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This arXiv preprint outlines a theoretical framework for interpreting quantum circuits through four progressive lenses: operational, diagrammatic, error-protected, and geometric. By bridging abstract ZX-calculus with physical implementations like hyperbolic superconducting lattices, the work offers researchers a new mathematical toolkit for designing fault-tolerant architectures. The study is particularly relevant for those tracking the convergence of topological code theory with tangible device physics experiments.
Rethinking Quantum Circuits
arXiv:2608.19370v1 Announce Type: new Abstract: These notes develop four interconnected ways of reading a quantum circuit. A circuit for us begins as an operational composition of gates; then, it becomes a diagram whose local equalities may be used as calculations; next, it becomes a protected process once errors, syndromes, and logical degrees of freedom are separated; and finally, it becomes geometric when its connectivity, topology, and boundary data are treated as physical design parameters. The development begins at the level of bits and qubits before appealing to Deutsch's and Grover's algorithms as basic examples of quantum circuits. With the basics in hand, we interpret quantum circuits diagramatically, leading us to compact closed string diagrams and the ZX-calculus. After that, we consider how to correct quantum circuits by introducing the Knill--Laflamme condition, homological surface codes, and related concepts with a view towards thinking of these as operations on diagrams. The lectures eventually arrive at the properties of hyperbolic quantum codes and the prospect of physical superconducting circuits emulating the negatively-curved lattices needed to support those codes. These mathematical ideas and physical experiments, taken together, represent one way to impart a geometric layer onto quantum circuits. By the very end, we bring the ideas nearly full circle by assessing the extent to which these device physics experiments operationalize the basic ZX diagrams encountered much earlier in the story. While the later material reports on original research, and while the discussion becomes increasingly mathematical as the sections progress, no prior knowledge of quantum information, quantum computing, or quantum error correction is actually assumed.
Abstract:These notes develop four interconnected ways of reading a quantum circuit. A circuit for us begins as an operational composition of gates; then, it becomes a diagram whose local equalities may be used as calculations; next, it becomes a protected process once errors, syndromes, and logical degrees of freedom are separated; and finally, it becomes geometric when its connectivity, topology, and boundary data are treated as physical design parameters. The development begins at the level of bits and qubits before appealing to Deutsch's and Grover's algorithms as basic examples of quantum circuits. With the basics in hand, we interpret quantum circuits diagramatically, leading us to compact closed string diagrams and the ZX-calculus. After that, we consider how to correct quantum circuits by introducing the Knill--Laflamme condition, homological surface codes, and related concepts with a view towards thinking of these as operations on diagrams. The lectures eventually arrive at the properties of hyperbolic quantum codes and the prospect of physical superconducting circuits emulating the negatively-curved lattices needed to support those codes. These mathematical ideas and physical experiments, taken together, represent one way to impart a geometric layer onto quantum circuits. By the very end, we bring the ideas nearly full circle by assessing the extent to which these device physics experiments operationalize the basic ZX diagrams encountered much earlier in the story. While the later material reports on original research, and while the discussion becomes increasingly mathematical as the sections progress, no prior knowledge of quantum information, quantum computing, or quantum error correction is actually assumed.
75 pages, 26 figures (incl. cover figure), 10 tables; based on a four-lecture series delivered by the author during the Niels Bohr Quantum Summer School in Odense, DK, 10-13 August 2026
Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Information Theory (cs.IT); Mathematical Physics (math-ph); Category Theory (math.CT)
arXiv:2608.19370 [quant-ph]
(or arXiv:2608.19370v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2608.19370 arXiv-issued DOI via DataCite (pending registration)
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