EO-GRAPE and EO-DRLPE: Open and Closed Loop Approaches for Energy Efficient Quantum Optimal Control

TL;DR

It is found that the EO-GRAPE method performs better than the EO-DRLPE methods with and without a warm start for most experimental settings, and the Pareto optimality in the trade-off between process fidelity and energetic cost is demonstrated.

Abstract

This research investigates the possibility of using quantum optimal control techniques to co-optimize the energetic cost and the process fidelity of a quantum unitary gate. The energetic cost is theoretically defined, and thereby, the gradient of the energetic cost for pulse engineering is derived. We empirically demonstrate the Pareto optimality in the trade-off between process fidelity and energetic cost. Thereafter, two novel numerical quantum optimal control approaches are proposed: (i) energy-optimized gradient ascent pulse engineering (EO-GRAPE) as an open-loop gradient-based method, and (ii) energy-optimized deep reinforcement learning for pulse engineering (EO-DRLPE) as a closed-loop method. The performance of both methods is probed in the presence of increasing noise. We find that the EO-GRAPE method performs better than the EO-DRLPE methods with and without a warm start for most experimental settings. Additionally, for one qubit unitary gate, we illustrate the correlation between the Bloch sphere path length and the energetic cost.

Figures (11)

• Figure 1: Overview of the system design of a quantum accelerator with classical control and various software modules required for research and development is shown on the left. The different abstraction layers for full-stack quantum computing are shown on the right. This research pertains to the optimal control layer highlighted with a dotted outline.
• Figure 2: Visual interpretation of an energy-efficient quantum unitary gate. The three paths (blue, red, and green) accomplish the same transformation between the two states. The optimal geodesic path (in blue) requires a rotation against a single axis along an angle $\le \pi$.
• Figure 3: Example control pulses generated by the EO-GRAPE algorithm for different weight settings. Parameters: $U_T =$RAND, $\hat{H}_D = \hat{H}_D^{2}$, $\hat{\sigma}_k \in \{ \hat{\sigma}_{x}^{1}, \hat{\sigma}_{x}^{2}, \hat{\sigma}_x^{1} \hat{\sigma}_x^{2} \}$, $T_1 = \infty$, $T_2 = \infty$, $w_f = [1, 0.1]$, $w_e = [0, 0.9]$, $N = 500$, $N_G = 500$
• Figure 4: (a) Infidelity and energetic cost values for $10$ different weight settings (indicated by diamond markers for each color) and learning rates $\epsilon_e$, showing the trade-off or Pareto front between fidelity and energetic cost. (b) Zoomed-in view of the Pareto front between fidelity and energetic cost. Parameters: $U_T =$RAND, $\hat{H}_D = \hat{H}_D^{2}$, $\hat{\sigma}_k \in \{ \hat{\sigma}_{x}^{1}, \hat{\sigma}_{x}^{2}, \hat{\sigma}_x^{1} \hat{\sigma}_x^{2} \}$, $T_1 = \infty$, $T_2 = \infty$, $w_f = [1, 0.1]$, $w_e = [0, 0.9]$, $N = 500$, $N_G = 100$
• Figure 5: Fidelity (red) and Energetic Cost (green) of EO-GRAPE generated control pulses as a function of decreasing decoherence time, or increasing noise level. The moving average (MA) is shown in bold colors. Parameters: $U_T =$Hadamard, $\hat{H}_D = \hat{H}_D^{1}$, $\hat{\sigma}_k \in \{ \hat{\sigma}_{x}^{1} \}$, $T_1 = [100T, 1T]$, $T_2 = [100T, 1T]$, $w_f = 0.7$, $w_e = 0.3$, $N = 100$, $N_G = 500$.