Full quantum theory of control-not gate in ion-trap quantum computation

TL;DR

This paper derives an exact expression for failure probability in ion-trap quantum CNOT gates considering field quantization, providing insights into error thresholds and operational limits for fault-tolerant quantum computing.

Contribution

It presents a precise calculation of failure probabilities in ion-trap CNOT gates accounting for field quantization effects, informing error correction strategies.

Findings

Failure probability increases with operation number and certain initial states.

Failure probability exceeds 10^-2 after about 100 CNOT operations, surpassing fault-tolerance thresholds.

Maximum of 100 CNOT gates per error-correction period for reliable quantum computation.

Abstract

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.