Robust and optimal control of open quantum systems

Abstract

Recent advancements in quantum technologies have highlighted the importance of mitigating system imperfections, including parameter uncertainties and decoherence effects, to improve the performance of experimental platforms. However, most of the previous efforts in quantum control are devoted to the realization of arbitrary unitary operations in a closed quantum system. Here, we improve the algorithm that suppresses system imperfections and noises, providing notably enhanced scalability for robust and optimal control of open quantum systems. Through experimental validation in a superconducting quantum circuit, we demonstrate that our approach outperforms its conventional counterpart for closed quantum systems with an ultra-low infidelity of about $0.60\%$, while the complexity of this algorithm exhibits the same scaling, with only a modest increase in the prefactor. This work represents a notable advancement in quantum optimal control techniques, paving the way for realizing quantum-enhanced technologies in practical applications.

Figures (9)

• Figure 1: The schematic of the optimal control algorithms. The shadow area represents the general model of the open quantum system considering both the uncertainty of the system Hamiltonian and the decoherence. Exhibiting low computational complexity, Closed-GRAPE algorithm (the blue part) solely focuses on the dynamics of the ideal closed system when optimizing control pulses. In contrast, Open-GRAPE algorithms (the red part) consider these two types of perturbations during the optimization process, leading to an enhancement of the operation fidelity. The purple part corresponds to the approximate Open-GRAPE algorithm that shows both of the advantages mentioned above.
• Figure 2: The numerical simulation results. (A) The schematic of the numerical simulation. The upper part displays the control pulses obtained by the Closed-GRAPE algorithm (blue) and the approximate Open-GRAPE algorithm (red). The middle part is a superconducting circuit with a three-dimensional cavity and a transmon qubit, where the green line represents the dominant extrinsic noise. The black box at the bottom represents a typical Wigner function of the target quantum state, while the blue and red arrows point to the Wigner functions of the quantum states prepared by the respective pulses from the Closed-GRAPE and approximate Open-GRAPE algorithms. (B) The infidelity $\widetilde{f}_{\mathrm{open}}$ of $500$ Closed-GRAPE pulses (blue) and their corresponding approximate Open-GRAPE pulses (red). The objective functions for the Closed-GRAPE and approximate Open-GRAPE algorithms are $\Phi_\mathrm{close}$ and $\Phi_\mathrm{open}$ (see the main text), respectively. The black line represents the reference line where the objective function in the algorithm is equal to the infidelity considering perturbations. It can be seen that all the red points (from approximate Open-GRAPE) closely align with the reference line, while the blue points (from Closed-GRAPE) exhibit significant deviations. (C) The distribution of infidelity corresponding to the data shown in (B). The black line is a Gaussian distribution fitted to the data from the Closed-GRAPE algorithm. The dark gray vertical dashed line indicates the average value, while the distance between the gray dashed lines is the standard deviation $\sigma$ of the Gaussian distribution. In contrast, the data from the approximate Open-GRAPE shows notably lower average infidelity.
• Figure 3: The performance of the algorithms with varying noise strength. (A) The average infidelity as a function of the scaling $s_f$ of uncertainty in the Hamiltonian without decoherence noise. (B) The average infidelity as a function of the scaling $s_m$ of decoherence noise strength without Hamiltonian uncertainty. The average infidelities are calculated with $60$ Closed-GRAPE pulses (blue) and their corresponding approximate Open-GRAPE pulses (red). The error bars for the blue points are smaller than the marker size, making them difficult to distinguish visually.
• Figure 4: The experimental results. (A) Quantum circuit for the experiment. (B) Scatter diagram of infidelities in the initialization experiment with $30$ randomly chosen initial pulses. The horizontal and vertical axes represent the outcomes after optimization with the Closed-GRAPE algorithm and the subsequent refinement through the approximate Open-GRAPE algorithm, respectively. The red dashed line indicates the boundary where the infidelity remains the same with the two algorithms, and the lower half-space is the region where approximate Open-GRAPE is advantageous. (C) The distribution of infidelity corresponding to the data shown in (B). The black line is a Gaussian distribution fitted to the data from the Closed-GRAPE algorithm. The dark gray vertical dashed line indicates the average value and the distance between the gray dashed lines is the standard deviation $\sigma$ of the Gaussian distribution. (D) Infidelity versus the number of repetitive rotation gates. 13 random initial pulses are chosen in this experiment. The red (blue) line is the average infidelity corresponding to the approximate Open-GRAPE (Closed-GRAPE) algorithm. The infidelity deviations for different initial pulses are reflected by the error bars.
• Figure 5: The computational complexity of the algorithms. (A) The variation of the iteration time with an increasing state dimension $d$. The blue and red points represent the data of the Closed-GRAPE algorithm and the approximate Open-GRAPE algorithm with $f=2$ and $v=2$. The blue and red dashed lines show the scaling of $3.38\times10^{-4}\times d^{1.02}$ and $2.18\times10^{-3}\times d^{1.00}$, respectively, fitted from the rightmost five blue or red data points. (B) The variation of the iteration time with an increasing number of uncertain Hamiltonians $f$. (C) The variation of the iteration time with an increasing number of decoherence noise sources $v$. The green points and the orange points represent situations with a matrix dimension of $10^2$ and $10^5$, respectively.