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The ZX-calculus is complete for stabilizer quantum mechanics

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arxiv 1307.7025 v1 pith:G3PGZ7J3 submitted 2013-07-26 quant-ph

classification quant-ph
keywords derivedmechanicspurequantumzx-calculuscalculuscompleteequality
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The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics, meaning any pure state, unitary operation and post-selected pure projective measurement can be expressed in the ZX-calculus. The calculus is also sound, i.e. any equality that can be derived graphically can also be derived using matrix mechanics. Here, we show that the ZX-calculus is complete for pure qubit stabilizer quantum mechanics, meaning any equality that can be derived using matrices can also be derived pictorially. The proof relies on bringing diagrams into a normal form based on graph states and local Clifford operations.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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    Simpler qudit flow definition yields O(n^3) flow-finding algorithm and flow-preserving operations for measurement-based quantum computing on prime-dimensional qudits.

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    physics.pop-ph 2026-08 conditional novelty 3.0 of 10

    Misty-state terms are assigned an unnormalized-amplitude semantics with scoped normalization, canonical normal forms, and branch-based measurement, so the notation's rewrites become exactly checkable.

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