Quantum Signal Processing and optimal Hamiltonian simulation using Rydberg atoms

Quantum algorithms promise an immense improvement to our current information processing capabilities by utilizing the interference phenomena in an exponentially large Hilbert space. However, the large size of the Hilbert space also poses a crucial challenge to the experimentalists, who strive to design control sequences that navigate this space using only semiclassical fields. A set of novel frameworks consisting of Linear Combination of Unitaries (LCU) and Quantum Signal Processing (QSP) provide effective solutions to this control problem by constructing ever more complicated operators starting from simply implemented multi-qubit Paulis. Here, we introduce an efficient and scalable toolbox for realizing these solutions on the Rydberg atom platform. 

The performance of our proposal relies heavily on the efficient realization of controlled unitaries with multiple control and target atoms. To highlight the advantages of the blockade-based gates, we show that the time complexity of the blockade-based implementation of such controlled unitaries exhibits exponentially better scaling with respect to the number of control atoms than the best-known implementation that uses arbitrary time-dependent and infinite-range two-qubit interactions. Moreover, we provide protocols for realizing the gates of our toolbox in a modular and distributed manner, as well as a parallelization scheme that reduces the circuit depth for signal operators that can be decomposed into geometrically local Pauli operators. To showcase our approach, we construct an explicit blueprint to implement a provably optimal Quantum Hamiltonian simulation algorithm by Haah et al. on the Rydberg atom platform. We demonstrate that the implementation overhead is an order of magnitude smaller than that of previous approaches. Our results reveal the unique advantages of the Rydberg atom platform for realizing a vast variety of quantum algorithms.