The ZX Seminar

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: Click here (passcode: x6P.CW^?)

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: Click here (passcode: dV9Z.Tk+)

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: Click here (passcode: K5JN?$5c)

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: Click here (passcode: 3kpn#^yF)

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 Berlin/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: Click here (passcode: 1ydqJLw#)

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: Click here (It is in Google drive, no passcode needed)

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: Click here (passcode: =9c#nTF3)

Joint-Work With: Song Cheng and Bei Zeng

Speaker: Cole Comfort (University of Oxford)

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.

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.

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

Speakers: Nicklas Botö and Fabian Forslund (Chalmers University of Technology)

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

Abstract: We propose a quantum programming language that generalizes the λ-calculus. The language is non-linear; duplicated variables denote, not cloning of quantum data, but sharing a qubit's state; that is, producing an entangled pair of qubits whose amplitudes are identical with respect to a chosen basis. The language has two abstraction operators, ζ and ξ, corresponding to the Z- and X-bases; each abstraction operator is also parameterised by a phase, indicating a rotation that is applied to the input before it is shared. We give semantics for the language in the ZX-calculus and prove its equational theory sound. We show how this language can provide a good representation of higher-order functions in the quantum world.