  Implementing time evolution operators on quantum circuits is important for
quantum simulation. However, the standard way, Trotterization, requires a huge
numbers of gates to achieve desirable accuracy. Here, we propose a local
variational quantum compilation (LVQC) algorithm, which allows to accurately
and efficiently compile a time evolution operators on a large-scale quantum
system by the optimization with smaller-size quantum systems. LVQC utilizes a
subsystem cost function, which approximates the fidelity of the whole circuit,
defined for each subsystem as large as approximate causal cones brought 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 $O(L^0)$- or $O(\log L)$-size quantum systems depending on
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 dimension involving finite-, short-, and long-ranged
interactions. We also numerically demonstrate the LVQC algorithm for
one-dimensional systems. Employing classical simulation by time-evolving block
decimation, we succeed in compressing the depth of a 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 will
shed light on applications of intermediate-scale quantum devices in
implementing algorithms in larger-scale quantum devices.
