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Pivoting makes the ZX-calculus complete for real stabilizers
Ross Duncan, Simon Perdrix
TL;DR
The paper examines whether graph-state pivoting and local complementation follow from the ZX-calculus axioms. It adds an H-loop axiom to capture pivoting, establishes that the resulting calculus is intermediate between the plain and Euler-augmented calculi, and presents an angle-free calculus complete for real stabilizer quantum mechanics.
Problem
Method
The paper adds an H-loop axiom for pivoting, then replaces π vertices with loops to define an angle-free calculus.
Source: §3 · p. 7, §4 · p. 10Results
The H-loop calculus is strictly intermediate between the plain ZX-calculus and the calculus with Euler decomposition, and the angle-free calculus is complete for real stabilizer quantum mechanics.
Source: §3 · p. 9, §4 · p. 10, §5 · p. 12Takeaways & Limitations
For 0- and π-rotations, the calculus has a simple equivalent angle-free form that is complete for real stabilizer quantum mechanics.
Source: §4 · p. 10, §5 · p. 12
Abstract
from arXiv · showhide
We show that pivoting property of graph states cannot be derived from the axioms of the ZX-calculus, and that pivoting does not imply local complementation of graph states. Therefore the ZX-calculus augmented with pivoting is strictly weaker than the calculus augmented with the Euler decomposition of the Hadamard gate. We derive an angle-free version of the ZX-calculus and show that it is complete for real stabilizer quantum mechanics.
1 The Graphical Formalism
The ZX-calculus represents diagrams as open graphs and interprets them as linear maps. Its graphical axioms support basic diagram rewrites, but the Euler decomposition of the Hadamard gate is not derivable from them.
Syntax and semantics
- Diagrams are arrows in a free category generated by prime graphs, with phases labeling vertices.Open graphs have input and output boundary vertices; composition identifies outputs with inputs, and tensor product juxtaposes graphs.Source: §1 · p. 2
- The interpretation functor maps a diagram with n inputs and m outputs to a linear map from C⊗2n to C⊗2m.Z and X vertices are defined using the corresponding Z and X bases.Source: §1 · p. 2
Axioms and consequences
- The ZX-calculus is formed by quotienting the free category by Figure 1’s equations, which are sound under the interpretation.The spider rules also make connected diagrams containing only Z or only X vertices equivalent to prime graphs.Source: §1 · pp. 3–4
- The axioms imply that diagrams without H are equivalent to simple bipartite graphs, and that any equation remains valid when Z and X are exchanged.Any diagram can also be rewritten with no Z vertices or with no X vertices.Source: §1 · p. 4
- The Euler decomposition of H cannot be derived from the ZX-calculus axioms.An alternative interpretation satisfies the Figure 1 equations while the Euler equation fails.Source: §1 · p. 4
2 Graph states and Local complementation
Graph states are represented by ZX diagrams, and local complementation changes edges among a vertex’s neighbors. The graph-state local-complementation property is equivalent to the Euler decomposition of H in the ZX-calculus.
Graph states
- A graph state can be represented by a diagram with one Z vertex per graph vertex and one H vertex per edge.Each Z vertex connects to an output, and each H vertex connects to the Z vertices at the edge’s endpoints.Source: §2 · p. 4
Local complementation
- Local complementation at a vertex toggles edges between its neighbors while leaving other graph edges unchanged.A neighbor pair is connected afterward exactly when it was not connected beforehand.Source: §2 · p. 5
- The local-complementation property for graph states is equivalent to the Euler decomposition of H and therefore cannot be proved in the ZX-calculus.The equivalence is stated as Proposition 2.5’s equivalence to Equation (EU).Source: §2 · p. 5
3 Pivoting
Pivoting swaps the endpoints of an edge and complements edges among the common and exclusive neighbourhoods. The section shows how pivoting relates to local complementation and establishes that an added H-loop axiom is sufficient for arbitrary pivoting but weaker than Euler decomposition.
3 Pivoting
- Pivoting exchanges the endpoints of an edge and toggles edges among common and exclusive neighbours of those endpoints.The three groups are common neighbours, neighbours exclusive to u, and neighbours exclusive to v.Source: §3 · p. 6, Figure 2 · p. 6
- Pivoting is a sequence of three local complementations and can be implemented on graph states using Hadamards on the endpoints and Z on their common neighbours.The paper also notes applications of graph-state pivoting to measurement-based computing and linear-code distance.Source: §3 · p. 6
- When u and v have no common neighbour, their graph-state pivoting is derivable in the plain ZX-calculus.For triangle-free or bipartite graphs, pivoting can be derived in plain ZX-calculus; pivoting preserves bipartiteness but not triangle freeness.Source: §3 · pp. 6–7
- The H-loop axiom lets the ZX-calculus prove pivoting for arbitrary graphs, placing it between plain ZX-calculus and ZX-calculus + Euler.The paper shows that H-loop cannot be derived from the plain rules, and Euler decomposition cannot be derived from ZX-calculus + H-loop.Source: §3 · pp. 7–9
- Pivoting arbitrary graphs is proved by adding the H-loop axiom, which supports splitting common neighbours before applying the proof for the no-common-neighbour case.The proof then uses spider and Hopf laws to complete the transformation.Source: §3 · p. 7
- The triangle pivoting property implies the H-loop equation, and the paper states that pivoting can be derived if and only if that equation holds.The H-loop equation itself cannot be derived from the plain ZX-calculus rules.Source: §3 · p. 8
4 Angle-free calculus for Real Stabilizers
The paper restricts phases to 0 or π, then replaces π vertices with loops to obtain an angle-free calculus. It proves completeness for real stabilizer quantum mechanics by reducing diagrams to graph-state forms and showing that equivalent reduced forms are identical.
4 Angle-free calculus for Real Stabilizers
- Restricting phases to 0 or π gives the real-valued stabilizer fragment, and the angle-free calculus is stronger than the weak calculus but weaker than restricted ZX-calculus + Euler.With (HL), π vertices can be replaced by loops.Source: §4 · p. 10
- The angle-free calculus is complete for real-valued stabilizers, whose semantics are real Clifford operations with |0⟩ initialisations and projections.The calculus replaces π vertices with loops and replaces axiom (C2) with (L).Source: §4 · p. 10, §4.1 · p. 11
4.1 Real stabilizer quantum mechanics
- Every angle-free diagram state can be transformed into a GS-RLC diagram and then into a reduced GS-RLC diagram.The proof sketch uses spider fusion, Hopf-law edge removal, and pivoting to remove disconnected internal structure or internal vertices.Source: §4.1 · p. 11
- Pivoting can simplify pairs of reduced GS-RLC diagrams by moving an H from one adjacent vertex to the other without changing H operators elsewhere.The simplification is applied inductively until the pair satisfies the stated simplified condition.Source: §4.1 · pp. 11–12
- For a simplified pair of reduced GS-RLC diagrams, equal semantics hold if and only if the diagrams are identical.The proof establishes matching H locations, Z operators, and underlying graphs.Source: §4.1 · p. 12
5 Conclusion and Perspectives
The conclusion positions the H-loop calculus between plain ZX and ZX with Euler decomposition, and says its restricted language is complete for real stabilizer quantum mechanics. It also identifies simplifications and scope limits of the real setting.
5 Conclusion and Perspectives
- The H-loop axiom cannot be derived in plain ZX and is strictly weaker than the Euler decomposition of H.The new calculus is intermediate between plain ZX and ZX augmented with Euler decomposition.Source: §5 · p. 12
- Restricting the calculus to 0- and π-rotations yields completeness for real stabilizer quantum mechanics and an equivalent angle-free calculus.Source: §5 · p. 12
- Real quantum mechanics offers diagrammatic simplifications, including the same interpretation for an object and its dual and an intuitive X/Z account of unbiased bases.The paper says the third, complex mutually unbiased qubit basis is less intuitive to axiomatise.Source: §5 · p. 12
- Applications requiring complex numbers, such as local tomography, are outside the intermediate language and require additional axioms such as Euler decomposition of H.Source: §5 · p. 12