Fig. 1: Quantum-frequency discrimination.

a The sensor state is tailored to depend on each frequency so that the sensor is driven in orthogonal directions. Readout of the state provides a “Yes/No” answer. The task is to correctly identify the frequency with highest fidelity in the shortest time. b Phase accumulation for frequency discrimination. For each Hamiltonian \({\cal{H}}_i\left( t \right) = B\;{\mathrm{sin}}\left( {\omega _i{\mathrm{t}} + \theta } \right)\sigma _{\mathrm{Z}} = {\mathrm{H}}_i\left( t \right)\sigma _{\mathrm{Z}}\), the sensor accumulates a different phase. Left: without control, the phase difference \(\varphi _{\Delta}\) oscillates and increases only slowly. Right: optimal control in this scenario implies applying π-pulses whenever the sign of H₁–H₂ changes, leading to a monotone phase increase with t². c Geometrical picture of the optimal protocol. The states of the sensor can be thought of as two runners, where the aim is to maximize the gap between them, which is equivalent to the angle between the states. The speed of each runner is proportional to \({\mathrm{H}}_i\left( t \right)\). As soon as the initially slower runner becomes faster, we change their direction of motion to ensure an increasing gap. Note that the sketch just demonstrates the idea and the runners actually move on circular orbits. d Error probability as a function of time for three different strategies: optimal control and a single measurement (solid blue), this strategy is optimal and sets the fundamental error limit (see Eqs. 3 and 4). Correlated measurements (dashed red), in this illustration measurement is applied every \(\frac{{2\pi }}{{\omega _1 + \omega _2}}\). No control (dotted green). More details on this comparison are found in Supplementary Note 1.