Robustness of non-Abelian holonomic quantum gates against parametric noise

We present a numerical study of the robustness of a specific class of non-Abelian holonomic quantum gates. We take into account the parametric noise due to stochastic fluctuations of the control fields which drive the time-dependent Hamiltonian along an adiabatic loop. The performance estimator used is the state fidelity between noiseless and noisy holonomic gates. We carry over our analysis with different correlation times and we find out that noisy holonomic gates seem to be close to the noiseless ones for “short” and “long” noise correlation times. This result can be interpreted as a consequence of the geometric nature of the holonomic operator. Our simulations have been performed by using parameters relevant to the excitonic proposal for the implementation of holonomic quantum computation [P. Solinas et al., Phys. Rev. B 67, 121307 (2003)].

Figure 1

Fidelity for holonomic gate with ${\Omega }^{-1}=50\mathrm{fs}$, $T_{\mathrm{ad}}=7.5\mathrm{ps}\left(\Omega T_{\mathrm{ad}}=150\right)$, and $\delta \Omega =0.1\Omega$. (a) Slowly varying fluctuations and (b) fast varying fluctuations.

Figure 2

Loop in the parameter space for the holonomic gate with (solid line) and without (dashed line) noise, with extraction of $2\left(T_n=T_{\mathrm{ad}}∕2\right)$ (a) and 70 random numbers $\left(T_n=T_{\mathrm{ad}}∕70\right)$ (b). The sphere radius is normalized to $\Omega =1$.

Figure 3

Loop in the parameter space for the holonomic gate with noise and the extraction of 5000 random numbers $\left(T_n=T_{\mathrm{ad}}∕5000\right)$. The sphere radius is normalized to $\Omega =1$.

Figure 4

Fidelity for small perturbations $\delta \Omega =0.01\Omega$. Parameters as in Fig. 1.

Figure 5

Fidelity for the “phase” noise. Parameters as in Fig. 1.

Figure 6

Fidelity for holonomic gates when the system is subject to “intensity” and “phase” noise. Parameters as in Fig. 1.

Figure 7

Population of the nonlogical states $\mid G⟩$ and $\mid E_L^0⟩$ after the holonomic gate application when the system is subject to “intensity.” Parameters as in Fig. 1.

Figure 8

Comparison between holonomic (solid line) and dynamical (dashed line) gates with (a) $\delta \Omega =0.1\Omega$ and (b) $\delta \Omega =0.01\Omega$. On the top $T_{\mathrm{ad}}∕T_n$ for the holonomic gate and on the bottom $T_{\mathrm{dyn}}∕T_n$ for dynamical gates are reported.

Figure 9

Fidelity for the single qubit phase shift gate with intensity noise. The parameters are as in Fig. 1.

Figure 10

Fidelity for the two qubit phase shift with intensity noise. Parameters are: $\delta =5$ meV, $\mid \mathrm{\Omega }\mid =\delta ∕15$, and $T_{\mathrm{ad}}=0.8$ ns.