Quantum computation in noiseless subsystems with fast non-Abelian holonomies

Quantum-information processing requires a high degree of isolation from the detrimental effects of the environment as well as an extremely precise level of control of the way quantum dynamics unfolds in the information-processing system. In this paper we show how these two goals can be ideally achieved by hybridizing the concepts of noiseless subsystems and of holonomic quantum computation. An all-geometric universal computation scheme based on nonadiabatic and non-Abelian quantum holonomies embedded in a four-qubit noiseless subsystem for general collective decoherence is proposed. The implementation details of this synergistic scheme along with the analysis of its stability against symmetry-breaking imperfections are presented.

Figure 1

Gate fidelity in the presence of a noncollective environment with $g$ controlling the degree of symmetry breaking. A square pulse with magnitude $\Omega$ and duration $\pi /\Omega$ is used. The dephasing rate $\Gamma$ and dissipation rate $\gamma$ are chosen to satisfy $\Gamma =\gamma =0.1\Omega$. The logical unitary gate is the standard Pauli $Z$ operator and the fidelity is averaged over the six axial pure states on the Bloch sphere as input states. The dashed line and the solid line show the gate fidelity in the NS for mean number of environmental quanta $\overline{n}=0$ and 1, respectively. The inset shows the plot of ${log}_{10}(1-F)$ as a function of ${log}_{10}g$; the dashed line and the solid line are for $\overline{n}=0$ and 1, respectively.