Weighted graph states as a resource for quantum metrology

TL;DR

This work demonstrates that weighted graph states, especially star and complete geometries, can serve as practical resources for quantum metrology by achieving precision beyond the standard quantum limit and approaching the Heisenberg limit under homogeneous encodings. It combines numerical optimization over Pauli-type measurements with analytic expressions for the quantum Fisher information and estimator variance, revealing robust regimes where imperfect weightings still confer advantages. The main contributions include identifying the star and complete weighted-graph sub-classes, deriving closed-form QFI and expectation-value formulas, and showing scalability to large N while maintaining below-SQL performance. The findings suggest that weakly entangled, structurally simple graph states can enable quantum-enhanced sensing in realistic experimental settings, with reduced entanglement requirements and tolerance to weight imperfections.

Abstract

Quantum metrology exploits quantum mechanical effects to increase the precision of measurements of physical quantities. A wide variety of applications are currently being developed for scientific and technological purposes, however, most research relies on the use of highly entangled resource states that are challenging to generate and control in a given physical system. Here, we study the use of weighted graph states as more accessible resources for quantum metrology, which yield a favorable precision beyond the classical limit, approaching the Heisenberg limit. We find a notable robustness to variation in weights and less challenging weight requirements compared to standard graph states, which require a maximal weight at all edges. Both of these aspects reduce the practical demands in a physical setup, with the latter implying significantly less entanglement is required to gain a quantum advantage in metrology. We study the quantum Fisher information and optimized estimator variance of two identified sub classes of weighted graph states for an arbitrary number of N qubits, providing analytical forms and investigating their scaling. Our work opens up opportunities for using weakly entangled states in quantum-enhanced metrology.

Figures (12)

• Figure 1: The main steps of the phase estimation scheme which utilize the example star and complete weighted graph state sub-classes (4-qubit class representatives shown here): (I) State generation - the vertices correspond to qubits initialized to $|+\rangle$ and the $\phi_{ab}$-edges connecting the vertices represent controlled phase-$\phi$ gates. Note that for the analytical study in this section, we assume uniform edge weights, i.e., $\phi_{ab} = \phi$ for all edges, whereas in the subsequent numerical analysis we allow randomly chosen edge weights $\phi_{ab}$; (II) Perturbation - encode the phase $\theta$ by a unitary transformation; (III) Measurement - execute the optimal (Pauli) measurement operators which minimizes the uncertainty, that is the variance $(\Delta \theta)^2$ in estimating $\theta$. More specifically, the optimal measurement operators are found to be $\hat{Y} \otimes \hat{Z} \otimes \hat{Y} \otimes \hat{Z}$ and $\hat{Y} \otimes \hat{Y} \otimes \hat{Y} \otimes \hat{Y}$, for the $N=4$ star and complete weighted graph states, respectively. The right hand side depicts the generalization to $N$ qubits for the two sub-classes. The two sub-classes are locally equivalent only when all edge weights are maximal (see the main text for details).
• Figure 2: The QFI of weighted star graph states, with $\hat{H}$ given by Eq. (\ref{['C']}). The dashed horizontals denote the corresponding QFI upper bounds for separable states (SQL) of size $N \in \{2,3,4,5\}$.
• Figure 3: The QFI of weighted complete graph states, with $\hat{H}$ given by Eq. (\ref{['C']}). The dashed horizontals denote the corresponding QFI upper bounds for separable states (SQL) of size $N \in \{2,3,4,5\}$.
• Figure 4: The optimal estimator variance $(\Delta \theta)^2$, restricting to Pauli measurements, over the set of connected uniformly weighted graph states which go below the SQL (assuming $\theta =0.001$ and system size $N \in \{2,3,4,5\}$). It is important to note that small $\theta$ is the most interesting regime for quantum metrology and that any $\theta$ can be shifted to this point in principle via an offset phase. Dashed horizontal green and red lines indicate the SQL and HL, respectively, while the dotted curves show the Cramér-Rao bounds computed from the analytical QFI expressions derived for star and complete weighted graph states in Section IV. The points of discontinuity are discussed in the main text.
• Figure 5: Variance of the estimator for the weighted star graph state, using the measurement operator given in Eq. (\ref{['evenstar']}), for $N \in \{4,6,8,10\}$ and $\phi_{ab} = \phi \in [0, 2 \pi)$ for all $\{a,b\} \in E$. The horizontal lines denote the SQL.

Theorems & Definitions (4)

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