  Environmental fluctuations degrade the performance of solid-state qubits but
can in principle be mitigated by real-time Hamiltonian estimation down to time
scales set by the estimation efficiency. We implement a physics-informed and an
adaptive Bayesian estimation strategy and apply them in real time to a
semiconductor spin qubit. The physics-informed strategy propagates a
probability distribution inside the quantum controller according to the
Fokker-Planck equation, appropriate for describing the effects of nuclear spin
diffusion in gallium-arsenide. Evaluating and narrowing the anticipated
distribution by a predetermined qubit probe sequence enables improved dynamical
tracking of the uncontrolled magnetic field gradient within the singlet-triplet
qubit. The adaptive strategy replaces the probe sequence by a small number of
qubit probe cycles, with each probe time conditioned on the previous
measurement outcomes, thereby further increasing the estimation efficiency. The
combined real-time estimation strategy efficiently tracks low-frequency nuclear
spin fluctuations in solid-state qubits, and can be applied to other qubit
platforms by tailoring the appropriate update equation to capture their
distinct noise sources.
