Coin-State Metrology in Quantum Walks

• Coin-state metrology is a quantum parameter estimation paradigm that encodes unknown parameters in the coin degree of freedom of a discrete-time quantum walk.
• It exploits coin dimensionality and entanglement to achieve quadratic scaling in quantum Fisher information, mapping physical properties like mass and charge to coin settings.
• Adaptive protocols and joint walker–coin measurements enable simultaneous estimation of multiple parameters, paving the way for advanced quantum sensors.

Coin-state metrology is a quantum parameter estimation paradigm in which the unknown parameter is encoded in the internal “coin” degree of freedom of a quantum walker undergoing a discrete-time quantum walk (DTQW). The walker’s state evolution, dictated by coin-dependent unitary operators, facilitates the extraction of metrological information via projective or adaptive measurements. Coin-state metrology leverages quantum resources such as entanglement and coin dimensionality, enabling precision enhancements that surpass classical bounds and offering robust multi-parameter estimation protocols, including those where physical Hamiltonian parameters (e.g., mass, charge) are mapped to coin parameters.

1. Coin-State Encoding in Discrete-Time Quantum Walks

In DTQWs, the Hilbert space is decomposed as a tensor product $\mathcal{H} = \mathcal{H}_p \otimes \mathcal{H}_c$, with $\mathcal{H}_p = \mathrm{span}\{\ket{x}_p\}$ and $\mathcal{H}_c = \mathrm{span}\{\ket{j}_c\}$, where the coin space may be $D$-dimensional. The single-step walk is governed by a propagator $\mathcal{U}(\theta) = \mathcal{S}[\openone_p \otimes \mathcal{C}(\theta)]$, where $\mathcal{C}(\theta)$ is a coin operator parametrized by the target unknown parameter $\theta$, and $\mathcal{S}$ is a conditional shift in position based on the coin outcome. For spin-$s$ coins, $\mathcal{C}(\theta)$ typically implements rotations around a fixed axis $\hat{n}$: $$\mathcal{C}(\theta) = R_{\hat{n}}^{(D)}(\theta) = \exp[-i\theta \mathcal{T}_{\hat{n}}^{(D)}]$$, where $\{\mathcal{T}_{\hat{n}}^{(D)}\}$ are SU(2) generators (Cavazzoni et al., 2023).

The coin parameters can realize physical quantities, e.g., mixing angles and relative phases corresponding to Dirac mass and charge (via Trotterization) in the DTQW (Annabestani et al., 2021).

2. Quantum Fisher Information and Precision Scaling

The metrological power of a quantum protocol is quantified by the quantum Fisher information (QFI). For a pure state $\ket{\psi(\theta;t)}$ after $t$ steps:

$$\mathcal{F}_Q(\theta) = 4\left[ \langle \partial_{\theta} \psi | \partial_{\theta} \psi \rangle - |\langle \partial_{\theta} \psi | \psi \rangle|^2 \right]$$

For rotation encoding about $z$, the QFI is:

$$\mathcal{F}_Q^{(D)}(\theta;t) = 4 t^2 \left[ \sum_{m_s=-s}^{+s} m_s^2 |\chi_{m_s}|^2 - \left( \sum_{m_s=-s}^{+s} m_s |\chi_{m_s}|^2 \right)^2 \right]$$

where $|\chi_{m_s}|^2$ are the coin state population amplitudes (Cavazzoni et al., 2023). The maximal QFI is attained for $|\chi_{+s}| = |\chi_{-s}| = 1/\sqrt{2}$, yielding $\mathcal{F}_Q^\mathrm{opt} = (D-1)^2 t^2$, exhibiting quadratic scaling in both coin dimension $D$ and step number $t$.

In multi-parameter metrology with the general 2$\times$2 coin operator $C(\lambda_1,\lambda_2,\lambda_3)$, only the mixing angle ($\lambda_2$) and relative phase ($\lambda_1$) parameters can be accessed (rank-2 QFI), with Fisher matrix components given in closed form:

• $$F_{11} = 4 t^2 \frac{\sin \lambda_2}{1 + \sin \lambda_2}$$
• $F_{22} = 4 t^2 (1 - \sin \lambda_2)$ and $F_{12} = 0$ at the optimum (Annabestani et al., 2021).

Grover-like coin encodings can result in divergent QFI as the parameter approaches its boundary (e.g., $\theta \to 1$ in $\mathcal{C}_G^{(2)}(\theta)$), with optimal probes depending on coin structure (Cavazzoni et al., 2023).

3. Optimal Probe States and Measurement Strategies

The optimal initial state for maximal QFI in rotation encoding is $\ket{\psi(0)} = \ket{0}_p \otimes \ket{\Phi_c}$, with $$\ket{\Phi_c} = (\ket{-s}_c + e^{i\gamma}\ket{+s}_c)/\sqrt{2}$$ for $z$-axis rotation, generalizing to higher-dimensional coins.

For the accessible two-parameter subspace of a general coin, a suitably entangled walker-coin state or a localized position state with optimal Bloch vector achieves classical error scaling ($O(t^{-2})$). The projective measurement in the eigenbasis of commuting SLDs $L_{\lambda_1},L_{\lambda_2}$ saturates the quantum Cramér–Rao bound; these measurements can be implemented via joint walker-coin projections or coin-only interferometry after Fourier-sampled $k$-space readout (Annabestani et al., 2021).

Position-only measurement is optimal for $x$/$y$ coin rotations only near $\theta=0$, with classical Fisher information $\mathcal{F}_C \approx \mathcal{F}_Q$ in that regime, but becomes suboptimal for larger angles or for $z$-rotations where $\mathcal{F}_C=0$ (Cavazzoni et al., 2023).

4. Multi-Parameter Estimation and Asymptotic Classicality

Coin-state metrology supports the simultaneous estimation of multiple parameters, notably when coin parameterization relates directly to physical quantities (e.g., mass and charge in the Dirac model). The accessible subspace leads to a rank-2 quantum Fisher information matrix, with the remaining parameter (global phase $\lambda_3$) uncoupled from the metrological protocol: $O_3(k) \propto \mathcal{A}_k^{\mathbb{1}}[O_3] = 0$ and $F_{i3}=0$ for all $i$ (Annabestani et al., 2021).

The two-parameter coin model exhibits vanishing Uhlmann curvature, rendering the asymptotic estimation problem classical; the SLDs commute and the protocol admits a common projective measurement basis, ensuring saturation of both matrix and scalar Cramér–Rao bounds.

For Dirac-walker mass ($m$) and charge ($q$), the coin parameters are linearly mapped: $\lambda_2 \simeq \theta \approx -m\epsilon$, $\lambda_1 = \alpha \approx -qA_x\epsilon$ ($\epsilon$ is a time step), and the estimation variance $$\mathrm{Var}(m)+\mathrm{Var}(q) \ge \frac{1}{4t^2\epsilon^2 (1+1/A_x^2)}$$ (Annabestani et al., 2021).

5. Coin Dimensionality as a Metrological Resource

Increasing the coin dimension $D$ in DTQW-based metrology provides a systematic enhancement of attainable precision. For rotation encoding about $z$, the quantum Fisher information scales as $(D-1)^2 t^2$, with higher $D$ directly amplifying sensitivity (Cavazzoni et al., 2023). For $x$/$y$ rotations, similar but weaker $D$-dependence is observed, with asymptotic prefactors (e.g., $1/2$, $2$, $7/2$ for $D=2,3,4$ respectively).

Grover-like coin operators also exhibit QFI enhancement with increasing $D$ and parameter-dependent singularities, contingent on the probe state's structure.

Experimental realizations of high-dimensional coins utilize multi-port interferometers (path-encoded qudits, orbital angular momentum modes), higher-spin physical systems (atoms, ions), and photonic lattices. Challenges include precise implementation of coin rotations, maintenance of coherence, and operational complexity of joint walker–coin measurements.

6. Adaptive Coin-Bias Estimation in Quantum Walks

Adaptive quantum-enhanced metrology (AQEM) leverages real-time feedback to iteratively refine parameter estimates in multipartite quantum walks. For coin-bias estimation, a decision-tree framework is implemented, where successive measurement outcomes modulate the coin parameter for each walker, forming a single-shot estimator $\hat{p}$ of the bias $p$.

To circumvent exponential memory requirements, outcome coarse-graining is used (partitioning into four positional bins), combined with a logarithmic-search parametrization, reducing memory to $O(N)$ for $N$ walkers (Lovett et al., 2013). Policy optimization is performed via differential evolution (DE) over candidate feedback laws, yielding estimation error scaling $\Delta_{p,N} \sim O(N^{-0.74})$, beating the standard quantum limit ($N^{-1/2}$) and outperforming particle swarm optimization (PSO), which becomes intractable for $N>35$. DE policies remain robust up to $N=75$ walkers.

An optical implementation utilizes photonic polarization for coin states, half-wave plates for controllable bias, and fiber loops for multi-step walks. Real-time feedback and measurement update are realized with MHz-rate electronics. Numerical benchmarks demonstrate $>200\times$ speed-up and extended scaling regimes compared to PSO (Lovett et al., 2013).

7. Practical Impact and Experimental Considerations

Coin-state metrology via DTQW protocols establishes a versatile framework for quantum parameter estimation, supporting scalable error scaling, multi-parameter access, and adaptability to diverse physical systems. While joint walker–coin measurements are generally required to saturate the theoretical quantum bounds, realistic protocols may exploit high-dimensional coins and temporal feedback to optimize performance under experimental constraints.

Robust adaptive protocols and enhanced quantum Fisher information via coin-state engineering open avenues for quantum sensors, Hamiltonian diagnostics, and fundamental studies in quantum information processing, provided coherence and unitarity of coin operations can be maintained throughout the metrological sequence.