  Simulating the dynamic evolutions of physical and molecular systems in a
quantum computer is of fundamental interest in many applications. Its
implementation requires efficient quantum simulation algorithms. The
Lie-Trotter-Suzuki approximation algorithm, also well known as the
Trotterization, is a basic algorithm in quantum dynamic simulation. A
multi-product algorithm that is a linear combination of multiple
Trotterizations has been proposed to improve the approximation accuracy.
Implementing such multi-product Trotterization in quantum computers however
remains experimentally challenging and its success probability is limited.
Here, we modify the multi-product Trotterization and combine it with the
oblivious amplitude amplification to simultaneously reach a high simulation
precision and high success probability. We experimentally implement the
modified multi-product algorithm in an integrated-photonics programmable
quantum simulator in silicon, which allows the initialization, manipulation and
measurement of four-qubit states and a sequence of linearly combined
controlled-unitary gates, to emulate the dynamics of a coupled electron and
nuclear spins system. Theoretical and experimental results are in good
agreement, and they both show the modified multi-product algorithm can simulate
Hamiltonian dynamics with a higher precision than conventional Trotterizations
and a nearly deterministic success probability. We certificate the
multi-product algorithm in a small-scale quantum simulator based on linear
combinations of operations, and this work promises the practical
implementations of quantum dynamics simulations.
