Quantum group,

Institute for Computing and Information Sciences,

Radboud University Nijmegen

Toernooiveld 212

6525EC Nijmegen, the Netherlands

Email:

You can find me on Google Scholar, ORCID, arXiv and LinkedIn.

I am a Phd student at the Institute for Computing and Information Sciences of the Radboud University Nijmegen where I also did a Masters programme in Mathematical Physics. I was hired in 2016 as part of of the ERC advanced grant on Quantum Computation, Logic and Security. I am jointly supervised by Aleks Kissinger and Bart Jacobs.

I spent the first two years of my PhD focussing on the foundations of quantum theory and quantum computation, in particular, the question *why* nature is described by the laws of quantum mechanics.

I am currently working on optimising quantum circuits using graphical calculi like the ZX-calculus. In particular, together with Aleks, I am developing the open source optimising quantum compiler PyZX, a Python tool for doing quantum circuit optimization using the ZX-calculus.

I jointly teach the Masters course Quantum Processes and Computation with Aleks Kissinger at the Radboud University.

- Universal MBQC with generalised parity-phase interactions and Pauli measurements, Aleks Kissinger and John van de Wetering,
*Quantum 3, 134*(2019).**Abstract:**We introduce a new family of models for measurement-based quantum computation which are deterministic and approximately universal. The resource states which play the role of graph states are prepared via 2-qubit gates of the form \(\exp(-i π2nZ⊗Z)\). When \(n=2\), these are equivalent, up to local Clifford unitaries, to graph states. However, when \(n>2\), their behaviour diverges in two important ways. First, multiple applications of the entangling gate to a single pair of qubits produces non-trivial entanglement, and hence multiple parallel edges between nodes play an important role in these generalised graph states. Second, such a state can be used to realise deterministic, approximately universal computation using only Pauli Z and X measurements and feed-forward. Even though, for \(n>2\), the relevant resource states are no longer stabiliser states, they admit a straightforward, graphical representation using the ZX-calculus. Using this representation, we are able to provide a simple, graphical proof of universality. We furthermore show that for every \(n>2\) this family is capable of producing all Clifford gates and all diagonal gates in the \(n\)-th level of the Clifford hierarchy. - Reducing T-count with the ZX-calculus, Aleks Kissinger and John van de Wetering,
*arXiv preprint arXiv:1903.10477*(2019).**Abstract:**Reducing the number of non-Clifford quantum gates present in a circuit is an important task for efficiently implementing quantum computations, especially in the fault-tolerant regime. We present a new method for reducing the number of T-gates in a quantum circuit based on the ZX-calculus, which matches or beats previous approaches to T-count reduction on the majority of our benchmark circuits in the ancilla-free case, in some cases yielding up to 50% improvement. Our method begins by representing the quantum circuit as a ZX-diagram, a tensor network-like structure that can be transformed and simplified according to the rules of the ZX-calculus. We then show that a recently-proposed simplification strategy can be extended to reduce T-count using a new technique called phase teleportation. This technique allows non-Clifford phases to combine and cancel by propagating non-locally through a generic quantum circuit. Phase teleportation does not change the number or location of non-phase gates and the method also applies to arbitrary non-Clifford phase gates as well as gates with unknown phase parameters in parametrised circuits. Furthermore, the simplification strategy we use is powerful enough to validate equality of many circuits. In particular, we use it to show that our optimised circuits are indeed equal to the original ones. We have implemented the routines of this paper in the open-source library PyZX. - Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus, Ross Duncan, Aleks Kissinger, Simon Pedrix and John van de Wetering,
*arXiv preprint arXiv:1902.03178*(2019).**Abstract:**We present the theoretical foundations for a new quantum circuit optimisation technique based on an equational theory called the ZX-calculus. We first interpret quantum circuits as ZX-diagrams, which provide a flexible, lower-level language for describing quantum computations graphically. Then, using the rules of the ZX-calculus, we derive a terminating simplification procedure for ZX-diagrams based on the two graph transformations of local complementation and pivoting. While little is known about extracting an equivalent quantum circuit from an arbitrary ZX-diagram, we show that our simplification procedure preserves a graph property called focused gFlow, and use this property to derive a deterministic strategy for circuit re-extraction. Hence, this approach enables us to temporarily break the rigid quantum circuit structure and 'see around' obstructions (namely non-Clifford quantum gates) to produce significant reductions, while still maintaining enough data to extract a reduced quantum circuit at the end. - Pure Maps between Euclidean Jordan Algebras, Abraham Westerbaan, Bas Westerbaan and John van de Wetering,
*Proceedings of the 15th International Conference on Quantum Physics and Logic, Halifax, Canada, 3-7th June 2018*(2019).**Abstract:**We propose a definition of purity for positive linear maps between Euclidean Jordan Algebras (EJA) that generalizes the notion of Kraus rank one channels (\(A↦B^*AB\)). We show that this definition of purity is closed under composition and taking adjoints and thus that the pure maps form a dagger category. This dagger yields a notion of positivity: maps of the form \(f=g^\dagger∘g\) are called \(\dagger\)-positive. We show that such a \(\dagger\)-positive map \(f\) is completely determined by \(f(1)\) and equal to \(Q_{\sqrt{f(1)}}\), the Jordan algebraic version of the sequential measurement map \(A↦\sqrt{f(1)}A\sqrt{f(1)}\). The notion of \(\dagger\)-positivity therefore characterises the sequential product. These results hold for the general reason that the opposite category of EJAs with positive contractive linear maps is a \(\dagger\)-effectus. The notion of \(\dagger\)-effectus was introduced to prove similar results for the opposite category of normal completely positive contractive maps between von Neumann algebras. - Ordering quantum states and channels based on positive Bayesian evidence, John van de Wetering,
*J. Math. Phys. Vol.59-10*(2018).**Abstract:**In this paper we introduce a new partial order on quantum states that considers which states can be achieved from others by updating on 'agreeing' Bayesian evidence. We prove that this order can also be interpreted in terms of minimising worst case distinguishability between states using the concept of quantum max-divergence. This order solves the problem of which states are optimal approximations to their more pure counterparts and it shows in an explicit way that a proposed quantum analogue of Bayes' rule leads to a Bayesian update that changes the state as little as possible when updating on positive evidence. We prove some structural properties of the order, specifically that the order preserves convex mixtures and tensor products of states and that it is a domain. The uniqueness of the order given these properties is discussed. Finally we extend this order on states to one on quantum channels using the Jamiolkowski isomorphism. This order turns the spaces of unital/non-unital trace-preserving quantum channels into domains that, unlike the regular order on channels, is not trivial for unital trace-preserving channels. - Three characterisations of the sequential product, John van de Wetering,
*J. Math. Phys. Vol. 59-8*(2018).**Abstract:**It has already been established that the properties required of an abstract sequential product as introduced by Gudder and Greechie are not enough to characterise the standard sequential product \(a∘b = \sqrt{a}b\sqrt{b}\) on an operator algebra. We give three additional properties, each of which characterises the standard sequential product on either a von Neumann algebra or a Euclidean Jordan algebra. These properties are (1) invariance under application of unital order isomorphisms, (2) symmetry of the sequential product with respect to a certain inner product, and (3) preservation of invertibility of the effects. To give these characterisations we first have to study convex sigma-sequential effect algebras. We show that these objects correspond to unit intervals of spectral order unit spaces with a homogeneous positive cone. - Sequential Product Spaces are Jordan Algebras, John van de Wetering,
*arXiv preprint arXiv:1803.11139*(2018).**Abstract:**We show that finite-dimensional order unit spaces equipped with a continuous sequential product as defined by Gudder and Greechie are homogeneous and self-dual. As a consequence of the Koecher-Vinberg theorem these spaces therefore correspond to Euclidean Jordan algebras. We remark on the significance of this result in the context of reconstructions of quantum theory. In particular, we show that sequential product spaces that have locally tomographic tensor products, i.e. their vector space tensor products are also sequential product spaces, must be C* algebras. Finally we remark on a couple of ways these results can be extended to the infinite-dimensional setting of JB- and JBW-algebras and how changing the axioms of the sequential product might lead to a new characterisation of homogeneous cones. - An effect-theoretic reconstruction of quantum theory, John van de Wetering,
*arXiv preprint arXiv:1801.05798*(2018).**Abstract:**An often used model for quantum theory is to associate to every physical system a C*-algebra. From a physical point of view it is unclear why operator algebras would form a good description of nature. In this paper, we retrieve the category of finite-dimensional C*-algebras using concepts from effectus theory, a categorical logical framework generalizing classical and quantum logic. As these concepts have a physical interpretation, this motivates the usage of operator algebras as a model for quantum theory. In contrast to other reconstructions of quantum theory, we do not start with the framework of generalised probabilistic theories and instead use effect theories where no convex structure and no tensor product needs to be present. The lack of this structure in effectus theory has led to a different notion of pure maps. A map in an effectus is pure when it is a composition of a compression and a filter. These maps satisfy particular universal properties and respectively correspond to 'forgetting' and measuring the validity of an effect. We define a pure effect theory (PET) to be an effect theory where the pure maps form a dagger-category and filters and compressions are adjoint. We characterise the category of Euclidean Jordan algebras as the most general convex, finite-dimensional PET. If we in addition assume monoidal structure, then any such PET must embed into either the category of real or complex C*-algebras, which completes our reconstruction. As a PET must be quantum theory in the probabilistic convex setting, a PET without those properties can be viewed as a new type of generalisation of quantum theory. - Quantum Theory is a Quasi-stochastic Process Theory, John van de Wetering,
*Proceedings 14th International Conference on Quantum Physics and Logic, Nijmegen, The Netherlands, 3-7 July 2017*(2018).**Abstract:**There is a long history of representing a quantum state using a quasi-probability distribution: a distribution allowing negative values. In this paper we extend such representations to deal with quantum channels. The result is a convex, strongly monoidal, functorial embedding of the category of trace preserving completely positive maps into the category of quasi-stochastic matrices. This establishes quantum theory as a subcategory of quasi-stochastic processes. Such an embedding is induced by a choice of minimal informationally complete POVM's. We show that any two such embeddings are naturally isomorphic. The embedding preserves the dagger structure of the categories if and only if the POVM's are symmetric, giving a new use of SIC-POVM's, objects that are of foundational interest in the QBism community. We also study general convex embeddings of quantum theory and prove a dichotomy that such an embedding is either trivial or faithful. - Entailment Relations on Distributions, John van de Wetering,
*Proceedings of the 2016 Workshop on Semantic Spaces at the Intersection of NLP, Physics and Cognitive Science, Glasgow, Scotland, 11th June 2016*(2016).**Abstract:**In this paper we give an overview of partial orders on the space of probability distributions that carry a notion of information content and serve as a generalisation of the Bayesian order given in (Coecke and Martin, 2011). We investigate what constraints are necessary in order to get a unique notion of information content. These partial orders can be used to give an ordering on words in vector space models of natural language meaning relating to the contexts in which words are used, which is useful for a notion of entailment and word disambiguation. The construction used also points towards a way to create orderings on the space of density operators which allow a more fine-grained study of entailment. The partial orders in this paper are directed complete and form domains in the sense of domain theory.

- T-count optimization of quantum circuits using graph-theoretical rewriting of ZX-diagrams (slides). Given at EQTC2019.
- PyZX: Graph-theoretic optimization of quantum circuits (slides). Given at FOSDEM 2019.
- PyZX: Quantum circuit optimization using the ZX-calculus (slides). Given at SYCO 2.
- Reconstruction of Quantum Theory from Universal Filters (slides). Given at Foundations 2018.
- Sequential Measurement characterises Quantum Theory (slides). Given at QPL 2018.
- Purity in Euclidean Jordan Algebras (slides). Given at QPL 2018.
- Quantum Theory is a Quasi-Stochastic Process Theory (slides). Given at QPL 2017.
- Poster: Universal measurement-based quantum computation with Mølmer-Sørensen interactions and just two measurement bases (poster). Presented at TQC 2017.