Circuit Relations for Real Stabilizers: Towards TOF+H
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by
Cole Comfort
2019
Abstract
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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1904.10614v1
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