Hybrid Quantum-Classical Approach to Quantum Optimal Control

A central challenge in quantum computing is to identify more computational problems for which utilization of quantum resources can offer significant speedup. Here, we propose a hybrid quantum-classical scheme to tackle the quantum optimal control problem. We show that the most computationally demanding part of gradient-based algorithms, namely, computing the fitness function and its gradient for a control input, can be accomplished by the process of evolution and measurement on a quantum simulator. By posing queries to and receiving answers from the quantum simulator, classical computing devices update the control parameters until an optimal control solution is found. To demonstrate the quantum-classical scheme in experiment, we use a seven-qubit nuclear magnetic resonance system, on which we have succeeded in optimizing state preparation without involving classical computation of the large Hilbert space evolution.

Figure 1

(a) Hybrid quantum-classical approach to gradient-based optimal control iterative algorithms, wherein the quantum simulator is combined with classical computing devices to jointly implement the procedure of optimal control searching. Here, ${\rho }_{\mathrm{in}}$ is the input state, ${\rho }_{\text{out}}$ is the output state, double-lined arrows signify quantum information, and $\mathbf{M}$ represents quantum measurement. (b) Schematic diagram of an NMR based implementation of the quantum-classical hybrid optimal control searching. The sample consists of an ensemble of spins and serves as a quantum processor. Query is encoded in input radio-frequency (rf) control pulse and the answer that the sample generates is extracted from observing the free induction decay (FID).

Figure 2

(a) Molecular structure of crotonic acid. (b) Pulse sequence scheme for our multiple-quantum coherence generation experiment. The gray part is designed to reset the system back into ${\rho }_i$, and the preparation part is an approximate circuit (in which cw: continuous wave; $G_z$: gradient pulse along $z$ axis; ${\varphi }_1=-18°$ and ${\varphi }_2=82°$) aimed for making the transform ${\rho }_i\to \overline{\rho }$. (c) Iterative results for our system. Here $u_c$ and $u_o$ denote the controls obtained by searching on a classical computer and on the sample, respectively. (d) NMR spectrum of ${\rho }_f$ after 10 times of iteration under the observation of ${\mathrm{C}}_2$. It is placed with the simulated ideal spectrum of target state $\overline{\rho }$ together for comparison.