Research interests

My work lies at the intersection of mathematical physics and computer science and concerns applications of techniques from noncommutative geometry to quantum computing and quantum information. In particular, I am interested in modelling higher-order quantum processes in terms of noncommutative structures. I am motivated by the quest to abstract away from the quantum circuit model of quantum computing. Quantum circuits can be regarded as “quantum assembly code”, and moving to higher-level paradigms is expected to have a similarly transformative effect on quantum computing as the development of high-level programming languages had on classical computation. The abstraction away from machine instructions in classical computing enabled modularity, compositionality, and the development of programming paradigms in which programs themselves can be treated as computational objects. Analogously, higher-level approaches to quantum computing could provide abstractions for composing and manipulating quantum processes, rather than merely specifying individual circuits. In particular, a higher-order perspective makes it possible to regard quantum processes themselves as first-class quantum objects. This raises the intriguing possibility of forming coherent superpositions of different quantum programs, so that the choice of computation itself becomes a quantum degree of freedom. Such higher-order phenomena go beyond the conventional circuit model, in which a circuit is typically fixed before the computation begins, and point towards a richer notion of quantum computation in which programs can themselves be manipulated, composed, and placed in superposition.

The first step towards such a higher-level perspective is understanding higher-order quantum processes, since these describe transformations of lower-order processes. Most current approaches describe higher-order processes via categorical constructions on a category of quantum channels (first-order quantum processes) such as presheaves. However, such constructions cannot properly describe the quantum correlations between higher-order processes.

For this reason, I strongly believe that higher-order quantum processes should be described completely in terms of noncommutative geometry since this mathematical framework provides a natural setting for the description of quantum phenomena. I pursue two approaches:

Operator spaces
Banach spaces form a model of intuitionistic linear logic, but Banach space tensor products fail to describe the composition of quantum systems. Operator spaces, spaces of linear operators on a Hilbert space, are regarded as noncommutative versions of Banach spaces. Together with Zamdzhiev, I showed that operator spaces form a model of intuitionistic linear logic that correctly describes the composition of quantum systems. Moreover, based on operator spaces, we built a model for full linear logic in which the duality between positive and negative polarity corresponds to the duality between the Schrödinger and Heisenberg picture of quantum physics. Finally, we showed that the quantum switch naturally emerges in the operator space model. The quantum switch is the prime example of an indefinite causal process that puts the order of application of subprocesses in a superposition, and is an active research subject. We showed that the Haagerup product, an operator space tensor product, can be used to detect whether a process exhibits indefinite causal order. The Haagerup product has no analog in Banach space theory, and is therefore an example of a quantum feature emerging from noncommutativity.

Discrete quantization
Discrete quantization is a systematic process that produces noncommutative analogues of classical categories by internalizing their defining structures in the category qRel, a noncommutative generalization of the category Rel of sets and binary relations. The objects of qRel are quantum sets, i.e., (possibly infinite) direct sums of matrix algebras, and its morphisms are noncommutative generalizations of binary relations called quantum relations. Established quantum structures that can be understood and studied via discrete quantization are discrete quantum groups, quantum metric spaces (used in quantum error correction) and quantum graphs (also originating in quantum error correction, but quantum graphs are also used to study nonlocality with homomorphism games).

Together with Kornell and Mislove, I found a noncommutative version of omega-complete posets (cpos), and showed that the resulting quantum cpos provide categorical models for quantum programming languages with recursion, including linear recursive type systems that I developed with Mislove and Zamdzhiev. Furthermore, with Kornell, I showed that the category of quantum graphs admits a symmetric monoidal closed structure, enabling a characterization of winning strategies in quantum homomorphism games.