  We present a novel, computationally efficient approach to accelerate quantum
optimal control calculations of large multi-qubit systems used in a variety of
quantum computing applications. By leveraging the intrinsic symmetry of finite
groups, the Hilbert space can be decomposed and the Hamiltonians
block-diagonalized to enable extremely fast quantum optimal control
calculations. Our approach reduces the Hamiltonian size of an $n$-qubit system
from 2^n by 2^n to O(n by n) or O((2^n / n) by (2^n / n)) under Sn or Dn
symmetry, respectively. Most importantly, this approach reduces the
computational runtime of qubit optimal control calculations by orders of
magnitude while maintaining the same accuracy as the conventional method. As
prospective applications, we show that (1) symmetry-protected subspaces can be
potential platforms for quantum error suppression and simulation of other
quantum Hamiltonians, and (2) Lie-Trotter-Suzuki decomposition approaches can
generalize our method to a general variety of multi-qubit systems.
