Feedback‐Assisted Quantum Search by Continuous‐Time Quantum Walks

The quantum search of a target node on a cycle graph by means of a quantum walk assisted by continuous measurement and feedback are addressed. Unlike previous spatial search approaches, where the oracle is described as a projector on the target state, a dynamical oracle implemented through a feedback Hamiltonian is considered. The idea is based on continuously monitoring the position of the quantum walker on the graph and then applying a unitary feedback operation based on the information obtained from measurement. The feedback changes the couplings between the nodes and it is optimized at each time via a numerical procedure. The stochastic trajectories describing the evolution for graphs of dimensions up to N=15$N=15$are numerically simulated, and the performance of the protocol is quantified via the average fidelity between the state of the walker and the target node. Different constraints on the control strategy are discussed. For unbounded controls, the protocol is able to quickly localize the walker on the target node. How the performance is lowered by posing an upper bound on the control couplings is then discussed. Finally, it is shown how a digital feedback protocol seems in general as efficient as the continuous bounded one. Continuous measurements and feedback are used in a quantum walk spatial search protocol on a cycle graph. The oracle is directly implemented in the feedback mechanism, which is able to change the couplings between the nodes. Different constraints on the feedback operations are considered and numerical results are reported showing advantages with respect to the standard spatial search protocol.