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arXiv:2601.14708

Scaling Enhancement in Distributed Quantum Sensing via Causal Order Switching

Binke Xia, Zhaotong Cui, Jingzheng Huang, Yuxiang Yang, Guihua Zeng

quant-phphysics.opticsSubmitted Jan 21, 2026• Announced Jan 22, 2026• 0 citations
v1 · Jan 21, 2026v2 · Jul 25, 2026
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Abstract

Sensing networks underpin applications from fundamental physics to real-world engineering. Recently, distributed quantum sensing (DQS) has been investigated to boost the sensing performance, yet current schemes typically rely on entangled probes that are fragile to noise and difficult to scale. Here, we propose a DQS protocol that incorporates a causal-order switch into a cyclic network, enabling a single probe to sequentially query N independent sensors in a coherent superposition or a probabilistic mixture of opposite causal orders. By exploiting the noncommutativity between propagation and sensing processes, our scheme achieves a 1/N^2-scaling precision limit without involving entangled probes. Importantly, our approach utilizes a classical mixture of causal orders rather than a quantum switch, making it more feasible for practical realization. We experimentally implement this scheme for distributed beam tilts sensing in a free-space quantum optical network comprising up to 9 sensors, achieving picoradian-scale precision in estimating tilt angle. Our results demonstrate a robust and scalable DQS protocol that surpasses the conventional 1/N Heisenberg scaling in precision, advancing the practical deployment of quantum sensing networks.

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Sensing networks underpin applications from fundamental physics to real-world engineering. Recently, distributed quantum sensing (DQS) has been investigated to boost the sensing performance, yet current schemes typically rely on entangled probes that are fragile to noise and difficult to scale. Here, we propose a DQS protocol that incorporates a causal-order switch into a cyclic network, enabling a single probe to sequentially query N independent sensors in a coherent superposition or a probabilistic mixture of opposite causal orders. By exploiting the noncommutativity between propagation and sensing processes, our scheme achieves a 1/N^2-scaling precision limit without involving entangled probes. Importantly, our approach utilizes a classical mixture of causal orders rather than a quantum switch, making it more feasible for practical realization. We experimentally implement this scheme for distributed beam tilts sensing in a free-space quantum optical network comprising up to 9 sensors, achieving picoradian-scale precision in estimating tilt angle. Our results demonstrate a robust and scalable DQS protocol that surpasses the conventional 1/N Heisenberg scaling in precision, advancing the practical deployment of quantum sensing networks.
Abstract A plethora of applications hinge on a network or an array of sensors to undertake measurement tasks. A rule of thumb for sensing is that a collective measurement taken by M independent sensors can improve the sensitivity by 1 / M , known as the standard quantum limit (SQL). Quantum resources such as entanglement and squeezed light can be harnessed to surpass the SQL. Distributed quantum sensing is an emerging subject dedicated to investigating the performance gain enabled by entangled states shared by multiple sensors in tackling different measurement problems. This review formulates distributed quantum sensing (DQS) from a quantum-information perspective and describes DQS protocols and their experimental demonstrations. The applications enabled by DQS and an outlook for future opportunities will also be discussed.
Quantum sensor networks promise precision advantages over classical and single-sensor strategies, in particular when the estimator is non-local. We address the problem of finding such estimators through a framework we connote spatial quantum sensing: given an underlying field interrogated by a network of quantum sensors at fixed positions, construct an estimator for a property of the field, for example, distinguishing a source of signal, or evaluating the field or its derivatives at an arbitrary point. We first treat polynomial fields, casting the task as an interpolation problem, and then generalize to fields modeled by analytic functions, which yields general least-squares estimators. A central and largely unaddressed question is under what conditions on sensor placement these estimators are well-defined and error-free. For mmm-dimensional arrays we give explicit constructions and proofs in the interpolation setting using algebraic geometry, and establish necessary and sufficient conditions in the general case. Comparing a non-local entangled protocol with the best local strategy, we show that entanglement yields maximal precision in distributed sensing under global resource constraints. Finally, we introduce error-free subspaces, a technique that translates prior knowledge of the field into a reduction in the number of required sensors. We expect these techniques to be broadly useful in sensing problems across scales, ranging from earth-scale experiments to local applications such as biological imaging.
The precision limit in quantum metrology, quantified by the root-mean-square error of parameter estimation, is conventionally expected to improve at most linearly with the total interrogation time T and with the number N of queried quantum gates. Although several metrological schemes have been shown to achieve precision scaling faster than linear in T and N, they typically rely on unbounded probe-side information resources, usually qualified by an increasingly large variance of the parameter generator. This requirement complicates the interpretation of the resulting scaling advantage and poses substantial technical challenges. In this work, we employ an indefinite-time-direction encoding process to achieve a nonlinear-scaling enhancement of the precision limit. Rather than relying on increasingly informative probe states, our method converts controllable noncommuting encoding operations into metrological gain. Experimentally, we implement this protocol for angular-rotation measurement in a quantum optical system and demonstrate a nonlinear-scaling improvement in practical precision without using probe-side information resources. These results establish a practical framework for surpassing conventional linear-scaling precision limits in quantum metrology and provide new insights into precision enhancement in realistic quantum metrological and sensing applications.
Abstract A distributed sensing protocol uses a network of local sensing nodes to estimate a global feature of the network, such as a weighted average of locally detectable parameters. In the noiseless case, continuous-variable (CV) multipartite entanglement shared by the nodes can improve the precision of parameter estimation relative to the precision attainable by a network without shared entanglement; for an entangled protocol, the root mean square estimation error scales like 1/M with the number M of sensing nodes, the so-called Heisenberg scaling, while for protocols without entanglement, the error scales like 1 / M . However, in the presence of loss and other noise sources, although multipartite entanglement still has some advantages for sensing displacements and phases, the scaling of the precision with M is less favorable. In this paper, we show that using CV error correction codes can enhance the robustness of sensing protocols against imperfections and reinstate Heisenberg scaling up to moderate values of M. Furthermore, while previous distributed sensing protocols could measure only a single quadrature, we construct a protocol in which both quadratures can be sensed simultaneously. Our work demonstrates the value of CV error correction codes in realistic sensing scenarios.
Quantum metrology takes advantage of nonclassical resources such as squeezing and entanglement to achieve a sensitivity level below the standard quantum limit (SQL). To date, most demonstrations of quantum sensing are restricted to a single sensor, nevertheless, many sensing tasks rely on an array of sensors working collectively. The measurement sensitivity of separable sensors scales as 1/sqrt(M) known as the SQL where M is the number of sensors. Shared entanglement among all sensors can be harnessed to surpass the SQL. Recent theoretical advances in distributed quantum sensing have shown that multipartite entangled states give rise to an improvement of 1/sqrt(M) in measurement sensitivity over separable states when estimating a global parameter. We develop a reconfigurable entangled sensor network based on continuous variable multipartite entangled states. The demonstrated entangled sensor network is composed of three sensor nodes, each equipped with an electro-optic transducer for the detection of radio-frequency (RF) signals. By properly tailoring the entangled states through a variational quantum circuit (VQC), the entangled sensor network can be reconfigured to minimize the measurement noise by more than 3dB below the SQL in different distributed RF sensing tasks, e.g., measuring the angle of arrival and mean amplitude of incident RF waves. Such a capability of capturing global features of interrogating objects by a reconfigurable entangled sensor network with less quantum noise further creates opportunities to enable a quantum advantage in data-processing problems. VQCs in conjunction with classical processing constitutes a promising architecture for quantum simulations, classical optimization, and machine learning. We train the VQC by classical machine learning algorithms to optimize the entanglement shared by the sensors for solving practical data processing problems. We show an entanglement-enabled reduction in error probability for classification of multidimensional RF signals. Our work establishes the quantum advantage of distributed quantum sensing and would lead to applications in ultrasensitive positioning, navigation, and data processing.
Quantum metrology has emerged as a powerful tool for timekeeping, field sensing, and precision measurements in fundamental physics. With the advent of distributed quantum metrology, its capabilities have extended to probing spatially distributed parameters across networked quantum systems. However, scalable implementations of distributed quantum metrology with multiparameter estimation remain limited, particularly due to the challenges of generating and distributing entanglement across a quantum network and dealing with incompatibilities in multiparameter quantum metrology. Here we demonstrate distributed multiparameter quantum metrology on a modular superconducting quantum network with low-loss microwave interconnects, a platform that uniquely combines fast gate operations, adaptive control, and deterministic non-local entanglement generation. Using a control-enhanced sequential protocol, we estimate all three components of a remote vector field, achieving up to 13.72 dB improvement in precision over the individual strategy. We further perform direct estimation of vector field gradients along two directions across spatially separated nodes, realizing a 3.44 dB gain over local entanglement strategies. These results establish superconducting quantum networks as a competitive and reconfigurable platform for scalable multiparameter distributed quantum metrology.