presentation of the non-invertible duality operators (including non-invertible parity/reflection symmetries) of generalized Ising models based on graphs, encompassing the \(1+1d\) Ising model, the three-spin Ising model, the Ashkin-Teller model, and the \(2+1d\) plaquette Ising model. The mixing (or lack thereof) with spatial symmetries is understood from a unifying perspective based on graph theory.

Joint-Work With: Shu-Heng Shao and Nathanan Tantivasadakarn

Recording: https://youtu.be/A6tjao_PEK8

Speaker: Julio C. Magdalena de la Fuente (Freie Universität Berlin)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Germany/Netherlands

Abstract: Many routines that one might want to run on a quantum computer can benefit from adaptive circuits, relying on mid-circuit measurements and feedforward operations. Any such measurement has to be compiled into a sequence of elementary gates involving only a small number of qubits. In this work, we formalize dynamical weight reduction (DWR) schemes in which a high-weight Pauli measurement is decomposed into a sequence of measurements of smaller weight at the cost of adding additional auxiliary qubits. We first present our main method, deforming a ZX diagram that represents the measurement we want to compile. We then construct a general recipe that constructs a DWR on a given connectivity whenever the underlying connectivity graph fulfills certain necessary conditions. Further, we construct a family of DWR schemes using a given number of auxiliary qubits with indications that the schemes we present are optimal in terms of spacetime resource overheads needed for a DWR. We highlight three examples that achieve a constant time or a constant space overhead, respectively. Finally, we discuss different trade-offs of space and time overhead and how they might be chosen differently on different levels of abstraction within a (fault-tolerant) quantum computation. This work showcases the flexibility in compiling a measurement circuit in terms of lower-weight measurements using deformations of ZX diagrams and can find applications in quantum error correction, quantum simulation as well as near-term quantum computing tasks where the quality of a computation highly depends on the physical implementation of a given logical operation.

Recording: https://youtu.be/F9TCCpZ76RA

Speaker: Matthew Sutcliffe (University of Oxford)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: It is known that a quantum circuit may be simulated with classical hardware via stabilizer state (T-)decomposition in \(O(2^{\alpha t})\) time, given \(t\) non-Clifford gates and a decomposition efficiency \(\alpha\). The past years have seen a number of papers presenting new decompositions of lower \(\alpha\) to reduce this runtime and enable simulation of ever larger circuits. More recently, it has been demonstrated that well placed applications of apparently weaker (higher \(\alpha\)) decompositions can in fact result in better overall efficiency when paired with the circuit simplification strategies of ZX-calculus.

In this work, we take the most generalized T-decomposition (namely vertex cutting), which achieves a poor efficiency of \(\alpha = 1\), and identify common structures to which applying this can, after simplification via ZX-calculus rewriting, yield very strong effective efficiencies \(\alpha_{\text{eff}} \ll 1\). By taking into account this broader scope of the ZX-diagram and incorporating the simplification facilitated by the well-motivated cuts, we derive a handful of efficient T-decompositions whose applicabilities are relatively frequent. In benchmarking these new ‘dynamic’ decompositions against the existing alternatives, we observe a significant reduction in overall α and hence overall runtime for classical simulation, particularly for certain common circuit classes.

Joint-Work With: Wira Azmoon Ahmad

Recording: https://youtu.be/m-UVX_eO2BU

Speaker: Jordi Arnau Montañà López (Max-Planck-Institut für Quantenoptik & University of Washington)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Germany/Netherlands

Abstract: Quantum systems can not be efficiently simulated classically due to the presence of entanglement and nonstabilizerness, also known as quantum magic. Here we study the generation of magic under evolution by a quantum circuit. To be able to provide exact solutions, we focus on the dual-unitary XXZ model and a measure of magic called stabilizer Rényi entropy (SRE). Moreover, we focus also on long-range SRE, which cannot be removed by short-depth quantum circuits. To obtain exact solutions we use a ZX-calculus representation and graphical rules for the evaluation of the required expressions. We obtain exact results for SRE after short-time evolution in the thermodynamic limit and for long-range SRE for all times and all Rényi parameters for a particular partition of the state. Since the numerical evaluation of these quantities is exponentially costly in the Rényi parameter, we verify this numerically for low Rényi parameters and accessible system sizes and provide numerical results for the long-range SRE in other partitions.

Joint-Work With: Pavel Kos

Recording: https://youtu.be/yoRfKzBmAyg

Chaired by: John van de Wetering (University of Amsterdam)

Time: 8 - 9:30 am ET = 1 - 2:30 pm UK = 2 - 3:30 pm Netherlands

Abstract: In this special edition, leading minds from different aspects of ZX research will share the state of the art and discuss interesting open problems. We aim to provide a bird's eye view of different research directions and trigger interdisciplinary discussions.

Below are invited speakers, their discussion topics and slides, as well as the recordings. This is displayed according to the order of presentation.

Speakers		Topics		Recording		Slides
Lia Yeh		Education		🎥		🕮
Miriam Backens		Measurement-Based Quantum Computing		🎥		🕮
John van de Wetering		Quantum Compilation		Pending		🕮
Razin Shaikh		ZX Completeness		🎥		🕮
Matthew Sutcliffe and Richie Yeung		Machine Learning		🎥		🕮
Aleks Kissinger		Quantum Error Correction		🎥		🕮

Speaker: Qunsheng Huang (Technical University of Munich)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands/Germany

Abstract: In quantum computing, the efficient optimization of Pauli string decompositions is a crucial aspect for the compilation of quantum circuits for many applications, such as chemistry simulations and quantum machine learning. In this paper, we propose a novel algorithm for the synthesis of trotterized time-evolution operators that results in circuits with significantly fewer gates than previous solutions. Our synthesis procedure takes the qubit connectivity of a target quantum computer into account. As a result, the generated quantum circuit does not require routing, and no additional CNOT gates are needed to run the resulting circuit on a target device. We compare our algorithm against Paulihedral and TKET, and show a significant improvement for randomized circuits and different molecular ansatzes. We also investigate the Trotter error introduced by our ordering of the terms in the Hamiltonian versus default ordering and the ordering from the baseline methods and conclude that our method on average does not increase the Trotter error.

Joint-Work With: David Winderl, Arianne Meijer–van de Griend, Richie Yeung

Recording: https://youtu.be/LlNTQqqwn4g

Speaker: Simon Burton (Quantinuum)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: A great deal of work has been done developing quantum codes with varying overhead and connectivity constraints. However, given the such an abundance of codes, there is a surprising shortage of fault-tolerant logical gates supported therein. We define a construction, such that given an input \([[n, k, d]]\) code, yields a \([[2n, 2k, \geq d]]\) symplectic double code with naturally occurring fault-tolerant logical Clifford gates. As applied to 2-dimensional \(D(\mathbb{Z}_2)\)-topological codes with genons (twists) and domain walls, we find the symplectic double is genon free, and of possibly higher genus. Braiding of genons on the original code becomes Dehn twists on the symplectic double. Such topological operations are particularly suited for architectures with all-to-all connectivity, and we demonstrate this experimentally on Quantinuum’s H1-1 trapped-ion quantum computer.

Joint-Work With: Elijah Durso-Sabina and Natalie C. Brown

Recording: https://youtu.be/mpdJdBFZNIA

Speakers: Renaud Vilmart (Université Paris-Saclay)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands/France

Abstract: 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 qu\(\mathbf{d}\)it version of the language, where all systems have the same arbitrary finite dimension d, 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.

Joint-Work With: Marc de Visme

Recording: https://youtu.be/-lUqneUsRYk

Speaker: Vivien Vandaele (Eviden Quantum Lab and Université de Lorraine)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands/France

Abstract: We propose several methods for optimizing the number of qubits in a quantum circuit while preserving the number of non-Clifford gates. One of our approaches consists in reversing, as much as possible, the gadgetization of Hadamard gates, which is a procedure used by some T-count optimizers to circumvent Hadamard gates at the expense of additional qubits. We prove the NP-hardness of this problem and we present an algorithm for solving it. We also propose a more general approach to optimize the number of qubits by showing how it relates to the problem of finding a minimal-width path-decomposition of the graph associated with a given ZX-diagram. This approach can be used to optimize the number of qubits for any computational model that can natively be depicted in ZX-calculus, such as the Pauli Fusion computational model which can represent lattice surgery operations. We also show how this method can be used to efficiently optimize the number of qubits in a quantum circuit by using the ZX-calculus as an intermediate representation.

Recording: https://youtu.be/7RkmBX_96ik

Speaker: Lia Yeh (Quantinuum and University of Oxford)

Time: 10 - 11 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: Quantum theory is often regarded as challenging to learn and teach, with advanced mathematical prerequisites ranging from complex numbers and probability theory to matrix multiplication, vector space algebra and symbolic manipulation within the Hilbert space formalism. It is traditionally considered an advanced undergraduate or graduate-level subject. In this work, we challenge the conventional view by proposing “Quantum Picturalism” as a new approach to teaching the fundamental concepts of quantum theory and computation. We establish the foundations and methodology for an ongoing educational experiment to investigate the question “From what age can students learn quantum theory if taught using a diagrammatic approach?”. We anticipate that the primary benefit of leveraging such a diagrammatic approach, which is conceptually intuitive yet mathematically rigorous, will be eliminating some of the most daunting barriers to teaching and learning this subject while enabling young learners to reason proficiently about high-level problems. We posit that transitioning from symbolic presentations to pictorial ones will increase the appeal of STEM education, attracting more diverse audience.

Joint-Work With: Selma Dundar-Coecke, Caterina Puca, Sieglinde M.-L. Pfaendler, Muhammad Hamza Waseem, Thomas Cervoni, Aleks Kissinger, Stefano Gogioso, and Bob Coecke

Recording: https://youtu.be/Je2I-Z74p7k

Speaker: John van de Wetering (University of Amsterdam)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: A catalysis state is a quantum state that is used to make some desired operation possible or more efficient, while not being consumed in the process. Recent years have seen catalysis used in state-of-the-art protocols for implementing magic state distillation or small angle phase rotations. In this paper we will see that we can also use catalysis to prove that certain gate sets are computationally universal, and to extend completeness results of graphical languages to larger fragments. In particular, we give a simple proof of the computational universality of the CS+Hadamard gate set using the catalysis of a T gate using a CS gate, which sidesteps the more complicated analytic arguments of the original proof by Kitaev. This then also gives us a simple self-contained proof of the computational universality of Toffoli+Hadamard. Additionally, we show that the phase-free ZH-calculus can be extended to a larger complete fragment, just by using a single catalysis rule (and one scalar rule).

Joint-Work With: Aleks Kissinger and Neil J. Ross

Recording: https://youtu.be/guZ5cVECJGI

Speaker: Alexander Koziell-Pipe (University of Oxford)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: In this work, we train a graph neural network with reinforcement learning to more efficiently simulate quantum circuits using the ZX-calculus. Our experiments show a marked improvement in simulation efficiency using the trained model over existing methods that do not incorporate AI. In this way, we demonstrate a machine learning model that can reason effectively within a mathematical framework such that it enhances scientific research in the important domain of quantum computing.

Joint-Work With: Richie Yeung and Matthew Sutcliffe

Recording: https://youtu.be/fTN_AiLbepg

Speakers: Julio C. Magdalena de la Fuente (Freie Universität Berlin) and Josias Old (Forschungszentrum Jülich, RWTH Aachen University)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands/Germany

Abstract: Analyzing and developing new quantum error-correcting schemes is one of the most prominent tasks in quantum computing research. In such efforts, introducing time dynamics explicitly in both analysis and design of error-correcting protocols constitutes an important cornerstone. In this work, we present a graphical formalism based on tensor networks to capture the logical action and error-correcting capabilities of any Clifford circuit with Pauli measurements. We showcase the functioning of the formalism on new Floquet codes derived from topological subsystem codes, which we call XYZ ruby codes. Based on the projective symmetries of the building blocks of the tensor network we develop a framework of Pauli flows. Pauli flows allow for a graphical understanding of all quantities entering an error correction analysis of a circuit, including different types of QEC experiments, such as memory and stability experiments. We lay out how to derive a well-defined decoding problem from the tensor network representation of a protocol and its Pauli flows alone, independent of any stabilizer code or fixed circuit. Importantly, this framework applies to all Clifford protocols and encompasses both measurement-based and circuit-based approaches to fault tolerance. We apply our method to our new family of dynamical codes which are in the same topological phase as the \(2+1\)-dimensional color code, making them a promising candidate for low-overhead logical gates. In contrast to its static counterpart, the dynamical protocol applies a \(\mathbb{Z}_3\) automorphism to the logical Pauli group every three timesteps. We highlight some of its topological properties and comment on the anyon physics behind a planar layout. Lastly, we benchmark the performance of the XYZ ruby code on a torus by performing both memory and stability experiments and find competitive circuit-level noise thresholds of \(\approx 0.18%\), comparable with other Floquet codes and \(2+1\)-dimensional color codes.

Joint-Work With: Alex Townsend-Teague, Manuel Rispler, Jens Eisert, and Markus Müller

Recording: https://youtu.be/JjcPwUOUpJk

Speaker: Razin A. Shaikh (University of Oxford)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: While the ZX and ZW calculi have been effective as graphical reasoning tools for finite-dimensional quantum computation, the possibilities for continuous-variable quantum computation (CVQC) in infinite-dimensional Hilbert space are only beginning to be explored. In this work, we formulate a graphical language for CVQC. Each diagram is an undirected graph made of two types of spiders: the Z spider from the ZX calculus defined on the reals, and the newly introduced Fock spider defined on the natural numbers. The Z and X spiders represent functions in position and momentum space respectively, while the Fock spider represents functions in the discrete Fock basis. In addition to the Fourier transform between Z and X, and the Hermite transform between Z and Fock, we present exciting new graphical rules capturing heftier CVQC interactions.

We ensure this calculus is complete for all of Gaussian CVQC interpreted in infinite-dimensional Hilbert space, by translating the completeness in affine Lagrangian relations by Booth, Carette, and Comfort. Applying our calculus for quantum error correction, we derive graphical representations of the Gottesman-Kitaev-Preskill (GKP) code encoder, syndrome measurement, and magic state distillation of Hadamard eigenstates. Finally, we elucidate Gaussian boson sampling by providing a fully graphical proof that its circuit samples submatrix hafnians.

Joint-Work With: Lia Yeh and Stefano Gogioso

Recording: https://youtu.be/12QCw7PJBU4

Speaker: Aleks Kissinger (University of Oxford)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: This is the second in a series of "graphical grokking" papers in which we study how stabiliser codes can be understood using the ZX-calculus. In this paper we show that certain complex rules involving ZX-diagrams, called spider nest identities, can be captured succinctly using the scalable ZX-calculus, and all such identities can be proved inductively from a single new rule using the Clifford ZX-calculus. This can be combined with the ZX picture of CSS codes, developed in the first "grokking" paper, to give a simple characterisation of the set of all transversal diagonal gates at the third level of the Clifford hierarchy implementable in an arbitrary CSS code.

Joint-Work With: John van de Wetering

Recording: https://youtu.be/l8U9pHzPHpw

Speaker: Dichuan (David) Gao (University of Oxford)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: We propose a generalization of the graphical ZH calculus to qudits of prime-power dimensions, implementing Galois field arithmetic over its computational basis. This is an extension of a previous result by Roy et.al. which implemented arithmetic of prime-sized fields; and an alternative to a result by de Beaudrap and East which extended the ZH to implement cyclic ring arithmetic rather than field arithmetic. We show this generalized ZH calculus to be universal over matrices with entries in the correct ring. We then use the Galois-qudit ZH calculus to develop a graphical theory of the qudit Pauli group and therefore of qudit error correction. In particular we demonstrate that the family of Quantum Reed-Solomon codes, which are known to saturate the quantum Singleton bound, has a particularly natural representation in the Galois-qudit ZH calculus.

Joint-Work With: Razin A. Shaikh, Lia Yeh, and Aleks Kissinger

Recording: https://youtu.be/o4biik1qI7g

Speaker: Daniel Bochen Tan (Harvard University)

Time: 9 - 10 am ET = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: Quantum error correction is necessary for largescale quantum computing. A promising quantum error correcting code is the surface code. For this code, fault-tolerant quantum computing (FTQC) can be performed via lattice surgery, i.e., splitting and merging patches of code. Given the frequent use of certain lattice-surgery subroutines (LaS), it becomes crucial to optimize their design in order to minimize the overall spacetime volume of FTQC. In this study, we define the variables to represent LaS and the constraints on these variables. Leveraging this formulation, we develop a synthesizer for LaS, LaSsynth, that encodes a LaS construction problem into a SAT instance, subsequently querying SAT solvers for a solution. Starting from a baseline design, we can gradually invoke the solver with shrinking spacetime volume to derive more compact designs. Due to our foundational formulation and the use of SAT solvers, LaSsynth can exhaustively explore the design space, yielding optimal designs in volume. For example, it achieves 8% and 18% volume reduction respectively over two states-of-the-art human designs for the 15-to-1 T-factory, a bottleneck in FTQC.

Joint-Work With: Murphy Yuezhen Niu and Craig Gidney

Recording: https://youtu.be/-8-6Vd3z5rc

Chaired by: John van de Wetering (University of Amsterdam)

Time: 10 - 11 am ET = 3 - 4 pm UK = 4 - 5 pm Netherlands

Speaker: John van de Wetering (University of Amsterdam)

Time: 10 - 11 am ET = 3 - 4 pm UK = 4 - 5 pm Netherlands

Abstract: I will show an elementary proof that optimising T-count in quantum circuits is NP-hard. The same argument also applies to showing that optimising Toffoli count in classical reversible circuits, and optimising parametrised phase gates in a parametrised circuit is NP-hard. Then I will discuss in more detail the problem of optimising the number of parameters in a parametrised circuit and show that while it is hard in general, if we restrict to circuits where each parameter occurs on a unique gate, and the only allowed reductions are affine, then an existing strategy based on the ZX-calculus finds the optimal number. This talk is based on work with Matthew Amy (https://arxiv.org/abs/2310.05958) and Richie Yeung, Tuomas Laakkonen and Aleks Kissinger (https://arxiv.org/abs/2401.12877).

Recording: https://youtu.be/nhWeIdAXYyU

Speaker: Robert Booth (University of Edinburgh)

Time: 10 - 11 am ET = 3 - 4 pm UK = 4 - 5 pm Netherlands

Abstract: We give generators and relations for the hypergraph props of Gaussian relations and positive affine Lagrangian relations. The former extends Gaussian probabilistic processes by completely-uninformative priors, and the latter extends Gaussian quantum mechanics with infinitely-squeezed states. These presentations are given by adding a generator to the presentation of real affine relations and of real affine Lagrangian relations which freely codiscards effects, as well as certain rotations. The presentation of positive affine Lagrangian relations provides a rigorous justification for many common yet informal calculations in the quantum physics literature involving infinite-squeezing. Our presentation naturally extends Menicucci et al.’s graph-theoretic representation of Gaussian quantum states with a representation for Gaussian transformations. Using this graphical calculus, we also give a graphical proof of Braunstein and Kimble’s continuous-variable quantum teleportation protocol. We also interpret the LOv-calculus, a diagrammatic calculus for reasoning about passive linear-optical quantum circuits in our graphical calculus. Moreover, we show how our presentation allows for additional optical operations such as active squeezing.

Joint-Work With: Titouan Carette and Cole Comfort

Recording: https://youtu.be/Ar_H0ra-45w

Speaker: Grace M. Sommers (Princeton University)

Time: 10 - 11 am ET = 3 - 4 pm UK = 4 - 5 pm Netherlands

Abstract: In chaotic quantum systems, the entanglement of a region A can be described in terms of the surface tension of a spacetime membrane pinned to the boundary of A. Here, we interpret the tension of this entanglement membrane in terms of the rate at which information “flows” across it. For any orientation of the membrane, one can define (generically nonunitary) dynamics across the membrane; we explore this dynamics in various space-time translation-invariant (STTI) stabilizer circuits in one and two spatial dimensions. We find that the flux of information across the membrane in these STTI circuits reaches a steady state. In the cases where this dynamics is nonunitary and the steady state flux is nonzero, this occurs because the dynamics across the membrane is unitary in a subspace of extensive entropy. This generalized unitarity is present in a broad class of STTI stabilizer circuits, and is also present in some special non-stabilizer models. The existence of multiple unitary (or generalized unitary) directions forces the entanglement membrane tension to be a piecewise linear function of the orientation of the membrane; in this respect, the entanglement membrane behaves like an interface in a zero-temperature classical lattice model. We argue that entanglement membranes in random stabilizer circuits that produce volume-law entanglement are also effectively at zero temperature.

Joint-Work With: Sarang Gopalakrishnan, Michael J. Gullans, and David A. Huse

Recording: https://youtu.be/q2UvZS3OmRI

Speaker: Matthew Sutcliffe (University of Oxford)

Time: 10 - 11 am ET = 3 - 4 pm UK = 4 - 5 pm Netherlands

Abstract: Clifford quantum circuits can be classically simulated very efficiently by reduction via the ZX-calculus. Classically simulating Clifford+T quantum circuits, on the other hand, is a notoriously inefficient task. Such circuits may be decomposed into sums of efficiently simulable Clifford diagrams, though the number of terms grows exponentially with the initial T-count. In a number of related tasks, such as weak simulation and computing marginal probabilities, this technique results in reductions over many similarly structured diagrams. In the first of two recent papers, we show how the work in reducing these similar diagrams can be shared via parameterisation of the diagrams and the rewriting rules of ZX-calculus, replacing complex reductions with simple GPU-parallelisable evaluations. With this approach, we are able, in many cases, to simulate Clifford+T quantum circuits upwards of 100x quicker. In the second paper, we offer a method - based on propagating weights from common local structures - that procedurally optimises a decomposition strategy to the specific circuit given, rather than relying on a “one-size-fits-all” strategy for all circuits. Applied to random structured circuits, we achieve a decomposition efficiency of 0.1≤ α ≤ 0.2, corresponding to an exponential improvement over the conventional approaches.

[1] Fast Classical Simulation of Quantum Circuits via Parametric Rewriting in the ZX-Calculus

[2] Procedurally Optimised ZX-Diagram Cutting for Efficient T-Decomposition in Classical Simulation

Joint-Work With: Aleks Kissinger

Recording: https://youtu.be/WP0ZVSOsOOM

Speaker: Léo Colisson (QuSoft and Centrum Wiskunde & Informatica)

Time: 10 - 11 am ET = 3 - 4 pm UK = 4 - 5 pm Netherlands

Abstract: Graph state verification protocols allow multiple parties to share a graph state while checking that the state is honestly prepared, even in the presence of malicious parties. Since graph states are the starting point of numerous quantum protocols, it is crucial to ensure that graph state verification protocols can safely be composed with other protocols, this property being known as composable security. Previous works conjectured that such a property could not be proven within the abstract cryptography framework: we disprove this conjecture by showing that all graph state verification protocols can be turned into a composably secure protocol with respect to the natural functionality for graph state preparation. Moreover, we show that any unchanged graph state verification protocols can also be considered as composably secure for a slightly different, yet useful, functionality. Finally, we show that these two results are optimal, in the sense that any such generic result, considering arbitrary black-box protocols, must either modify the protocol or consider a different functionality.

Along the way, we show a protocol to generalize entanglement swapping to arbitrary graph states that might be of independent interest.

Joint-Work With: Damian Markham and Raja Yehia

Recording: https://youtu.be/slZWkYTOKnI

Speaker: Korbinian Staudacher (Ludwig-Maximilians-Universität München)

Time: 10 - 11 am ET = 3 - 4 pm UK = 4 - 5 pm Germany/Netherlands

Abstract: Quantum circuit synthesis describes the process of converting arbitrary unitary operations into a gate sequence of a fixed universal gate set, usually defined by the operations native to a given hardware platform. Most current synthesis algorithms are designed to synthesize towards a set of single-qubit rotations and an additional entangling two-qubit gate, such as CX, CZ, or the Mølmer–Sørensen gate. However, with the emergence of neutral atom-based hardware and their native support for gates with more than two qubits, synthesis approaches tailored to these new gate sets become necessary. In this work, we present an approach to synthesize (multi-) controlled phase gates using ZX-calculus. By representing quantum circuits as graph-like ZX-diagrams, one can utilize the distinct graph structure of diagonal gates to identify multi-controlled phase gates inherently present in some quantum circuits even if none were explicitly defined in the original circuit. We evaluate the approach on a wide range of benchmark circuits and compare them to the standard Qiskit synthesis regarding its circuit execution time for neutral atom-based hardware with native support of multi-controlled gates. Our results show possible advantages for current state-of-the-art hardware and represent the first exact synthesis algorithm supporting arbitrary-sized multi-controlled phase gates.

Joint-Work With: Ludwig Schmid, Johannes Zeiher, Robert Wille, and Dieter Kranzlmüller

Recording: https://youtu.be/B1P6dcJrYfk

Speaker: Nicolas Heurtel (Quandela and Université Paris-Saclay)

Time: 10 - 11 am ET = 3 - 4 pm UK = 4 - 5 pm France/Netherlands

Abstract: Linear optical circuits can be used to manipulate the quantum states of photons as they pass through components including beam splitters and phase shifters. Those photonic states possess a particularly high level of expressiveness, as they reside within the bosonic Fock space, an infinite-dimensional Hilbert space. However, in the domain of linear optical quantum computation, these basic components may not be sufficient to efficiently perform all computations of interest. To address this limitation it is common to add auxiliary sources and detectors, which enable projections onto auxiliary photonic states and thus increase the versatility of the processes. In this paper, we introduce the \(LO_{fi}\)-calculus, a graphical language to reason on the infinite-dimensional bosonic Fock space with circuits composed of four core elements of linear optics: the phase shifter, the beam splitter, and auxiliary sources and detectors with bounded photon number. We present an equational theory that we prove to be complete: two \(LO_{fi}\)-circuits represent the same quantum process if and only if one can be transformed into the other with the rules of the \(LO_{fi}\)-calculus. We give a unique and compact universal form for such circuits.

Recording: https://youtu.be/_0l1KjI0nfA

Speaker: Daniel Azses (Tel Aviv University)

Time: 10 - 11 am ET = 3 - 4 pm UK = 4 - 5 pm Netherlands = 5 - 6 pm Israel

Abstract: Measurement-based quantum computation (MBQC) is a universal platform to realize unitary gates, only using measurements which act on a pre-prepared entangled resource state. By deforming the measurement bases, as well as the geometry of the resource state, we show that MBQC circuits always transmit and act on the input state but generally realize nonunitary logical gates. In contrast to the stabilizer formalism which is often used for unitary gates, we find that ZX calculus is an ideal computation method of these nonunitary gates. As opposed to unitary gates, nonunitary gates can not be applied with certainty, due to the randomness of quantum measurements. We maximize the success probability of realizing nonunitary gates, and discuss applications including imaginary time evolution, which we demonstrate on a noisy intermediate scale quantum device.

Joint-Work With: Jonathan Ruhman and Eran Sela

Recording: https://youtu.be/-ccLsj4McpA

Speaker: Piotr Mitosek (University of Birmingham)

Time: 10 - 11 am ET = 3 - 4 pm UK = 4 - 5 pm Netherlands

Abstract: The ZH calculus is a graphical language for quantum computation reasoning. The phase-free variant offers a simple set of generators that guarantee universality. ZH calculus is effective in MBQC and analysis of quantum circuits constructed with the universal gate set Toffoli+H. While circuits naturally translate to ZH diagrams, finding an ancilla-free circuit equivalent to a given diagram is hard. Here, we show that circuit extraction for phase-free ZH calculus is #P-hard, extending the existing result for ZX calculus. Another problem believed to be hard is comparing whether two diagrams represent the same process. We show that two closely related problems are NP#P-complete. The first problem is: given two processes represented as diagrams, determine the existence of a computational basis state on which they equalize. The second problem is checking whether the matrix representation of a given diagram contains an entry equal to a given number. Our proof adapts the proof of Cook-Levin theorem to a reduction from a non-deterministic Turing Machine with access to #P oracle.

Recording: https://youtu.be/EuqikLkSHX8

Speaker: Daniel Litinski (PsiQuantum)

Time: 10 - 11 am ET = 3 - 4 pm UK = 4 - 5 pm Germany/Netherlands

Abstract: We use Shor's algorithm for the computation of elliptic curve private keys as a case study for resource estimates in the silicon-photonics-inspired active-volume architecture. Here, a fault-tolerant surface-code quantum computer consists of modules with a logarithmic number of non-local inter-module connections, modifying the algorithmic cost function compared to 2D-local architectures. We find that the non-local connections reduce the cost per key by a factor of 300-700 depending on the operating regime. At 10% threshold, assuming a 10-μs code cycle and non-local connections, one key can be generated every 10 minutes using 6000 modules with 1152 physical qubits each. By contrast, a device with strict 2D-local connectivity requires more qubits and produces one key every 38 hours. We also find simple architecture-independent algorithmic modifications that reduce the Toffoli count per key by up to a factor of 5. These modifications involve reusing the stored state for multiple keys and spreading the cost of the modular division operation over multiple parallel instances of the algorithm.

Speaker: Tobias Stollenwerk (Institute for Quantum Computing Analytics (PGI-12), Jülich Research Center)

Time: 10 - 11 am ET = 3 - 4 pm UK = 4 - 5 pm Germany/Netherlands

Abstract: Parameterized quantum circuits are attractive candidates for potential quantum advantage in the near term and beyond. At the same time, as quantum computing hardware not only continues to improve but also begins to incorporate new features such as mid-circuit measurement and adaptive control, opportunities arise for innovative algorithmic paradigms. In this work we focus on measurement-based quantum computing protocols for approximate optimization, in particular related to quantum alternating operator ans¨atze (QAOA), a popular quantum circuit model approach to combinatorial optimization. For the construction and analysis of our measurement-based protocols we demonstrate that diagrammatic approaches, specifically ZX-calculus and its extensions, are effective for adapting such algorithms to the measurement-based setting. In particular we derive measurement patterns for applying QAOA to the broad and important class of QUBO problems. We further outline how for constrained optimization, hard problem constraints may be directly incorporated into our protocol to guarantee the feasibility of the solution found and avoid the need for dealing with penalties. Finally we discuss the resource requirements and tradeoffs of our approach to that of more traditional quantum circuits.

Joint-Work With: Stuart Hadfield

Recording: https://youtu.be/dwLxBdhT0ZU

Speaker: Maximilian Nägele (Max Planck Institute for the Science of Light)

Time: 10 - 11 am EST = 3 - 4 pm UK = 4 - 5 pm Germany/Netherlands

Abstract: ZX-diagrams are a powerful graphical language for the description of quantum processes with applications in fundamental quantum mechanics, quantum circuit optimization, tensor network simulation, and many more. The utility of ZX-diagrams relies on a set of local transformation rules that can be applied to them without changing the underlying quantum process they describe. These rules can be exploited to optimize the structure of ZX diagrams for a range of applications. However, finding an optimal sequence of transformation rules is generally an open problem. In this work, we bring together ZX-diagrams with reinforcement learning, a machine learning technique designed to discover an optimal sequence of actions in a decision-making problem and show that a trained reinforcement learning agent can significantly outperform other optimization techniques like a greedy strategy or simulated annealing. The use of graph neural networks to encode the policy of the agent enables generalization to diagrams much bigger than seen during the training phase.

Recording: https://youtu.be/jVu-7MBImvs

Joint-Work With: Florian Marquardt

Speaker: Jan Nogué Gómez (Qilimanjaro Quantum Tech)

Time: 10 - 11 am EST = 3 - 4 pm UK = 4 - 5 pm Spain/Netherlands

Abstract: We propose a novel Reinforcement Learning (RL) method for optimizing quantum circuits using the graph-like representation of a ZX-diagram. The agent, trained using the Proximal Policy Optimization (PPO) algorithm, employs Graph Neural Networks to approximate the policy and value functions. We test our approach for two differentiated circuit size regimes of increasing relevance, and benchmark it against the best-performing ZX-calculus based algorithm of the PyZX library, a state-of-the-art tool for circuit optimization in the field. We demonstrate that the agent can generalize the strategies learned from 5-qubit circuits to 20-qubit circuits of up to 450 Clifford gates, with enhanced compressions with respect to its counterpart while remaining competitive in terms of computational performance.

Recording: https://youtu.be/PwqKvMNVzj0

Joint-Work With: Jordi Riu, Gerard Vilaplana, Artur Garcia-Saez, and Marta P. Estarellas

Speaker: Shuxiang Cao (University of Oxford)

Time: 10 - 11 am EST = 3 - 4 pm UK = 4 - 5 pm Netherlands

Abstract: Blind quantum computation (BQC) protocols enable quantum algorithms to be executed on third-party quantum agents while keeping the data and algorithm confidential. The previous proposals for measurement-based BQC require preparing a highly entangled cluster state. In this paper, we show that such a requirement is not necessary. Our protocol only requires pre-shared Bell pairs between delegated quantum agents, and there is no requirement for any classical or quantum information exchange between agents during the execution. Our proposal requires fewer quantum resources than previous proposals by eliminating the need for a universal cluster state.

Recording: https://youtu.be/xFihqZ5cIZ0

Speaker: Tommy McElvanney (University of Birmingham)

Time: 10 - 11 am EST = 3 - 4 pm UK = 4 - 5 pm Netherlands

Abstract: In the one-way model of measurement-based quantum computation (MBQC), computation proceeds via measurements on a resource state. So-called flow conditions ensure that the overall computation is deterministic in a suitable sense, with Pauli flow being the most general of these. Computations, represented as measurement patterns, may be rewritten to optimise resource use and for other purposes. Such rewrites need to preserve the existence of flow to ensure the new pattern can still be implemented deterministically. The majority of existing work in this area has focused on rewrites that reduce the number of qubits, yet it can be beneficial to increase the number of qubits for certain kinds of optimisation, as well as for obfuscation.

In this work, we introduce several ZX-calculus rewrite rules that increase the number of qubits and preserve the existence of Pauli flow. These rules can be used to transform any measurement pattern into a pattern containing only (general or Pauli) measurements within the XY-plane. We also give the first flow-preserving rewrite rule that allows measurement angles to be changed arbitrarily, and use this to prove that the ‘neighbour unfusion’ rule of Staudacher et al. preserves the existence of Pauli flow. This implies it may be possible to reduce the runtime of their two-qubit-gate optimisation procedure by removing the need to regularly run the costly gflow-finding algorithm.

Recording: https://youtu.be/rKbtnOgzKZQ

Joint-Work With: Miriam Backens

Speaker: Robert Booth (University of Edinburgh)

Time: 10 - 11 am EST = 3 - 4 pm UK = 4 - 5 pm Netherlands

Abstract: We give complete presentations for the dagger-compact props of affine Lagrangian and coisotropic relations over an arbitrary field. This provides a unified family of graphical languages for both affinely constrained classical mechanical systems, as well as odd-prime-dimensional stabiliser quantum circuits. To this end, we present affine Lagrangian relations by a particular class of undirected coloured graphs. In order to reason about composite systems, we introduce a powerful scalable notation where the vertices of these graphs are themselves coloured by graphs. In the setting of stabiliser quantum mechanics, this scalable notation gives an extremely concise description of graph states, which can be composed via ``phased spider fusion.'' Likewise, in the classical mechanical setting of electrical circuits, we show that impedance matrices for reciprocal networks are presented in essentially the same way.

Recording: https://youtu.be/QJcEeZsL2S8

Joint-Work With: Titouan Carette and Cole Comfort

Speaker: Ben Caldwell (University of Chicago)

Time: 9 - 10 am CT = 10 - 11 am EST = 3 - 4 pm UK = 4 - 5 pm Netherlands

Abstract: Mathematical 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.

Recording: https://youtu.be/5cwd9u8jivo

Joint-Work With: Adrian Lehmann, Bhakti Shah, and Robert Rand

Speaker: Yehua Liu (PsiQuantum)

Time: 7 - 8 am PST = 10 - 11 am EST = 3 - 4 pm UK = 4 - 5 pm Netherlands

Abstract: Fault-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.

Recording: https://youtu.be/wCwkOX03hq8/a>

Joint-Work With: Hector Bombin, Chris Dawson, Terry Farrelly, Naomi Nickerson, Mihir Pant, Fernando Pastawski, and Sam Roberts

Speaker: Alejandro David Villoria Gonzalez (Leiden Institute of Advanced Computer Science)

Time: 10 - 11 am EST = 3 - 4 pm UK = 4 - 5 pm Netherlands

Abstract: In 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.

Recording: https://youtu.be/yJCaEdw_Ots

Joint-Work With: Henning Basold and Alfons Laarman

Speaker: Boldizsár Poór (Quantinuum)

Time: 10 - 11 am EST = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: Finite-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.

Recording: https://youtu.be/AhEEOqpzADY

Joint-Work With: Quanlong Wang and Razin A. Shaikh

Speaker: Richie Yeung (Quantinuum, University of Oxford)

Time: 10 - 11 am EST = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: ZX calculus is a graphical language for reasoning about linear maps. Maps are represented as graphs, and reasoning amounts to graph rewrites. The main applications of ZX calculus are in quantum computation. We train small transformers to simplify ZX graphs, i.e. perform resource optimisation of quantum circuits. Preliminary experiments show that transformers can be trained to simplify CNOT and Clifford circuits with high accuracy. These are the simplest kinds of ZX graphs, in the sense that there exists an efficient rewrite strategy. We also show evidence that transformers learn to simplify the more complex Clifford + T graphs, for which in general there does not exist an efficient simplification algorithm.

Recording: https://youtu.be/gW_ZzRoIUMU

Joint-Work With: Konstantinos Meichanetzidis, Alexandre Krajenbrink, and François Charton

Speaker: Alexandre Clément (Université Paris-Saclay)

Time: 10 - 11 am EST = 2 - 3 pm UK = 3 - 4 pm France/Netherlands

Abstract: We 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.

Recording: https://youtu.be/3q8eqihdtTs

Joint-Work With: Noé Delorme and Simon Perdrix

Speaker: Alex Townsend-Teague (Freie Universität Berlin)

Time: 9 - 10 am EDT = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: Floquet codes are a recently(-ish!) discovered type of quantum error correction code. Operationally, they work by repeatedly measuring multi-qubit Pauli operators, and thus can be thought of as generalising stabilizer codes. However, they have the defining property that representatives of logical Pauli operators in a Floquet code are dynamic - they change over time. In this talk, we will consider the following perspective, which unifies stabilizer and Floquet codes: every Clifford ZX-diagram is a quantum error correcting code. We will introduce tools needed to understand this view and discuss its applications.

Recording: https://www.youtube.com/watch?v=pP1DaKxlC_8

Joint-Work With: Julio Magdalena de la Fuente and Markus Kesselring

Chaired by: John van de Wetering (University of Amsterdam)

Time: 9 - 10 am EDT = 2 - 3 pm UK = 3 - 4 pm Netherlands

Speaker: Mark Koch (Quantinuum)

Time: 9 - 10 am EDT = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: Recent advances in classical simulation of Clifford+T circuits make use of the ZX calculus to iteratively decompose and simplify magic states into stabiliser terms. We improve on this method by studying stabiliser decompositions of ZX diagrams involving the triangle operation. We show that this technique greatly speeds up the simulation of quantum circuits involving multi-controlled gates which can be naturally represented using triangles. We implement our approach in the QuiZX library and demonstrate a significant simulation speed-up (up to multiple orders of magnitude) for random circuits and a variation of previously used benchmarking circuits. Furthermore, we use our software to contract diagrams representing the gradient variance of parametrised quantum circuits, which yields a tool for the automatic numerical detection of the barren plateau phenomenon in ansätze used for quantum machine learning. Compared to traditional statistical approaches, our method yields exact values for gradient variances and only requires contracting a single diagram. The performance of this tool is competitive with tensor network approaches, as demonstrated with benchmarks against the quimb library.

Recording: https://youtu.be/8L6jqiNg9iM

Joint-Work With: Richie Yeung and Quanlong Wang

Speaker: David Winderl (Technical University of Munich)

Time: 9 - 10 am EDT = 2 - 3 pm UK = 3 - 4 pm Germany

Abstract: The synthesis of quantum circuits from so-called Pauli-Polynomials in the ZX Calculus facilitates quantum circuit optimization. In this work, we will develop three algorithms that can perform this task in an architecture-aware fashion. On top, we develop an architecture-aware synthesis method for synthesizing Clifford-Tableaus. Our results show an increase in the Reduction of CNOT-Gates for larger circuits for both Clifford circuits and circuits described by Pauli-Polynomials, which outlines the capabilities of architecture-aware synthesis in the circuit optimization realm.

Recording: https://youtu.be/V8dlZgSQYiU

Joint-Work With: Qunsheng Huang, Arianne Meijer-van de Griend, and Richie Yeung

Speaker: Boldizsár Poór (Quantinuum)

Time: 9 - 10 am EDT = 1 - 2 pm UK = 2 - 3 pm Netherlands

Abstract: In this paper, we develop a graphical calculus to rewrite photonic circuits involving light-matter interactions and non-linear optical effects. We introduce the infinite ZW calculus, a graphical language for linear operators on the bosonic Fock space which captures both linear and non-linear photonic circuits. This calculus is obtained by combining the QPath calculus, a diagrammatic language for linear optics, and the recently developed qudit ZXW calculus, a complete axiomatisation of linear maps between qudits. It comes with a 'lifting' theorem allowing to prove equalities between infinite operators by rewriting in the ZXW calculus. We give a method for representing bosonic and fermionic Hamiltonians in the infinite ZW calculus. This allows us to derive their exponentials by diagrammatic reasoning. Examples include phase shifts and beam splitters, as well as non-linear Kerr media and Jaynes-Cummings light-matter interaction.

Recording: https://youtu.be/nOli0jB2hoY

Joint-Work With: Giovanni de Felice, Razin A. Shaikh, Lia Yeh, Quanlong Wang, and Bob Coecke

Speaker: Fernando Pastawski (PsiQuantum)

Time: 9 - 10 am EDT = 2 - 3 pm UK = 3 - 4 pm Germany/Netherlands

Abstract: There are several models of quantum computation which exhibit shared fundamental fault-tolerance properties. This article makes commonalities explicit by presenting these different models in a unifying framework based on the ZX calculus. We focus on models of topological fault tolerance – specifically surface codes – including circuit-based, measurement-based and fusion-based quantum computation, as well as the recently introduced model of Floquet codes. We find that all of these models can be viewed as different flavors of the same underlying stabilizer fault-tolerance structure, and sustain this through a set of local equivalence transformations which allow mapping between flavors. We anticipate that this unifying perspective will pave the way to transferring progress among the different views of stabilizer fault-tolerance and help researchers familiar with one model easily understand others.

Joint-Work With: Hector Bombin, Daniel Litinski, Naomi Nickerson, and Sam Roberts

Speaker: Calum Holker (University of Oxford)

Time: 9 - 10 am EDT = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: Optimising quantum circuits to reduce physical implementation resources is crucial, especially in the context of near term hardware, which is limited by quantum volume. This paper aims to minimise T-count and two-qubit gate count by building on ZX-calculus-based circuit optimisation strategies. By translating a circuit into a ZX-diagram it can be simplified before being extracted back into a circuit. We assert that simplifications preserve a graph-theoretic property called causal flow. This has the advantage that qubit lines are well defined throughout simplification, permitting a trivial extraction procedure and in turn enabling the calculation of an individual transformation's impact on the resulting circuit. A general algorithm for a decision strategy is introduced, inspired by the heuristic based method of Staudacher et. al. (2022). Also, both phase teleportation and the neighbour unfusion rule are generalised. In particular allowing unfusion of multiple neighbours is shown to lead to significant improvements in optimisation. When run on a set of benchmark circuits, the algorithm developed reduces the two qubit gate count by an average of 19.6%, beating both the previous best ZX-based strategy (14.2%) as well as the previous best non-ZX strategy (18.9%). This lays the foundation for improvements such as more advanced decision models.

Recording: https://youtu.be/9wO_-QcSAaM

Joint-Work With: Aleks Kissinger

Speaker: Razin A. Shaikh (University of Oxford & Quantinuum)

Time: 9 - 10 am EDT = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: ZXLive is a user-friendly, interactive tool designed for reasoning with the ZX calculus. It offers a simple interface to draw ZX diagrams or import them from various file formats. Once you have your diagram, switch to proof mode where you can easily apply the ZX rules and construct proofs. The tool supports exporting proofs to Tikzit for adding it into papers. These proofs can also be saved as lemmas, which can be used later as rewrite rules in other proofs. With support for custom rewrites, you can add new rules to ZXLive that are relevant for your current project. Beyond the vanilla ZX calculus, ZXLive supports the generators and rewrite rules of ZH, ZW, and ZXW calculi.

ZXLive is available as an open-source project on GitHub: https://github.com/Quantomatic/zxlive. Under the hood, it is powered by PyZX, which is an open-source python library for ZX. Along with being a useful tool for ZX practitioners, ZXLive is also a great educational tool for learning and experimenting with the ZX calculus.

Recording: https://youtu.be/J--c2q-KOc8

Speaker: Zipeng Wu (Hong Kong University of Science and Technology)

Time: 9 - 10 am EDT = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: Quantum Error-Correcting Codes (QECCs) are vital for ensuring the reliability of quantum computing and quantum communication systems. Among QECCs, stabilizer codes, particularly graph codes, have attracted considerable attention due to their unique properties and potential applications. Concatenated codes, which combine multiple layers of quantum codes, offer a powerful technique for achieving high levels of error correction with a relatively low resource overhead. In this paper, we examine the concatenation of graph codes using the powerful and versatile graphical language of ZX-calculus. We establish a correspondence between the encoding map and ZX-diagrams, and provide a simple proof of the equivalence between encoding maps in the Pauli X basis and the graphic operation "generalized local complementation" (GLC) as previously demonstrated in [J. Math. Phys. 52, 022201]. Our analysis reveals that the resulting concatenated code remains a graph code only when the encoding qubits of the same inner code are not directly connected. When they are directly connected, additional Clifford operations are necessary to transform the concatenated code into a graphcode, thus generalizing the results in [J. Math. Phys. 52, 022201]. We further explore concatenated graph codesin different bases, including the examination of holographic codes as concatenated graph codes. Our findings showcase the potential of ZX-calculus in advancing the field of quantum error correction.

Recording: https://youtu.be/mbqx0OudY5c

Joint-Work With: Song Cheng and Bei Zeng

Speaker: Cole Comfort (MOCQUA team, LORIA, Université de Lorraine)

Time: 9 - 10 am EDT = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: The ZX-calculus is a family of graphical languages used to reason about fragments of complex-valued matrices under the bilinear tensor product. However, by restricting oneself to the phase-free fragment of the ZX-calculus, one obtains a graphical calculus for reasoning about linear subspaces over the integers modulo p under the direct sum. In order to expose this correspondence, I will review the scalable notation and discuss how this allows one to perform basic constructions in linear algebra using the ZX-calculus. I will also discuss how one can capture more general notions of subspaces by adding more generators to the ZX-calculus. In particular, I will discuss how extending this correspondence between the phase-free ZX-calculus and linear subspaces over Z/pZ has can be applied to stabilizer codes as well as SAT solving. I will also discuss open problems which may be interesting to the ZX-calculus community.

Recording: https://youtu.be/80rx6LzWCWo

Speaker: Jiaxin Huang (The University of Hong Kong)

Time: 9 - 10 am EDT = 2 - 3 pm UK = 3 - 4 pm Netherlands

Abstract: In this work, we present a generic approach to transform CSS codes by building upon their equivalence to phase-free ZX diagrams. Using the ZX calculus, we demonstrate diagrammatic transformations between encoding maps associated with different codes. As a motivating example, we give explicit transformations between the Steane code and the quantum Reed-Muller code, since by switching between these two codes, one can obtain a fault-tolerant universal gate set. To this end, we propose a bidirectional rewrite rule to find a (not necessarily transversal) physical implementation for any logical ZX diagram in any CSS code.

Then we focus on two code transformation techniques: code morphing, a procedure that transforms a code while retaining its fault-tolerant gates, and gauge fixing, where complimentary codes can be obtained from a common subsystem code (e.g., the Steane and the quantum Reed-Muller codes from the [[15,1,3,3]] code). We provide explicit graphical derivations for these techniques and show how ZX and graphical encoder maps relate several equivalent perspectives on these code-transforming operations.

Recording: https://youtu.be/ZhfQxdjodNs

Joint-Work With: Sarah Meng Li, Lia Yeh, Aleks Kissinger, Michele Mosca, and Michael Vasmer

Speaker: Nicklas Botö (Chalmers University of Technology)

Time: 9 - 10 am EDT = 2 - 3 pm UK = 3 - 4 pm Sweden

Abstract: Working from the internal type theory of compact closed symmetric monoidal categories, we extend the linear multiplicative typed λ-calculus with observable structures to obtain a programming language suitable for the categorical view of quantum computation. At the current stage of development, the phase groups and cocommutative comonoids of observable structures are embedded into the typing and structural rules of the language to provide basis-dependent non-linearity as well as a notion of rotational computing. We show that this language provides novel programming techniques and ways of constructing categorical quantum programs, notably in the ZX-calculus. The generality of the categorical definition of the language allows it to be instantiated in any model of compact closed symmetric monoidal categories with any number of observable structures, or internal cocommutative comonoids. We hope that this creates new avenues in which to explore what quantum programming might be, and which theories it might describe.

Recording: https://youtu.be/iUHEy3PZCso

Joint-Work With: Fabian Forslund