The ZX-calculus is a powerful graphical language for manipulating quantum circuits, which has recently found many applications in quantum error correction. We extend this language to handle Floquet and other dynamical stabilizer codes via the connection between measurement-based code switching and gauge fixing (arXiv:1810.10037). We combine gauge-fixing steps to implement a closed loop in the space of stabilizer codes, returning to the original codespace up to a logical Clifford gate. These measurement-based paths in the space of stabilizer codes can be viewed as shortcuts, or "chutes and ladders", relative to single-qubit Clifford operations and qubit permutations. This yields a machine-interpretable method for constructing dynamical automorphisms and facilitates the search for implementations of desired logical gates. As an example, we implement a logical phase gate via distance-preserving code switching for the seven-qubit code bare code (arXiv:1702.01155), which has no non-trivial logical Clifford gates based on single-qubit Clifford operations and qubit permutations (arXiv:2409.18175).
The ZX-calculus is a powerful graphical language for manipulating quantum circuits, which has recently found many applications in quantum error correction. We extend this language to handle Floquet and other dynamical stabilizer codes via the connection between measurement-based code switching and gauge fixing (arXiv:1810.10037). We combine gauge-fixing steps to implement a closed loop in the space of stabilizer codes, returning to the original codespace up to a logical Clifford gate. These measurement-based paths in the space of stabilizer codes can be viewed as shortcuts, or "chutes and ladders", relative to single-qubit Clifford operations and qubit permutations. This yields a machine-interpretable method for constructing dynamical automorphisms and facilitates the search for implementations of desired logical gates. As an example, we implement a logical phase gate via distance-preserving code switching for the seven-qubit code bare code (arXiv:1702.01155), which has no non-trivial logical Clifford gates based on single-qubit Clifford operations and qubit permutations (arXiv:2409.18175).
We study the implementation of fault-tolerant logical Clifford gates on stabilizer quantum error-correcting codes based on their symmetries. Our approach is to map the stabilizer code to a binary linear code, compute its automorphism group, and impose constraints based on the Clifford operators permitted. We provide a rigorous formulation of the method for finding automorphisms of stabilizer codes and generalize the Z X dualities previously introduced for Calderbank-Shor-Steane (CSS) codes to non-CSS codes. We provide a package implementing our algorithms that uses the computational-algebra software system for certain subroutines. Our algorithms map automorphism-group generators to physical circuits, calculate Pauli corrections based on the destabilizers of the code, and determine their logical action. We discuss the fault tolerance of the circuits and include examples of gates through automorphisms for the [ [ 4 , 2 , 2 ] ] and perfect [ [ 5 , 1 , 3 ] ] codes, bivariate bicycle codes, and the best-known distance codes.
We propose a new model of quantum computation comprised of low-weight measurement sequences that simultaneously encode logical information, enable error correction, and apply logical gates. These measurement sequences constitute a new class of quantum error-correcting codes generalizing Floquet codes, which we call dynamic automorphism (DA) codes. We construct an explicit example, the DA color code, which is assembled from short measurement sequences that can realize all 72 automorphisms of the 2D color code. On a stack of N triangular patches, the DA color code encodes N logical qubits and can implement the full logical Clifford group by a sequence of two- and, more rarely, three-qubit Pauli measurements. We also make the first step towards universal quantum computation with DA codes by introducing a 3D DA color code and showing that a non-Clifford logical gate can be realized by adaptive two-qubit measurements.
Stabiliser codes with large weight measurements can be challenging to implement fault-tolerantly. To overcome this, we propose a Floquetification procedure which, given a stabiliser code, synthesises a novel Floquet code that only uses single- and two-qubit operations. Moreover, this procedure preserves the distance and number of logicals of the original code. The new Floquet code requires additional physical qubits. This overhead is linear in the weight of the largest measurement of the original code. Our method is based on the ZX calculus, a graphical language for representing and rewriting quantum circuits. However, a problem arises with the use of ZX in the context of rewriting error-correcting codes: ZX rewrites generally do not preserve code distance. Tackling this issue, we define the notion of distance-preserving rewrite that enables the transformation of error-correcting codes without changing their distance. These distance-preserving rewrites are used to decompose arbitrary weight stabiliser measurements into quantum circuits with single- and two-qubit operations. As we only use distance-preserving rewrites, we are guaranteed that a single error in the resulting circuit creates at most a single error on the data qubits. These decompositions enable us to generalise the Floquetification procedure of Townsend-Teague et al \cite{townsend-teagueFloquetifyingColourCode2023} to arbitrary stabiliser codes, provably preserving the distance and number of logicals of the original code.
Graphical languages offer intuitive and rigorous formalisms for quantum physics. They can be used to simplify expressions, derive equalities, and do computations. Yet in order to replace conventional formalisms, rigour alone is not sufficient: the new formalisms also need to have equivalent deductive power. This requirement is captured by the property of completeness, which means that any equality that can be derived using some standard formalism can also be derived graphically. In this thesis, I consider the ZX-calculus, a graphical language for pure state qubit quantum mechanics. I show that it is complete for pure state stabilizer quantum mechanics, so any problem within this fragment of quantum theory can be fully analysed using graphical methods. This includes questions of central importance in areas such as error-correcting codes or measurement-based quantum computation. Furthermore, I show that the ZX-calculus is complete for the single-qubit Clifford+T group, which is approximately universal: any single-qubit unitary can be approximated to arbitrary accuracy using only Clifford gates and the T-gate. [...] Lastly, I extend the use of rigorous graphical languages outside quantum theory to Spekkens' toy theory, a local hidden variable model that nevertheless exhibits some features commonly associated with quantum mechanics. [...] I develop a graphical calculus similar to the ZX-calculus that fully describes Spekkens' toy theory, and show that it is complete. Hence, stabilizer quantum mechanics and Spekkens' toy theory can be fully analysed and compared using graphical formalisms. Intuitive graphical languages can replace conventional formalisms for the analysis of many questions in quantum computation and foundations without loss of mathematical rigour or deductive power.
Gauge theories are important descriptions for many physical phenomena and systems in quantum computation. Automorphism of gauge group naturally gives global symmetries of gauge theories. In this work we study such symmetries in gauge theories induced by automorphisms of the gauge group, when the gauge theories have nontrivial topological actions in different spacetime dimensions. We discover the automorphism symmetry can be extended, become a higher group symmetry, and/or become a non-invertible symmetry. We illustrate the discussion with various models in field theory and on the lattice. In particular, we use automorphism symmetry to construct new transversal non-Clifford logical gates in topological quantum codes. In particular, we show that 2+1d ZN qudit Clifford stabilizer models can implement non-Clifford transversal logical gate in the 4th level ZN qudit Clifford hierarchy for N≥3, extending the generalized Bravyi-König bound proposed in the companion paper [arXiv:2511.02900] for qubits.
A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-König bound for n-dimensional topological stabilizer codes. In this work, we extend topological Pauli stabilizer codes to a broad class of n-dimensional Clifford hierarchy stabilizer codes. These codes correspond to the (n+1)D Dijkgraaf-Witten gauge theories with non-Abelian topological order. We construct transversal non-Clifford gates through automorphism symmetries represented by cup products. In 2D, we obtain the first transversal non-Clifford logical gates including T and CS for Clifford stabilizer codes, using the automorphism of the twisted Z23 gauge theory (equivalent to D4 topological order). We also combine it with the just-in-time decoder to fault-tolerantly prepare the logical T magic state in O(d) rounds via code switching. In 3D, we construct a transversal logical T gate in a non-Clifford stabilizer code at the third level of the Clifford hierarchy, located on a tetrahedron corresponding to a twisted Z24 gauge theory. Our constructions surpass the Bravyi-König bound by achieving the logical gates in the (n+1)-th level of Clifford hierarchy in n spatial dimension.