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arXiv · quant-ph/0501048

Four-level and two-qubit systems, sub-algebras, and unitary integration

A. R. P. Rau·G. Selvaraj·D. Uskov
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Abstract

Four-level systems in quantum optics, and for representing two qubits in quantum computing, are difficult to solve for general time-dependent Hamiltonians. A systematic procedure is presented which combines analytical handling of the algebraic operator aspects with simple solutions of classical, first-order differential equations. In particular, by exploiting $su(2) \oplus su(2)$ and $su(2) \oplus su(2) \oplus u(1)$ sub-algebras of the full SU(4) dynamical group of the system, the non-trivial part of the final calculation is reduced to a single Riccati (first order, quadratically nonlinear) equation, itself simply solved. Examples are provided of two-qubit problems from the recent literature, including implementation of two-qubit gates with Josephson junctions.

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A. R. P. Rau, G. Selvaraj, D. Uskov. 2005-01-11. Four-level and two-qubit systems, sub-algebras, and unitary integration. https://doi.org/10.1103/physreva.71.062316

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