Full quantum theory of control-not gate in ion-trap quantum computation
Biyao Yang, Yang Li · arXiv (Cornell University) · 2015
We investigate the exact effect on ion trap quantum computation after field quantization. First an exact expression of failure probability from field quantization after many CNOT operations in Cirac-Zoller scheme is given. It is proportional to operation number and the amplitude of $|1\rangle_x |0\rangle_y$ or $|1\rangle_x |1\rangle_y$ in initial state, and inverse proportional to mean number of photons and amplitude of $|0\rangle_x |0\rangle_y$ or $|0\rangle_x |1\rangle_y$ in initial state. Then we calculate the failure probability when the limitation to mean number of photons in sideband transition is considered. When the initial state is $|1\rangle_x |0\rangle_y$ or $|1\rangle_x |1\rangle_y$, after about $10^2$ times of CNOT operations, failure probability is no less than $10^{-2}$, while $10^{-2}$ is the known maximum threshold in fault-tolerant quantum computation. Then when the initial state is $|1\rangle_x |0\rangle_y$ or $|1\rangle_x |1\rangle_y$, the number of CNOT gates on the same pair of physical qubits should be no more than $10^2$ in one error-correction period, or else the computation cannot be implemented reliably. This conclusion can help to determine the number of CNOT operations between coding and decoding in one error-correction period in fault-tolerant quantum computation.