Pivoting makes the ZX-calculus complete for real stabilizers

Abstract

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.The ZX-calculus is a formal theory for reasoning about quantum computational systems [3]. It consists of a graphical language based on the Pauli Z and X observables, and a collection of axioms expressed as graph rewrite rules. The ZX-calculus is expressive enough to represent any quantum circuit, and its equations are complete for the stabilizer fragment of quantum mechanics [1]. Due to its graphical nature, and its close relationship to the Z and X observables, the ZX-calculus is particularly well adapted to the study of graph states and measurement-based quantum computation [9,7].In addition to the two observables, the ZX-calculus also contains an operator for the Hadamard map: this is the map which exchanges the Z and X bases, and thus provides a duality principle for the graphical language. In previous work [8] the authors showed that if the Hadamard can be expressed in terms of Z and X rotations-that is, as an Euler decomposition-then Van Den Nest's theorem [14] about local complementation of graph states follows, and vice versa. Furthermore, these results cannot be derived from the original axioms, hence the theory "ZX-calculus + Euler" is strictly stronger than the plain ZXcalculus.In this paper we find a theory intermediate between the two, albeit having a similar flavour. We consider an operation on graph states called pivoting and show that its defining property is equivalent to the possibility to express (one of) the Pauli matrices in terms of the Hadamard. Since pivoting can be done via local complementation, "ZX-calculus + Euler" is stronger than "ZX-calculus + Pivot". However, we will show that, once again, these equations cannot be derived from the plain ZX-calculus.The theory "ZX-calculus + Euler" is known to be complete for the stabiliser fragment of quantum mechanics [1]: the stabiliser fragment corresponds to the sub-calculus where all angles are multiples of π/2. We show that the intermediate calculus "ZX-calculus + Pivot" is complete for the real stabiliser fragment of quantum mechanics, and that this fragment admits an angle-free axiomatisation.Real quantum mechanics is sufficient for quantum computing [2], in the sense that any unitary evolution on n-qubits can be simulated (using a simple encoding) by a real unitary evolution acting on n + 1 qubits. As a consequence, while not complete for (complex) quantum mechanics, the intermediate calculus "ZX-calculus + Pivot" might be useful and simpler than the the whole "ZX-calculus + Euler" for proving properties of quantum systems, for example via rewriting.Remark. There is some variation about which axioms compris...

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