Circuit Relations for Real Stabilizers: Towards TOF+H
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The real stabilizer fragment of quantum mechanics was shown to have a complete axiomatization in terms of the angle-free fragment of the ZX-calculus. This fragment of the ZXcalculus--although abstractly elegant--is stated in terms of identities, such as spider fusion which generally do not have interpretations as circuit transformations. We complete the category CNOT generated by the controlled not gate and the computational ancillary bits, presented by circuit relations, to the real stabilizer fragment of quantum mechanics. This is performed first, by adding the Hadamard gate and the scalar sqrt 2 as generators. We then construct translations to and from the angle-free fragment of the ZX-calculus, showing that they are inverses. We remove the generator sqrt 2 and then prove that the axioms are still complete for the remaining generators. This yields a category which is not compact closed, where the yanking identities hold up to a non-invertible, non-zero scalar. We then discuss how this could potentially lead to a complete axiomatization, in terms of circuit relations, for the approximately universal fragment of quantum mechanics generated by the Toffoli gate, Hadamard gate and computational ancillary bits.
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