Measurement-based feedback steers a quantum system using information gained while it evolves. A measurement record is processed in real time, often into an estimate of the quantum state, and the controls are adjusted accordingly. This lets control react to stochastic events: random measurement back-action, noise, errors. Open-loop pulses cannot do that. It is the quantum analogue of classical feedback control, with one essential difference: measuring disturbs the system.

Continuous measurement and quantum filtering

Under continuous weak measurement — for example homodyne detection of a field leaking from a cavity — the state conditioned on the measurement record evolves stochastically. In one common form,

$$d\rho = -i[H,\rho]\,dt + \mathcal D[c]\rho\,dt + \sqrt{\eta}\,\mathcal H[c]\rho\,dW,$$

where \(c\) is the measurement operator, \(\mathcal D[c]\rho = c\rho c^\dagger - \tfrac12\{c^\dagger c, \rho\}\) the dissipator, \(\mathcal H[c]\rho = c\rho + \rho c^\dagger - \mathrm{Tr}\big[(c + c^\dagger)\rho\big]\rho\) the measurement back-action, \(\eta\) the detection efficiency and \(dW\) the innovation of the measured current [1]. The mathematically rigorous form of this state estimation is quantum filtering [2, 3].

Feedback strategies

  • Markovian (direct) feedback. The measured current is fed back directly as a control Hamiltonian, without explicit state estimation [4].
  • Estimation-based (Bayesian) feedback. The full conditional state is tracked and the control is a function of it. For linear systems this is closely analogous to classical linear-quadratic-Gaussian (LQG) control [5]. Global stabilization of states using filters and geometric control is analysed in [6]; see Lyapunov, tracking & model-predictive control.
  • Measurement as the control. Slowly varying the measured observable drags the state along its eigenstate through the quantum Zeno effect. This has been demonstrated with a superconducting qubit [7] and optimized as a control protocol [8].
  • Learned policies. Reinforcement learning discovers feedback strategies from data, including error correction and real-time feedback on FPGAs [9, 10]. Feedback-GRAPE optimizes measurement-conditioned strategies by gradients [11]. See Reinforcement & machine learning for control.

Landmark experiments

  • Cavity QED. Real-time feedback prepared and stabilized photon-number (Fock) states of a microwave cavity [12].
  • Superconducting qubits. Continuous weak measurement and feedback stabilized Rabi oscillations [13]. Parity measurement plus feedback created deterministic entanglement between two qubits [14]. Real-time error correction extended the lifetime of a bosonic qubit [15]; see Control for quantum error correction.
  • Mechanical motion. Measurement-based control has cooled and controlled the motion of mechanical resonators [16]. Real-time optimal control — Kalman filtering plus optimal feedback — has controlled the quantum trajectory of a levitated nanoparticle at room temperature [17].

Measurement versus coherent feedback

Instead of measuring, the output of a system can be processed by another quantum system and fed back coherently, avoiding measurement back-action [18]. When each approach is better is a question of linear systems theory; see Linear quantum systems & coherent feedback. Autonomous stabilization by engineered dissipation is the limit with no processing at all; see Reservoir engineering & dissipative preparation.

Practical constraints

  • Latency. The loop must be much faster than the system’s dynamics and decoherence, which drives the use of FPGAs and dedicated control hardware; see Control electronics & software stacks.
  • Efficiency. Lost measurement information (finite \(\eta\)) adds noise that feedback cannot remove.
  • Model accuracy. State estimation requires a good model of the measurement and the dynamics; see System identification for control.

Uses include state stabilization, cooling, error correction and adaptive sensing (Control for sensing & metrology).