  Simulation of quantum many-body systems is a promising application of quantum
computers. However, implementing the time-evolution operator as a quantum
circuit efficiently on near-term devices with limited resources is challenging.
Standard approaches like Trotterization often require deep circuits, making
them impractical. This paper proposes a hybrid quantum-classical algorithm
called Local Subspace Variational Quantum Compilation (LSVQC) for compiling the
time-evolution operator. The LSVQC uses variational optimization to reproduce
the action of the target time-evolution operator within a physically reasonable
subspace. Optimization is performed on small local subsystems based on the
Lieb-Robinson bound, allowing for cost function evaluation using small-scale
quantum devices or classical computers. Numerical simulations on a spin-lattice
model and an $\mathit{\text{ab initio}}$ effective model of strongly correlated
material Sr$_2$CuO$_3$ demonstrate the algorithm's effectiveness. It is shown
that the LSVQC achieves a 95% reduction in circuit depth compared to
Trotterization while maintaining accuracy. The subspace restriction also
reduces resource requirements and improves accuracy. Furthermore, we estimate
the gate count needed to execute the quantum simulations using the LSVQC on
near-term quantum computing architectures in the noisy intermediate-scale or
early fault-tolerant quantum computing era. Our estimation suggests that the
acceptable physical gate error rate for the LSVQC can be significantly larger
than for Trotterization.
