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Quantum optimal control represents a powerful technique to enhance the performance of quantum experiments by engineering the controllable parameters of the Hamiltonian. However, the computational overhead for the necessary optimization of these control parameters drastically increases as their number grows. We devise a novel variant of a gradient-free optimal-control method by introducing the idea of phase-modulated driving fields, which allows us to find optimal control fields efficiently. We numerically evaluate its performance and demonstrate the advantages over standard Fourier-basis methods in controlling an ensemble of two-level systems showing an inhomogeneous broadening. The control fields optimized with the phase-modulated method provide an increased robustness against such ensemble inhomogeneities as well as control-field fluctuations and environmental noise, with one order of magnitude less of average search time. Robustness enhancement of single quantum gates is also achieved by the phase-modulated method. Under environmental noise, an XY-8 sequence constituted by optimized gates prolongs the coherence time by 50% compared with standard rectangular pulses in our numerical simulations, showing the application potential of our phase-modulated method in improving the precision of signal detection in the field of quantum sensing.

We propose a novel strategy to reconstruct the quantum state of dark systems, i.e., degrees of freedom that are not directly accessible for measurement or control. Our scheme relies on the quantum control of a two-level probe that exerts a state-dependent potential on the dark system. Using a sequence of control pulses applied to the probe makes it possible to tailor the information one can obtain and, for example, allows us to reconstruct the density operator of a dark spin as well as the Wigner characteristic function of a harmonic oscillator. Because of the symmetry of the applied pulse sequence, this scheme is robust against slow noise on the probe. The proof-of-principle experiments are readily feasible in solid-state spins and trapped ions.Introduction.-The measurement of the quantum state of a system is a prerequisite ingredient in most modern quantum experiments, ranging from fundamental tests of quantum mechanics [1, 2] to various quantuminformation-processing tasks [3][4][5]. However, even with the rapid progress in the coherent manipulation and quantum-state tomography of several quantum systems, such as photons [6,7], electron spins [8-10], atomic qubits [11], superconducting circuits [12,13], and mechanical resonators [14,15], many quantum systems still remain difficult to access for a direct observation of their state, systems we will refer to as dark. In order to circumvent the requirement of such a direct access, a promising technique is to employ an auxiliary quantum system as a measurement probe, on which measurements as well as coherent manipulations can be performed [16][17][18][19][20][21][22][23]. Interferometry [24] based on such a measurement probe allows us to extract information on a target system [25][26][27][28][29][30]. Nevertheless, it still remains a key challenge to achieve a full quantum-state tomography of dark systems without requiring any direct control.

NV centers are among the most promising platforms in the field of quantum sensing. Magnetometry based on NV centers, especially, has achieved concrete development in areas of biomedicine and medical diagnostics. Improving the sensitivity of NV center sensors under wide inhomogeneous broadening and fieldamplitude drift is a crucial issue of continuous concern that relies on the coherent control of NV centers with high average fidelity. Quantum optimal control (QOC) methods provide access to this target; nevertheless, the high time consumption of current methods due to the large number of needful sample points as well as the complexity of the parameter space has hindered their usability. In this paper, we propose the Bayesian estimation phase-modulated (B-PM) method to tackle this problem. In the case of the state transforming of an NV center ensemble, the B-PM method reduced the time consumption by more than 90% compared with the conventional standard Fourier basis (SFB) method while increasing the average fidelity from 0.894 to 0.905. In the AC magnetometry scenario, the optimized control pulse obtained with the B-PM method achieved an eight-fold extension of coherence time T2 compared with the rectangular π pulse. Similar application can be made in other sensing situations. As a general algorithm, the B-PM method can be further extended to the open- and closed-loop optimization of complex systems based on a variety of quantum platforms.