Collective measurements on identically prepared quantum systems can extract more information than local measurements, thereby enhancing information-processing efficiency. Although this nonclassical phenomenon has been known for two decades, it has remained a challenging task to demonstrate the advantage of collective measurements in experiments. Here, we introduce a general recipe for performing deterministic collective measurements on two identically prepared qubits based on quantum walks. Using photonic quantum walks, we realize experimentally an optimized collective measurement with fidelity 0.9946 without post selection. As an application, we achieve the highest tomographic efficiency in qubit state tomography to date. Our work offers an effective recipe for beating the precision limit of local measurements in quantum state tomography and metrology. In addition, our study opens an avenue for harvesting the power of collective measurements in quantum information-processing and for exploring the intriguing physics behind this power.

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 under noncommuting dynamics. 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 pre-fixed without adaptation, which simplifies the experimental protocols significantly. We successfully implement the scheme up to eight controls in an optical platform, 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.Introduction.-Improving the measurement precision [1-6] is one of the major driving forces for technology and science. The precision of a measurement scheme is ultimately bounded by the available resources [7,8], which are typically quantified by the number of uses of a discrete-time dynamics, N , or by the evolution time of of a continuous-time dynamics, T . The best precision of a classical scheme, known as the quantum Shot-Noise-Limit (SNL), scales as 1/ √ N or 1/ √ T for discrete and continuous dynamics respectively. The SNL is already constraining the performance of current state-ofart precision measurements, such as LIGO interferometer [7,[9][10][11]. By exploring quantum effects, quantum metrology can surpass the SNL [1,4,[12][13][14]. For example, by preparing the probe state as the NOON state in the entangled parallel scheme [15,16], it can achieve the Heisenberg precision which scales as 1/N [1,2,4,5,17]. In practise, however, preparing large entangled states are extremely challenging. To date, the largest NOON state prepared deterministically in optical experiment is N = 5 [18], while the largest NOON state that has been implemented for quantum metrology is only N = 4(2) with(without) the postselection [2,5,17].

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