The ZXcalculus is complete for the singlequbit Clifford+T group
Abstract
The ZXcalculus is a graphical calculus for reasoning about pure state qubit quantum mechanics. It is complete for pure qubit stabilizer quantum mechanics, meaning any equality involving only stabilizer operations that can be derived using matrices can also be derived pictorially. Stabilizer operations include the unitary Clifford group, as well as preparation of qubits in the state 0>, and measurements in the computational basis. For general pure state qubit quantum mechanics, the ZXcalculus is incomplete: there exist equalities involving nonstabilizer unitary operations on single qubits which cannot be derived from the current rule set for the ZXcalculus. Here, we show that the ZXcalculus for single qubits remains complete upon adding the operator T to the singlequbit stabilizer operations. This is particularly interesting as the resulting singlequbit Clifford+T group is approximately universal, i.e. any unitary singlequbit operator can be approximated to arbitrary accuracy using only Clifford operators and T.
 Publication:

arXiv eprints
 Pub Date:
 December 2014
 arXiv:
 arXiv:1412.8553
 Bibcode:
 2014arXiv1412.8553B
 Keywords:

 Quantum Physics
 EPrint:
 In Proceedings QPL 2014, arXiv:1412.8102