Adaptive quantum algorithms are a relatively new class of quantum algorithms that utilize mid-circuit measurements and feed-forward operations conditioned on the results. [1, 2]. Such algorithms employ a closed-loop control strategy that is naturally robust to noise because deviations from the intended evolution can be detected and remedied during the execution of the algorithm, in the same way that Google Maps knows when you make a wrong turn and recalculates the route accordingly. Adaptive algorithms — really all quantum algorithms — are typically constructed and thought about in a gate-based setting. A “gate” is a discrete operation that instantaneously performs a transformation on the state of the system, analogous to Boolean logic gates such as AND and XOR used for classical digital computation. However, this digital gate-based paradigm for designing quantum algorithms is only an approximation to the underlying analog, continuous-time dynamics that actually occur in the hardware.
It is an extremely detailed and difficult computational problem to design analog control pulses that mimic a particular gate with high fidelity, but we can avoid this step by instead working natively in the analog setting where weak measurements are performed continuously and experimental control pulses are determined by the weak measurement output. Such dynamics are described by stochastic differential equations, since the state of the system is randomly collapsed by measurement backaction during each infinitesimal timestep. Any attempts to control such dynamics in a closed-loop fashion must contend with this randomness and use only the available information. By applying the Lyapunov control framework in designing these algorithms, one can analytically derive intuitive rules for the controllable parameters that ensure that the system asymptotically approaches the target state. It is most natural to think about utilizing this framework for state preparation, and we have applied it to ground state preparation for arbitrary qubit Hamiltonians as well as preparation of topological edge states of bosonic atoms hopping in an optical lattice.
[1] Y. Dong, D. An, and M. Y. Niu, Feedforward Quantum Singular Value Transformation, arXiv:2408.07803.
[2] M. Iqbal et al., Topological order from measurements and feed-forward on a trapped ion quantum computer, Commun Phys 7, 205 (2024).