Optimal Control of a Qubit Coupled to a Non-Markovian Environment

A central challenge for implementing quantum computing in the solid state is decoupling the qubits from the intrinsic noise of the material. We investigate the implementation of quantum gates for a paradigmatic, non-Markovian model: a single-qubit coupled to a two-level system that is exposed to a heat bath. We systematically search for optimal pulses using a generalization of the novel open systems gradient ascent pulse engineering algorithm. Next to the known optimal bias point of this model, there are optimal pulses which lead to high-fidelity quantum operations for a wide range of decoherence parameters.

Figure 1

Top: Gate error versus pulse time $t_g$ for optimal $Z$-gate pulses in the presence of a non-Markovian environment with dissipation strength $\kappa$. A periodic sequence of minima at around $t_n=n\pi /\Delta$ with $n\ge 1$ is obtained. Middle: The gate error of optimized pulses approaches a limit set by $T_1$ and $2T_1$, as shown with $\kappa =0.005$. Bottom: Optimized pulses reduce the error rate by approximately 1  order of magnitude compared to Rabi pulses for $\kappa =0.005$. Pulses starting from zero bias and with realistic rise times (penalty) require only slightly longer gate times, also for $\kappa =0.005$. $E_2=0.1\Delta$, $\Lambda =0.1\Delta$ and $T=0.2\Delta$ in all panels.

Figure 2

Left: Rabi (dashed line) and optimized pulse (solid line) close to the optimal time at $t_g=3.375/\Delta$ and $\kappa =0.05$. Inset shows the evolution of initial state $\rho =|+⟩⟨+|$, $|\pm ⟩=(|0⟩\pm |1⟩)/\sqrt{2}$ on the Bloch sphere under these pulses. Right: Comparison of pulse shape and qubit entropy between Rabi (dashed line), optimized (solid line), and penalty-constrained pulse ($-·-$) at $t_g=5.0/\Delta$ and $\kappa =0.005$. $E_2=0.1\Delta$, $\Lambda =0.1\Delta$, $T=0.2\Delta$ in all panels.

Figure 3

Gate error versus TLF rate $\gamma$ at various temperatures for an optimized pulse with $t_g=5.0/\Delta$. The left inset is a magnification of the low-$\gamma$ part of the main plot and reveals the linear behavior. The right inset shows the maximum of the curves of the main plot versus temperature. ($E_2=0.1\Delta$ and $\Lambda =0.1\Delta$).