Distributed multi-parameter quantum metrology with a superconducting quantum network

Abstract

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 multi-parameter estimation remain limited, particularly due to the challenges of generating and distributing entanglement across a quantum network and dealing with incompatibilities in multi-parameter quantum metrology. Here we demonstrate distributed multi-parameter 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 multi-parameter distributed quantum metrology.

Figures (5)

• Figure 1: Schematic of distributed quantum metrology with a modular superconducting sensor network. The platform comprises a central module $\mathcal{A}$ and multiple spatially separated sensor modules ($\mathcal{B}$, $\mathcal{C}$), each hosting several transmon qubits (blue and light blue spheres). The sensor modules are connected to the central module via low-loss coaxial cables, enabling high-fidelity microwave state transfer (orange arrows) and the generation of non-local entanglement across remote nodes. This architecture supports two key sensing protocols: (1) estimation of remote vector fields encoded by locally non-commuting generators (e.g., $\mathbf{B}_1$, $\mathbf{B}_2$), and (2) estimation of spatial gradients $\nabla \mathbf{B}$ via distributed entangled probes. Each sensor qubit undergoes a sequence of signal and control unitaries with interrogation time $T$ (top right), implementing a sequential metrological strategy. The blue background indicates local connectivity within a module, while the purple shading highlights modules linked by non-local entanglement. The inclusion of both sensor and ancilla qubits reflects their complementary roles in enabling tailored entangled state preparation, signal encoding, and joint measurement for multi-parameter sensing.
• Figure 2: Multi-parameter quantum metrology of a remote magnetic field with the sensor-ancilla network.a, Schematic diagram of the controlled-enhanced sequential strategy. The protocol measures a local magnetic field at the sensor's position using a sequential strategy with one ancilla qubit. The three-component field $\mathbf{B}$ is parameterized by ${B, \theta, \phi}$. b, Measurement probability $P_{00}$ for $N=8$ cycles as a function of signal encoding time $T$ (from $0$ to $2\pi$), scanned across control parameters $B_c$ (left panel) and $\phi_c$ (right panel). c, Measurement probability $P_{00}$ for a fixed encoding time $T=1.5\pi$ at different cycle numbers $N=1,2,4,8$, scanned across control parameters ${ B_c, \theta_c, \phi_c }$. Control parameters not being scanned in b and c are fixed to their optimal values. Error bars denote the standard deviation. d, Results of parameter estimation. Main panel: Distribution of estimator $\phi_{\rm{est}}$ for $N=1,2,4,6,8$. Dashed lines represent Gaussian fitting of the estimator histogram; circles and the error bars mark the average value of $\phi_{\rm{est}}$ and the standard deviation. Upper inset: Landscape of the likelihood function at $N=4$ (left) and $N=8$ (right) for parameters $\phi$ and $\theta$. The star marks the optimal control setting that maximizes the likelihood. Lower inset: Assessed precision (variance $\delta^2$) of the three parameters (dots) compared to the $1/N^2$ scaling limits (solid lines) and the $1/N$ scaling bounds of the individual measurement strategy (dashed lines), extracted from $M=600$ sets of estimators, each derived from $n=600$ repeats of single-shot measurements. The definition of the error bars is described in Methods. The dashed arrow marks the reduction of $\delta \theta_{\rm{est}}$ over its classical bound.
• Figure 3: Multi-parameter quantum metrology of the magnetic field gradient. a, Distributed sensing scheme for directly estimating the gradients of two vector fields with a non-local entangled state. Left section shows the probe state initialization. The system consists of a central module (yellow box) and two sensor modules (light blue boxes within the grey area). Blue spheres within the modules represent qubits, the purple stick-ball model represents CNOT gates, and the circular arrow denotes a local rotation. Four sensor qubits are initialized in a non-local entangled state, $|\Psi_{0}\rangle = \frac{1}{\sqrt{2}}(|0011\rangle - |1100\rangle)$. The middle section illustrates the signal control encoding process, while the right section depicts simultaneous Bell measurements on the two encoded sensor modules. Sensor qubits in the same colored stripes are encoded with identical vector field signals. b, Benchmarking the performance of the gradiometer. Main panel: Distribution of estimator $\nabla B_{y_{\rm{est}}}$ for $N=1$ to $N=4$. Dashed lines represent the Gaussian fit of the estimator histograms, while circles and error bars indicate the mean values and standard deviations ($\delta \nabla B_{y_{\rm{est}}}$). Inset: Landscape of the log-likelihood function $\mathcal{L^{\prime}}$ at $N=2$ (left) and $N=4$ (right) (see Methods). The star marks the optimal control point and the red dashed contour denotes the parameter-estimation region. c. The gradient estimation precision, evaluated by the sum of variances, $\sum_{i \in \{x, y, z\} \cup \{x,y\}} \delta^2 \nabla B_{i_{\text{est}}}$, obtained from $M$ sets of estimators ($M=600$), each derived from $n$ measurement shots ($n=600$). Top panel: 2-component vector field. Bottom panel: 3-component vector field. Solid lines indicate the $1/N^2$ scaling limit ($1/N^2$ S.L.). Precision estimated at $T=0.5\pi$ is marked in green, while precision at $T=1.5\pi$ is marked in blue. The definition of the error bars is described in Methods.
• Figure 4: Strategies comparison for gradient estimation of a 2-component vector field.a-b, Schematic diagrams of different strategies: ( a) Distributed sensing with non-local entanglement (NLE); ( b) Sensing with local entanglement (LE). c-e, Comparison of the precision ($\sum_{i\in \{x,y\}} \delta^2 \nabla\! B_{i_\text{est}}$) of the two strategies. ( c) Precision as a function of field strength $|B|$ at $T\ =\ 1.5\pi$ and $N=1$. ( d) Precision as a function of encoding time $T$ at $|B| = 1$ and $N = 1$. ( e) Precision versus number of cycles $N$ at $T = 1.5\pi$ and $|B| = 1$. The solid curve represents the theoretical precision bound for the local entanglement strategy with the probe state and measurement taken as the Bell state and Bell measurement, labelled as LE(B). The dashed curve represents the theoretical precision bound for the local entanglement strategy with the optimal probe state and optimal measurement, labelled as LE(O). The definition of the error bars in (c)-(e) is described in Methods.
• Figure 5: Adaptive control-enhanced metrology for simultaneous three-parameter estimation with signal parameters $(B,\theta,\phi) = (1, \frac{\pi}{4},\frac{\pi}{4})$. Here $R_\text{iter}$ denotes the iteration index of the adaptive estimation-control loop. a, The results of adaptive iterations starting from different initial guess values (10 sets randomly chosen within the boundary of the landscape) with $N=1$. b, The results of adaptive iterations with different sequential copies $N=1$ to $6$ and fixed initial guess (1 set randomly chosen within the boundary of the landscape for $N=1$). c, The calculated likelihood function landscape for $N=4$ after iteration cycle $1,2,3,5,10$. Stars indicate the locations of the optimal control parameters.