The Cirac-Zoller mechanism is generally used for the implementation of CNOT gates in ion trap quantum computers. We present here a highly detailed quantum physical description of this approach. To keep the presentation simple, the qubits are modeled as rigid rotors and the radiation field of each of the lasers is treated as a classical field. In combination with the corresponding first part (Ion Trap Quantum Computer Part 1: 1-Qubit Gates), this results in the complete quantum theory of a universal set of quantum gates. This presentation is intended for readers with basic knowledge of quantum physics and assumes familiarity with Part 1.
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⟩x∣0⟩y or
∣1⟩x∣1⟩y in initial state, and inverse proportional to mean number of photons and amplitude of
∣0⟩x∣0⟩y or
∣0⟩x∣1⟩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⟩x∣0⟩y or
∣1⟩x∣1⟩y, after about
102 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⟩x∣0⟩y or
∣1⟩x∣1⟩y, the number of CNOT gates on the same pair of physical qubits should be no more than
102 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.