Quantum Group · University of Oxford

Robert I. Booth

The mathematics of fault-tolerant quantum software.

I develop the algebra, diagrammatic calculi, and programming-language semantics needed to compile reliable computations onto error-corrected quantum computers.

What I work on Download CV
Robert I. Booth
three CNOTs = one swap, one rewrite at a time

About

I am a Research Associate in Quantum Software in the Quantum Group at the University of Oxford. My research is about the mathematical foundations of quantum computing: I have worked on diagrammatic reasoning, measurement-based quantum computation, measures of non-classicality, and quantum computation with continuous variables.

The problem that most excites me at the moment is fault-tolerant quantum compilation: any useful quantum computation will have to run on error-corrected, logical qubits, and we still lack the mathematical tools to compile programs onto them systematically — across different codes and architectures, with guarantees that the result still works. I attack this with the tools I know best: algebra (especially symplectic and stabiliser structure), category theory, graphical calculi, and the design and semantics of quantum programming languages.

Academic path

2024 – present Research Associate in Quantum Software University of Oxford · with Aleks Kissinger
2022 – 2025 Research Associate University of Edinburgh & University of Bristol · with Chris Heunen and Noah Linden
2018 – 2022 PhD in Quantum Computation Sorbonne Université (LIP6, Paris) & Loria (Nancy) · “Measurement-based quantum computation beyond qubits” · supervised by Damian Markham and Simon Perdrix
2016 – 2018 MSc in Fundamental Physics Université Pierre et Marie Curie, Paris
2016 – 2018 Research internships Quantum information theory (LIP6, Paris) · experimental quantum optics (Roma Tre, with Marco Barbieri)

News

Research

Quantum computers will only ever be useful if they are fault-tolerant: computations must run on logical qubits, encoded in error-correcting codes, using operations that keep errors under control. Compiling a program down to this level is nothing like ordinary circuit compilation — the target is a code, the gates are surgeries and measurements, and correctness is a theorem, not a test. My research builds the mathematics that makes this tractable: algebraic structures that describe codes and logical operations compositionally, graphical calculi to reason and compute with them, and semantics that say precisely what a quantum program means and what structure needs to be preserved during compilation.

Fault-tolerant compilation

How do we implement logic on encoded qubits — in any code, not just the surface code?

The question driving the whole programme: what does it take to compile a logical circuit onto a quantum error-correcting code — any code — and know that it still works?

Entangling logic between arbitrary CSS codes, via generalised surgery CSS surgery
Synthesis & optimisation of logical circuits across stabiliser codes open
Quantitative guarantees on how errors propagate through compiled protocols open

Graphical & algebraic calculi

What is the right algebra for quantum computation, and can we reason about it diagrammatically?

Complete graphical languages turn quantum-mechanical reasoning into diagram rewriting — and symplectic structure turns out to be the common backbone of stabiliser and Gaussian quantum mechanics.

Complete ZX-calculi for qudit stabiliser theory MFCS 2022 · QPL 2023
A graphical symplectic algebra spanning stabiliser & Gaussian QM GSA
Complete equational theories for Gaussian relations arXiv:2403.10479
The algebra of codes & their logical operations open

Quantum programming languages & semantics

What does a quantum program mean, and how do its pieces compose?

Compilation you can trust needs a precise, compositional account of what programs mean — semantics is what turns compilation into something you can verify.

Denotational semantics for stabiliser quantum programs FSCD 2026
How quantum instruments compose arXiv:2606.28291
Semantics for recursive hybrid quantum–classical programs arXiv:2607.09605
Semantics for quantum error-correcting programs open
Verified fault-tolerant compilation open

Continuous variables & bosonic systems

How do we reason about infinite-dimensional quantum systems?

Bosonic systems are infinite-dimensional — yet computation, non-classicality, and error correction all make sense there, and increasingly in hardware.

Flow & determinism for CV measurement-based computation Quantum 2023
Contextuality ≡ Wigner negativity for CV measurements PRL 2022
Effective descriptions of bosonic systems are complete Nat. Comms 2025
Bosonic codes inside the compilation framework open
Graphical languages for bosonic computing open

Earlier work. My PhD extended measurement-based quantum computation beyond qubits and before that I worked on experimental quantum thermodynamics and metrology in Rome.

The long game: these threads converge on a single algebraic framework for fault-tolerant compilation — one in which logical operations on any code can be described, synthesised, optimised, and formally verified, with quantitative guarantees on how errors propagate.

Publications

Peer-reviewed

FSCD 2026

Denotational semantics for stabiliser quantum programs

Robert I. Booth, Cole Comfort

What stabiliser quantum programs mean, compositionally.

FSCD 2026

Graphical Symplectic Algebra

Robert I. Booth, Titouan Carette, Cole Comfort

A graphical calculus built on symplectic structure, spanning stabiliser and Gaussian quantum mechanics.

Nature Communications · 2025

Effective descriptions of bosonic systems can be considered complete

Francesco Arzani, Robert I. Booth, Ulysse Chabaud

Finite effective descriptions capture everything operationally accessible about bosonic systems.

Quantum · 2023

Flow conditions for continuous variable measurement-based quantum computing

Robert I. Booth, Damian Markham

When continuous-variable measurement patterns define deterministic computations.

QPL 2023

The Qupit Stabiliser ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification

Boldizsár Poór, Robert I. Booth, Titouan Carette, John van de Wetering, Lia Yeh

A toolkit of results for qudit stabiliser ZX-calculus in odd prime dimensions.

J. Phys. A · 2023

Outcome determinism in measurement-based quantum computation with qudits

Robert I. Booth, Aleks Kissinger, Damian Markham, Clément Meignant, Simon Perdrix

Generalising flow-based determinism to qudit measurement-based computation.

Phys. Rev. Lett. · 2022

Contextuality and Wigner negativity are equivalent for continuous-variable quantum measurements

Robert I. Booth, Ulysse Chabaud, Pierre-Emmanuel Emeriau

Two central notions of quantumness coincide in the continuous-variable setting.

MFCS 2022

Complete ZX-Calculi for the Stabiliser Fragment in Odd Prime Dimensions

Robert I. Booth, Titouan Carette

Complete equational theories for qudit stabiliser quantum mechanics.

Phys. Rev. A · 2018

Bridging thermodynamics and metrology in nonequilibrium quantum thermometry

Vasco Cavina, Luca Mancino, Antonella De Pasquale, Ilaria Gianani, Marco Sbroscia, Robert I. Booth, Emanuele Roccia, Roberto Raimondi, Vittorio Giovannetti, Marco Barbieri

Phys. Rev. Lett. · 2018

Geometrical Bounds on Irreversibility in Open Quantum Systems

Luca Mancino, Vasco Cavina, Antonella De Pasquale, Marco Sbroscia, Robert I. Booth, Emanuele Roccia, Ilaria Gianani, Vittorio Giovannetti, Marco Barbieri

Preprints

arXiv:2606.28291

Composing Quantum Instruments

Robert I. Booth, Dominik Leichtle, Alex Rice, Kim Worrall

The compositional structure of quantum operations with classical input and output.

arXiv:2607.09605

Quantum Orchestras: a Concrete Semantics for Recursive Hybrid Programs

Alex Rice, Dominik Leichtle, Kim Worrall, Robert I. Booth

Concrete semantics for recursive hybrid quantum–classical programs.

arXiv:2505.01370

Engineering CSS surgery: compiling any CNOT in any code

Clément Poirson, Joschka Roffe, Robert I. Booth

Generalised surgery implementing entangling logical gates between arbitrary CSS codes.

arXiv:2403.10479

Complete equational theories for classical and quantum Gaussian relations

Robert I. Booth, Titouan Carette, Cole Comfort

A complete graphical calculus for the Gaussian, continuous-variable world.

Also on Google Scholar · ORCiD · arXiv.

Talks

Denotational semantics for stabiliser quantum programs
FSCD 2026 · slides
Denotational semantics for stabiliser quantum programs
LIQCS · slides
The algebra for quantum fault-tolerance
Algebraic and Combinatorial Perspectives in the Mathematical Sciences, online · video
Denotational semantics for stabiliser quantum programs
6th Scottish Programming Language Seminar, Glasgow · slides
CSS surgery: compiling any CNOT in any code
QPL 2025, Varna
A convenient setting for fault-tolerant compilation
Quantum Lunch seminar, University of Oxford
Finding a convenient setting for fault-tolerant quantum compilation
University College London
CSS surgery: compiling any CNOT in any CSS code
PhiQus team seminar, École Polytechnique, Saclay · slides
The nullifier theory of Gaussian quantum states
Qatalyze seminar, Paris · slides
Lattice surgery and the ZX-calculus
QEC Journal Club · slides
A graphical language for Gaussian quantum processes
QPL 2024, Buenos Aires · slides
Complete equational theories for classical and quantum Gaussian relations
ZX seminar, online · slides · video
A symplectic vision for the ZX-calculus
CATNIP workshop, University of Aberdeen · slides
Graphical reasoning for Gaussian quantum mechanics
Loria, Nancy · slides
Graphical Symplectic Algebra
ZX seminar, online · slides · video
Diagrammatic reasoning beyond qubits
Freie Universität Berlin · slides
ZX-calculus: (my) past, present… future?
LIP6 QI team seminar, Paris · slides
Distributed quantum computing
QCS Hub · slides
Distributing quantum circuits with the ZX-calculus
NQCC Launch Event, University of Edinburgh · slides
Complete ZX-calculi for the stabiliser fragment in odd prime dimensions
MFCS 2022, Vienna · slides
Complete ZX-calculi for the stabiliser fragment in odd prime dimensions
QPL 2022, Oxford · slides · video
Extracting reversible quantum circuits from measurement-based quantum computations with qudits
QPL 2022, Oxford · slides · video
Complete ZX-calculi for the stabiliser fragment in odd prime dimensions
ZX seminar, online · slides · video
Contextuality and Wigner negativity are equivalent for continuous-variable measurements
Online Quantum and Linear Optical Computation seminar · slides
Contextuality and Wigner negativity are equivalent for continuous-variable measurements
20th European Conference on Foundations of Physics, ENS Paris
Measurement-based quantum computation beyond qubits and related problems
EPiQC · slides
Measurement-based quantum computing and associated problems
Journées Scientifiques Inria · slides
Outcome determinism in measurement-based quantum computing with qudits
ZX seminar, online · slides
Outcome determinism in measurement-based quantum computing with qudits
Workshop on recent advances on quantum computing, Collège de France · slides
F-flow: determinism in measurement-based quantum computing with qudits
QPL 2021, University of Gdańsk · slides · video
Outcome determinism in measurement-based quantum computing for qudits
LIP6 group meeting, Paris · slides
F-flow: determinism in measurement-based quantum computing with qudits
YQIS 2021, Michigan State University · slides
Flow conditions for continuous-variable measurement-based quantum computation
ZX seminar, online · slides
Flow conditions for continuous variable measurement-based quantum computation
QPL 2020, Paris · slides · video
Flow conditions for continuous-variable measurement-based quantum computation
Workshop on Quantum Networks and Information 2020, NII Tokyo · slides
Flow conditions for continuous variable measurement-based quantum computing
LIP6, Paris · slides
Flow conditions for continuous variable measurement-based quantum computing
Atos · slides
Flow conditions for continuous variable measurement-based quantum computing
Loria, Nancy · slides
Convergence of continuous-variable measurement-based quantum computations
Edinburgh–Paris Joint Quantum Workshop, Edinburgh · slides
Flow conditions for continuous variables measurement-based quantum computation
LIP6, Paris · slides
Generalised flow for continuous variables
LIP6, Paris · slides
The complexity of non-Gaussian quantum circuits
Loria, Nancy · slides

If you would like me to speak at your seminar or event, get in touch.

Teaching & service

Categories and Quantum Informatics — lectures, computer labs, and course material, University of Edinburgh (2023). This work earned a University of Edinburgh Staff Award.

Introduction to Quantum Programming and Semantics — guest lectures and tutorials, University of Edinburgh (2024).

I have served on the programme committees of QPL, ACT, and PLanQC, review for Quantum, Compositionality, and LMCS among others, and organised the Quantum Software Lab seminar in Edinburgh (2023–2025).

Students supervised

  • Nathan Hall — MSc 2026
  • Axel Daboust — MSc 2026
  • Sascha Haupt — MSc 2024–2026
  • Boren Gu — MSc 2025, now PhD at Freie Universität Berlin
  • Clément Poirson — MSc 2024, now PhD at Inria Paris / Alice & Bob

I welcome students interested in the algebra of quantum computing, fault tolerance, or quantum programming languages — feel free to reach out.

Get in touch

Robert I. Booth
Quantum Group
Department of Computer Science
University of Oxford

Elsewhere

Google ScholarORCiDarXiv CV (PDF)