Beyond Penrose tensor diagrams with the ZX calculus: Applications to quantum computing, quantum machine learning, condensed matter physics, and quantum gravity

arXiv:2511.06012 · quant-ph · Submitted 2025-11-08 · Read on arXiv

quant-ph

Submitted: 2025-11-08

Updated: 2026-09-23

Comments: 52 pages

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Hilbert Space — 19× in this paper · explained in 2 episodes
• Core — 4× in this paper · explained in 2 episodes
• Quantum Circuits — 4× in this paper · explained in 2 episodes
• Quantum Physics — 4× in this paper · explained in 10 episodes
• IEEE Symposium on Logic in Computer Science — 3× in this paper
• Categorical Quantum Mechanics — 2× in this paper
• Trace — 2× in this paper · explained in 2 episodes
• Alignment — 1× in this paper · explained in 1 episode
• Anyons — 1× in this paper · explained in 2 episodes
• Axioms — 1× in this paper · explained in 2 episodes
• CSS Codes — 1× in this paper · explained in 2 episodes
• Degeneracy — 1× in this paper · explained in 3 episodes
• Redundancy — 1× in this paper · explained in 2 episodes
• Unified Framework — 1× in this paper · explained in 5 episodes
• Verification — 1× in this paper · explained in 2 episodes

Sources

• Categorical quantum mechanics
• Interacting Hopf Algebras
• Complete equational theories for classical and quantum Gaussian relations
• Generalised Compositional Theories and Diagrammatic Reasoning
• All you need is spin: SU(2) equivariant variational quantum circuits based on spin networks
• Spin-networks in the ZX-calculus
• Permutational Quantum Computing
• Phase-free ZX diagrams are CSS codes (...or how to graphically grok the surface code)
• Reducing T-count with the ZX-calculus
• Speedy Contraction of ZX Diagrams with Triangles via Stabiliser Decompositions
• A Spin Network Primer
• Introduction to SU(2) recoupling theory and graphical methods for loop quantum gravity
• A Primer of Group Theory for Loop Quantum Gravity and Spin-foams
• Computing Spin Networks
• ZX Graphical Calculus for Continuous-Variable Quantum Processes
• A universal completion of the ZX-calculus
• ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces
• Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus
• Floquetifying stabiliser codes with distance-preserving rewrites
• Fault Tolerance by Construction

Related papers

• Reconquering Bell sampling on qudits: stabilizer learning and testing, quantum pseudorandomness bounds, and more
• Encrypted clones can leak: Classification of informative subsets in Quantum Encrypted Cloning
• Polynomial-time classical and quantum simulation of quantum impurity models
• Theory of quantum-enhanced interferometry with general Markovian light sources
• A convergent hierarchy of spectral gap certificates for qubit Hamiltonians
• Universal Bound and Phase Transition in Many-Body Fermionic Non-Gaussianity