Quantum sensing estimates a parameter — a magnetic field, a frequency, a phase, a force — from its effect on a quantum system. Control enters at every stage: preparing a sensitive probe state, shaping its evolution so that it picks up the signal and rejects noise, and choosing the measurements. The tasks span magnetometry with spins, atomic clocks and interferometers, and optical phase estimation [1, 2].

The figure of merit

For \(\nu\) repetitions, the error of any unbiased estimator of a parameter \(\omega\) obeys the quantum Cramér–Rao bound \(\Delta\omega \ge 1/\sqrt{\nu F_Q}\), where \(F_Q\) is the quantum Fisher information of the final state [3]. \(F_Q\) is the natural objective for control. With \(N\) independent probes, the error scales as \(1/\sqrt N\), the standard quantum limit. Entangled probes can reach the Heisenberg scaling \(1/N\) in the ideal case [2]. Collective states of atomic ensembles, such as spin-squeezed states, are the best-developed route to this enhancement [4]. Preparing them is a State preparation & transfer problem.

Shaping the evolution

  • Filtering the signal. Pulse sequences make a sensor respond to fields at selected frequencies while averaging out noise. The same sequences that protect a qubit act as frequency filters (Filter functions & noise-tailored control [5]). This turns spins into lock-in detectors of oscillating fields: nanoscale magnetometers based on NV centres in diamond [6], sensors of individual weakly coupled nuclear spins [7], and noise spectrometers [8]. See Dynamical decoupling.
  • Control raises the bound. Usually the dynamics is taken as fixed and only the probe state and measurement are optimized. Adding controls during the evolution raises the attainable precision, beyond the limits the coherence time imposes on standard schemes [9]. Optimal feedback controls restore the expectation that a longer sensing time gives better precision [10]. For time-dependent Hamiltonians, the optimal control is generally adaptive. For a qubit in a rotating field, it gives a \(T^4\) scaling of the Fisher information in the rotation frequency, beyond the usual \(T^2\) [11].
  • Optimal control built for sensing. Standard gate or state fidelities are the wrong cost here. A sensing cost function accounts for both the unknown field and the noise. Optimized with it, controls on an NV centre improved the sensitivity to a broad set of time-varying fields [12]. Optimal control for NV-centre sensing has been reviewed in [13]. See Spins — NMR, ESR & colour centres.

Adaptive measurements

Choosing each measurement based on earlier outcomes is feedback control of the estimation. Adaptive phase measurements of optical modes beat non-adaptive ones [14]. With adaptive, multi-pass protocols, phase estimation reached Heisenberg-limited scaling without entangled states [15]. See Measurement-based feedback.

Noise and the Heisenberg limit

  • Decoherence caps the gain. For generic noise, such as dephasing, depolarization or loss, the asymptotic quantum advantage reduces to a constant factor rather than a change of scaling [16, 17]. In the presence of noise, entangled strategies and ancillas do help, unlike in the noiseless case [18].
  • Error correction can restore it. If the noise and the signal act differently, error correction can suppress the noise without erasing the signal and restore Heisenberg scaling [19, 20]. For Markovian noise there is a necessary and sufficient condition for this, and the optimal code follows from a semidefinite program [21]. See Control for quantum error correction.

Limitations

  • The quantum Cramér–Rao bound is asymptotic in the number of repetitions. Few-shot and prior-dependent problems need Bayesian figures of merit.
  • Control pulses have their own errors and bandwidth limits, which add noise and distort the filter.
  • Optimizing for a signal requires a model of it, and a sensor tuned for one signal can be worse for others.