Controlled Logic Gate Based on a Four-Node Linear Hybrid Cluster State

Abstract

In the case of a large amount of computation, a situation may arise when the intermediate results of calculations need to be stored for a given time. In this work, we show how to create a hybrid atomic-field cluster state and create a controlled gate in such a way that the results of calculations are recorded under long-lived degrees of freedom of an atomic ensemble and can be stored for a long time.

APPENDIX A. MATRIX QUADRATURE TRANSFORM COEFFICIENTS FOR PERFORMING A TWO-WAY OPERATION

The explicit form of the quadratures of matrix coefficients in expression (58), which describes the relationship between input and output quadratures in a two-way controlled logical operation on a linear four-node cluster, is given by

$$K = \left( {\begin{array}{*{20}{c}} { - \frac{{2\cos[{{\theta }_{1}}]\cos[{{\theta }_{2}}]}}{{\sin[{{\theta }_{1}} - {{\theta }_{2}}]}}}&0&{ - \frac{{\sin[{{\theta }_{1}} + {{\theta }_{2}}]}}{{\sin[{{\theta }_{1}} - {{\theta }_{2}}]}}}&0 \\ 0&{ - \frac{{2\cos[{{\theta }_{3}}]\cos[{{\theta }_{4}}]}}{{\sin[{{\theta }_{3}} - {{\theta }_{4}}]}}}&0&{ - \frac{{\sin[{{\theta }_{3}} + {{\theta }_{4}}]}}{{\sin[{{\theta }_{3}} - {{\theta }_{4}}]}}} \\ {\frac{{\sin[{{\theta }_{1}} + {{\theta }_{2}}]}}{{\sin[{{\theta }_{1}} - {{\theta }_{2}}]}}}&{ - \frac{{2\cos[{{\theta }_{3}}]\cos[{{\theta }_{4}}]}}{{\sin[{{\theta }_{3}} - {{\theta }_{4}}]}}}&{\frac{{2\sin[{{\theta }_{1}}]\sin[{{\theta }_{2}}]}}{{\sin[{{\theta }_{1}} - {{\theta }_{2}}]}}}&{ - \frac{{\sin[{{\theta }_{3}} + {{\theta }_{4}}]}}{{\sin[{{\theta }_{3}} - {{\theta }_{4}}]}}} \\ { - \frac{{2\cos[{{\theta }_{1}}]\cos[{{\theta }_{2}}]}}{{\sin[{{\theta }_{1}} - {{\theta }_{2}}]}}}&{\frac{{\sin[{{\theta }_{3}} + {{\theta }_{4}}]}}{{\sin[{{\theta }_{3}} - {{\theta }_{4}}]}}}&{ - \frac{{\sin[{{\theta }_{1}} + {{\theta }_{2}}]}}{{\sin[{{\theta }_{1}} - {{\theta }_{2}}]}}}&{\frac{{2\sin[{{\theta }_{3}}]\sin[{{\theta }_{4}}]}}{{\sin[{{\theta }_{3}} - {{\theta }_{4}}]}}} \end{array}} \right),$$

((A1))

$$F = \left( {\begin{array}{*{20}{c}} { - \sqrt 2 }&0&0&0 \\ 0&0&0&{\sqrt 2 } \\ 0&0&{ - \frac{{\sqrt 5 }}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}} \\ { - \frac{1}{{\sqrt 2 }}}&{ - \frac{{\sqrt 5 }}{{\sqrt 2 }}}&0&0 \end{array}} \right),$$

((A2))

$$G = \frac{{\sqrt 2 }}{{2{{\beta }_{0}}}}\left( {\begin{array}{*{20}{c}} {\frac{{\cos[{{\theta }_{2}}]}}{{\sin[{{\theta }_{1}} - {{\theta }_{2}}]}}}&{\frac{{\cos[{{\theta }_{1}}]}}{{\sin[{{\theta }_{1}} - {{\theta }_{2}}]}}}&0&0 \\ 0&0&{\frac{{\cos[{{\theta }_{4}}]}}{{\sin[{{\theta }_{3}} - {{\theta }_{4}}]}}}&{\frac{{\cos[{{\theta }_{3}}]}}{{\sin[{{\theta }_{3}} - {{\theta }_{4}}]}}} \\ { - \frac{{\sin[{{\theta }_{2}}]}}{{\sin[{{\theta }_{1}} - {{\theta }_{2}}]}}}&{ - \frac{{\sin[{{\theta }_{1}}]}}{{\sin[{{\theta }_{1}} - {{\theta }_{2}}]}}}&{ - \frac{{\cos[{{\theta }_{4}}]}}{{\sin[{{\theta }_{3}} - {{\theta }_{4}}]}}}&{ - \frac{{\cos[{{\theta }_{3}}]}}{{\sin[{{\theta }_{3}} - {{\theta }_{4}}]}}} \\ {\frac{{\cos[{{\theta }_{2}}]}}{{\sin[{{\theta }_{1}} - {{\theta }_{2}}]}}}&{\frac{{\cos[{{\theta }_{1}}]}}{{\sin[{{\theta }_{1}} - {{\theta }_{2}}]}}}&{ - \frac{{\sin[{{\theta }_{4}}]}}{{\sin[{{\theta }_{3}} - {{\theta }_{4}}]}}}&{ - \frac{{\sin[{{\theta }_{3}}]}}{{\sin[{{\theta }_{3}} - {{\theta }_{4}}]}}} \end{array}} \right),$$

((A3))

$$N = \left( {\begin{array}{*{20}{c}} 0&1&0&0&{ - 1}&0&0&0 \\ 0&0&1&0&0&0&0&{ - 1} \\ { - 1}&0&0&0&0&1&0&{ - 1} \\ 0&0&0&{ - 1}&{ - 1}&0&1&0 \end{array}} \right).$$

((A4))