Hamiltonian Dynamics Simulations Research Articles

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.

Low-cost quantum circuits for classically intractable instances of the Hamiltonian dynamics simulation problem

We develop circuit implementations for digital-level quantum Hamiltonian dynamics simulation algorithms suitable for implementation on a reconfigurable quantum computer, such as trapped ions. Our focus is on the codesign of a problem, its solution, and quantum hardware capable of executing the solution at the minimal cost expressed in terms of the quantum computing resources used, while demonstrating the solution of an instance of a scientifically interesting problem that is intractable classically. The choice for Hamiltonian dynamics simulation is due to the combination of its usefulness in the study of equilibrium in closed quantum mechanical systems, a low cost in the implementation by quantum algorithms, and the difficulty of classical simulation. By targeting a specific type of quantum computer and tailoring the problem instance and solution to suit physical constraints imposed by the hardware, we are able to reduce the resource counts by a factor of 10 in a physical-level implementation and a factor of 30–60 in a fault-tolerant implementation over state-of-the-art.

Energy corrections in Hamiltonian dynamics simulations

We study two energy-correction algorithms for numerical simulations of Hamiltonian systems. The first algorithm, which corrects a trajectory after-the-fact, can significantly improve symplectic integrators by achieving a higher order of energy accuracy. The second algorithm, which corrects a trajectory along-the-way, can drastically improve the conventional Runge-Kutta schemes both in energy conservation and trajectory accuracy. We present extended numerical simulation results for a nonlinear oscillator and the well-known Hénon-Heiles model that exhibits chaotic dynamics.

The NP-Completeness of Edge-Coloring

We show that it is NP-complete to determine the chromatic index of an arbitrary graph. The problem remains NP-complete even for cubic graphs.