The advantage of quantum metrology has been experimentally demonstrated for phase estimations where the dynamics are commuting. General noncommuting dynamics, however, can have distinct features. For example, the direct sequential scheme, which can achieve the Heisenberg scaling for the phase estimation under commuting dynamics, can have even worse performances than the classical scheme when the dynamics are noncommuting. Here we realize a scalable optimally controlled sequential scheme, which can achieve the Heisenberg precision under general noncommuting dynamics. We also present an intuitive geometrical framework for the controlled scheme and identify sweet spots in time at which the optimal controls used in the scheme can be prefixed without adaptation, which simplifies the experimental protocols significantly. We successfully implement the scheme up to eight controls in an optical platform and demonstrate a precision near the Heisenberg limit. Our work opens the avenue for harvesting the power of quantum control in quantum metrology, and provides a control-enhanced recipe to achieve the Heisenberg precision under general noncommuting dynamics.
The advantage of quantum metrology has been experimentally demonstrated for phase estimations where the dynamics are commuting. General noncommuting dynamics, however, can have distinct features. For example, the direct sequential scheme, which can achieve the Heisenberg scaling for the phase estimation under commuting dynamics, can have even worse performances than the classical scheme when the dynamics are noncommuting. Here we realize a scalable optimally controlled sequential scheme, which can achieve the Heisenberg precision under general noncommuting dynamics. We also present an intuitive geometrical framework for the controlled scheme and identify sweet spots in time at which the optimal controls used in the scheme can be prefixed without adaptation, which simplifies the experimental protocols significantly. We successfully implement the scheme up to eight controls in an optical platform and demonstrate a precision near the Heisenberg limit. Our work opens the avenue for harvesting the power of quantum control in quantum metrology, and provides a control-enhanced recipe to achieve the Heisenberg precision under general noncommuting dynamics.
A pivotal task in quantum metrology, and quantum parameter estimation in general, is to design schemes that achieve the highest precision with the given resources. Standard models of quantum metrology usually assume that the dynamics is fixed and that the highest precision is achieved by preparing the optimal probe states and performing optimal measurements. However, in many practical experimental settings, additional controls are usually available to alter the dynamics. Here we propose to use optimal control methods for further improvement of the precision limit of quantum parameter estimation. We show that, by exploring the additional degree of freedom offered by the controls, a higher-precision limit can be achieved. In particular, we show that the precision limit under the controlled schemes can go beyond the constraints put by the coherent time, which is in contrast with the standard scheme where the precision limit is always bounded by the coherent time.
Time is a valuable resource and it is expected that a longer time period should lead to better precision in Hamiltonian parameter estimation. However, recent studies in quantum metrology have shown that in certain cases more time may even lead to worse estimations, which puts this intuition into question. In this Letter we show that by including feedback controls this intuition can be restored. By deriving asymptotically optimal feedback controls we quantify the maximal improvement feedback controls can provide in Hamiltonian parameter estimation and show a universal time scaling for the precision limit under the optimal feedback scheme. Our study reveals an intriguing connection between noncommutativity in the dynamics and the gain of feedback controls in Hamiltonian parameter estimation.
The Hamiltonian, which determines the evolution of a quantum system, is fundamental in quantum physics. Therefore, it is crucial to implement high-precision generation and measurement of the Hamiltonian in a practical quantum system. Here, we experimentally demonstrate ultrahigh-precision Hamiltonian parameter estimation with a significant quantum advantage in a superconducting circuit via sequential control. We first observe the commutation relation for noncommuting operations determined by the system Hamiltonian, both with and without adding quantum control, verifying the commuting property of controlled noncommuting operations. Based on this control-induced commuting property, we further demonstrate Hamiltonian parameter estimation for polar and azimuth angles in superconducting circuits, achieving ultrahigh metrological gains in measurement precision exceeding the standard quantum limit by up to 16.0 and 16.1 dB at N=100, respectively.
Nature imposes fundamental limits on how precisely we can probe the physical world. Quantum metrology is the study of achieving higher precision and accuracy in estimating the physical parameters by harnessing quantum resources. It is considered one of the most promising quantum technologies, offering significant advantages over classical ones in both near‑term and long‑term applications. However, existing theoretical research in quantum metrology often falls short of providing a unified and rigorous understanding of the ultimate precision under realistic experimental limitations on available quantum resources. In this thesis, we conduct a systematic investigation of the optimal precision limits in quantum metrology and identify the metrological strategies that achieve optimality under specified resource constraints. We primarily focus on the single-parameter estimation of quantum processes, a cornerstone task in quantum metrology. We first establish a general framework of strictly optimal quantum metrology using semidefinite programming within the formalism of higher-order quantum operations. This framework reveals a strict hierarchy for the performance of different families of strategies---parallel, sequential, and indefinite-causal-order ones---each subject to specific causal constraints. Our results rigorously demonstrate a quantum metrological advantage of indefinite causal order in finite-dimensional systems, which has recently aroused rapidly growing interest. In realistic scenarios, a fully optimized quantum metrology protocol may be experimentally demanding due to the complexity of state preparation and control. Tailored to practical constraints on the allowed ancilla and control operations, we propose a highly efficient tensor network algorithm to optimize the probe state and a long sequence of control for quantum metrology. When the number of control steps tends to infinity, we develop an asymptotic theory of quantum metrology for achieving the ultimate precision limit---Heisenberg limit---without ancilla. This contrasts with the traditional approach using quantum error correction for metrology that typically requires noiseless and controllable ancilla. Our results pave the way for exploiting the full potential of quantum resources in metrology. In the near term, our methods enable high-precision estimation with limited interaction with the unknown process using experimentally feasible control techniques. In the long term, they lay the foundation for scalable and fault-tolerant quantum metrology.
Quantum metrology promises unprecedented precision of parameter estimation, but it is often vulnerable to noise. While significant efforts have been devoted to improving the metrology performance in Markovian environments, practical control schemes specifically designed for non-Markovian noises are much less investigated. Here, we propose two control-enhanced quantum metrology schemes that are suitable for tackling general non-Markovian noises described by noise channels or noise spectra. We conduct experiments to verify the efficacy of these schemes on a nuclear magnetic resonance system. The experimental results involving multiqubit probes show that the parameter estimation precision can be greatly improved, significantly surpassing the standard quantum limit, with our schemes. At present, non-Markovian noises are widely encountered on diverse quantum devices, the proposed schemes are relevant for realistic metrology applications on these platforms. Quantum metrology, a powerful paradigm for surpassing classical measurement precision, has been extensively studied for Markovian noise, while most practical physical processes obey non-Markovian dynamics. In this paper, the authors propose control-enhanced quantum metrology schemes to counteract non-Markovian noise and experimentally verify their efficacy.
A central feature of quantum metrology is the possibility of Heisenberg scaling, a quadratic improvement over the limits of classical statistics. This scaling, however, is notoriously fragile to noise. While for some noise types it can be restored through error correction, for other important types, such as dephasing, the Heisenberg scaling appears to be irremediably lost. Here we show that this limitation can sometimes be lifted if the experimenter has the ability to probe physical processes in a coherent superposition of alternative configurations. As a concrete example, we consider the problem of phase estimation in the presence of a random phase kick, which in normal conditions is known to prevent the Heisenberg scaling. We provide a parallel protocol that achieves Heisenberg scaling with respect to the probes' energy, as well as a sequential protocol that achieves Heisenberg scaling with respect to the total probing time. In addition, we show that Heisenberg scaling can also be achieved for frequency estimation in the presence of continuous-time dephasing noise, by combining the superposition of paths with fast control operations.