Local model dynamics for two qubits

Abstract

The density matrix model described in an earlier paper “Model Dynamics for Quantum Computing” is applied to the case of two qubits. The earlier paper provided a density matrix dynamics with unitary time-dependent gates including qubit energy splitting, entropy constraints, external bath and noise effects. The Lindblad formulation as extended by Beretta was adopted in designing this model. Visualization was provided by examining the time-evolution of the polarization vector. In this paper, the same ideas are applied to two-qubit systems, where there are now 15 time-dependent observables; namely, six polarization vectors (P→A,P→B), and nine spin-correlations (T↔). These time-dependent observables provide ready visualization of two qubit dynamics. Special emphasis is placed on the CNOT gate, which is implemented following recent outstanding developments in using silicon-based dots. By invoking a different splitting for each qubit, plus a spin–spin interaction and a carefully designed Rabi oscillation, a CNOT gate is generated. Careful analysis of the time-dependence of these aspects provides insight into CNOT dynamics. The model can be used to ascertain the sensitivity and efficacy of such a CNOT gate when subject to external environmental effects such as noise disturbances, and noise compensation, using the single-system formulation of steepest-entropy-ascent (SEA) non-equilibrium quantum thermodynamics, suitable to model controlled-gate implementations as long as the two qubits remain localized and effectively behave as a single physical four-level system. The role of careful timing in constructing an efficient CNOT gate is illustrated. The formation of Bell states and the evolution of entanglement including noise, bath and entropy considerations are examined. Extensions to the two-qubit swap, and to the three-qubit Toffoli gate dynamics are outlined. It is also shown that a simple Lindblad form can be used to introduce weak and distinct qubit noise pulses. A simple scheme for noise compensation is designed to increase purity and decrease entropy, without invocation of quantum correction methodology. It is based on using the preexisting entanglement of the environment with the quantum system and carefully designed non-Hermitian Lindblad pulses.