Problem
Research questions and friction points this paper is trying to address.
stabilizer codes
translation-invariant
ZX-calculus
quantum error correction
infinite systems
The Delayed Stabilizer ZX-Calculus
🤖 AI Summary
Existing stabilizer ZX-calculus struggles to characterize infinite quantum error-correcting codes exhibiting spacetime translation invariance, such as lattice codes and quantum convolutional codes. This work proposes delayed stabilizer ZX-calculus, introducing for the first time a "delay" generator to construct a finite diagrammatic language capable of expressing such infinite structures. By integrating odd-prime-dimensional stabilizer ZX-calculus, generalized Euler decompositions, color transformations, and graph-theoretic operations, the framework establishes a dual semantics based on equivalence classes of quantum channel sequences and formal power series in a generating variable. A complete axiomatization is provided, and every delayed ZX-diagram admits a unique normal form. Consequently, the calculus achieves completeness, universality, and soundness under the generating function semantics, enabling exact reconstruction of the corresponding infinite stabilizer group.
📝 Abstract
Many stabilizer quantum error-correcting codes are built from a finite pattern repeated across space or time, such as lattice codes, translation-invariant graph states, and quantum convolutional codes. Ordinary stabilizer ZX-diagrams capture only finite truncations of such systems, obscuring the repeated structure that defines them. We introduce the delayed stabilizer ZX-calculus, a finite graphical language for these infinite, translation-invariant processes. It extends the odd-prime-dimensional stabilizer ZX-calculus with a single new generator, the delay, which feeds data from one time step to the next. We equip the calculus with two semantics. In the first semantics, we interpret the behaviour of a delayed ZX-diagram as an equivalence class of sequences of quantum channels; where two sequences are identified if they have the same information content. We show that the behaviour of a delayed ZX-diagram uniquely determines an infinite stabilizer group. In the second semantics, we interpret the delay as a formal variable, encoding the translation-invariant families of Pauli operators as generating functions. This allows us to represent a delayed ZX-diagram in terms of a tableau of generating functions, from which the infinite stabilizer group can be recovered. Finally, we give a complete axiomatization of the delayed stabilizer ZX-calculus, featuring generalised Euler decomposition and colour change rules. Using generalised forms of local complementation and pivoting, we reduce every diagram to a unique normal form. This establishes soundness, universality, and completeness for the generating tableau semantics.
Innovation
Methods, ideas, or system contributions that make the work stand out.
delayed stabilizer ZX-calculus
translation-invariant quantum codes
generating functions
graphical calculus
infinite stabilizer groups
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