Protecting the $\sqrt{\mathit{SWAP}}$ operation from general and residual errors by continuous dynamical decoupling

We study the occurrence of errors in a continuously decoupled two-qubit state during a $\sqrt{\mathit{SWAP}}$ quantum operation under decoherence. We consider a realization of this quantum gate based on the Heisenberg exchange interaction, which alone suffices for achieving universal quantum computation. Furthermore, we introduce a continuous-dynamical-decoupling scheme that commutes with the Heisenberg Hamiltonian to protect it from the amplitude damping and dephasing errors caused by the system-environment interaction. We consider two error-protection settings. One protects the qubits from both amplitude damping and dephasing errors. The other features the amplitude damping as a residual error and protects the qubits from dephasing errors only. In both settings, we investigate the interaction of qubits with common and independent environments separately. We study how errors affect the entanglement and fidelity for different environmental spectral densities.

Figure 1

Amplitude damping plus dephasing: In (a) and (b) we show, respectively, the concurrence and fidelity for independent environments and, in (c) and (d), for the case of a common environment. The dotted (blue) line represents the dynamics of a $\sqrt{\mathit{SWAP}}$ quantum gate with protection. The solid (red) and dashed (green) lines represent the dynamics without protection for ohmic and superohmic spectral densities, respectively.

Figure 2

Amplitude damping as a residual error channel: In (a) and (b) we show, respectively, the concurrence and fidelity for independent environments and, in (c) and (d), for the case of a common environment. The solid (red) and dashed (green) lines represent the dynamics of a $\sqrt{\mathit{SWAP}}$ quantum gate for ohmic and superohmic environments, respectively.

Figure 3

The concurrence (a) and fidelity (b) at the end of the quantum gate operation, i.e., at $t=\tau$, as functions of the coupling constant $\lambda$ of the residual error channel. The solid (red) line represents the case of independent environments with ohmic spectra, the dotted (blue) line represents the case of a common environment with an ohmic spectrum, the dashed (green) line represents the case of independent environments with superohmic spectra, and the purple (closely dotted) line represents the case of a common environment with a superohmic spectrum.