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Quantum metrology

Quantum metrology is the science of using quantum mechanical phenomena to make precise measurements. The key insight is that quantum resources like entanglement and squeezing can provide measurement precision beyond what's possible with classical methods.

Quantum sensing protocols are generally composed of four stages:

  1. Probe State Preparation: Generation of some metrologically-useful probe state, \(|\psi_0\rangle\)
  2. Dynamical Interaction: The unknown parameter \(\varphi\) interacts with the probe, \(|\psi(\varphi)\rangle = U(\varphi)|\psi_0\rangle\)
  3. Detection & Measurement: The perturbed probe state is measured, yielding outcome probabilities, \(p(s_i|\varphi)\)
  4. Estimation: Estimate \(\bar{\varphi}\) from a finite set of classical measurement outcomes

Classical vs quantum limits

The precision of any unbiased estimator is bounded by the Cramér-Rao bound:

\[\Delta^2\bar{\varphi} \geq \frac{1}{m \mathcal{I}_\varphi}\]

where \(m\) is the number of measurements and \(\mathcal{I}_\varphi\) is the Fisher Information.

  • Classical sensors: \(\mathcal{I}_\varphi \propto n\) (Standard Quantum Limit)
  • Quantum sensors: \(\mathcal{I}_\varphi \propto n^2\) (Heisenberg Limit)

This quadratic improvement means quantum sensors can achieve the same precision with exponentially fewer resources.

Fisher Information

The Fisher Information quantifies how much information measurement data contains about an unknown parameter.

The Quantum Fisher Information (QFI) represents the maximum information extractable from a quantum state, achieved by the optimal measurement:

\[ \mathcal{I}_\varphi^{(Q)} = 4(\langle\partial_\varphi\psi|\partial_\varphi\psi\rangle - |\langle\psi|\partial_\varphi\psi\rangle|^2) \]

For mixed states,

\[\mathcal{I}_\varphi^{(Q)} = \sum_{i,j} \frac{2\text{Re}(\langle\lambda_i|\partial_\varphi\rho|\lambda_j\rangle)}{\lambda_i + \lambda_j}\]

The CFI quantifies the information in actual measurement outcomes:

\[\mathcal{I}_\varphi^{(C)} = \sum_i \frac{(\partial_\varphi p(s_i|\varphi))^2}{p(s_i|\varphi)}\]

Key relationship: \(\mathcal{I}_\varphi^{(C)} \leq \mathcal{I}_\varphi^{(Q)}\) always, with equality achieved by optimal measurements.

Variational quantum sensing

  1. Parameterizing all protocol components (state preparation, measurements)
  2. Optimizing parameters to maximize Fisher Information
  3. Adapting to hardware constraints and noise
  4. Learning optimal strategies from data

This approach enables:

  • Hardware-efficient protocols
  • Noise resilience
  • Automatic protocol discovery
  • End-to-end optimization

squint leverages JAX's automatic differentiation to compute, e.g., gradients of amplitudes and probability distributions, and to compute the Fisher Inforamtion itself. For a parameterized quantum state \(|\psi(\varphi)\rangle\), we can compute:

# State and its gradient
psi = circuit.amplitudes.forward(params)      # |ψ(θ)⟩
dpsi = circuit.amplitudes.grad(params)        # ∂|ψ(θ)⟩/∂θ

# Probabilities and their gradients  
p = circuit.probabilities.forward(params)     # p(s|θ)
dp = circuit.probabilities.grad(params)       # ∂p(s|θ)/∂θ

# Fisher Information Matrices
qfim = circuit.amplitudes.qfim(params)        # Quantum FIM
cfim = circuit.probabilities.cfim(params)     # Classical FIM