Local Variational Quantum Compilation of Large-Scale Hamiltonian Dynamics

The implementation of time-evolution operators on quantum circuits is important for quantum simulation. However, the standard method, Trotterization, requires a huge number of gates to achieve desirable accuracy. Here, we propose a local variational quantum compilation (LVQC) algorithm, which allows us to accurately and efficiently compile time-evolution operators on a large-scale quantum system by optimization with smaller-size quantum systems. LVQC utilizes a subsystem cost function, which approximates the fidelity of the whole circuit, defined for each subsystem that is as large as the approximate causal cones generated by the Lieb-Robinson (LR) bound. We rigorously derive its scaling property with respect to the subsystem size and show that the optimization conducted on the subsystem size leads to the compilation of whole-system time-evolution operators. As a result, LVQC runs with limited-size quantum computers or classical simulators that can handle such smaller quantum systems. For instance, finite-ranged and short-ranged interacting L-size systems can be compiled with O(L⁰)- or O(logL)-size quantum systems depending on the observables of interest. Furthermore, since this formalism relies only on the LR bound, it can efficiently construct time-evolution operators of various systems in generic dimensions involving finite-, short-, and long-ranged interactions. We also numerically demonstrate the LVQC algorithm for one-dimensional systems. Through the employment of classical simulation by time-evolving block decimation, we succeed in compressing the depth of the time-evolution operators up to 40 qubits by the compilation for 20 qubits. LVQC not only provides classical protocols for designing large-scale quantum circuits but also sheds light on applications of intermediate-scale quantum devices in implementing algorithms in larger-scale quantum devices.