Programmable silicon-photonic quantum simulator based on a linear combination of unitaries

Simulating the dynamic evolution of physical and molecular systems in a quantum computer is of fundamental interest in many applications. The implementation of dynamics simulation requires efficient quantum algorithms. The Lie-Trotter-Suzuki approximation algorithm, also known as the Trotterization, is basic in Hamiltonian dynamics simulation. A multi-product algorithm that is a linear combination of multiple Trotterizations has been proposed to improve the approximation accuracy. However, implementing such multi-product Trotterization in quantum computers remains challenging due to the requirements of highly controllable and precise quantum entangling operations with high success probability. Here, we report a programmable integrated-photonic quantum simulator based on a linear combination of unitaries, which can be tailored for implementing the linearly combined multiple Trotterizations, and on the simulator we benchmark quantum simulation of Hamiltonian dynamics. We modify the multi-product algorithm by integrating it with oblivious amplitude amplification to simultaneously reach high simulation precision and high success probability. The quantum simulator is devised and fabricated on a large-scale silicon-photonic quantum chip, which allows the initialization, manipulation, and measurement of arbitrary four-qubit states and linearly combined unitary gates. As an example, the quantum simulator is reprogrammed to emulate the dynamics of an electron spin and nuclear spin coupled system. This work promises the practical dynamics simulations of real-world physical and molecular systems in future large-scale quantum computers.