  Fault-tolerant quantum computation enables reliable quantum computation but
incurs a significant overhead from both time and resource perspectives. To
reduce computation time, Austin G. Fowler proposed time-optimal quantum
computation by constructing a quantum circuit for a fault-tolerant $T$ gate
without probabilistic $S$ gate correction. In this work, we introduce a
resource-compact quantum circuit that significantly reduces resource
requirements by more than 60% for a fault-tolerant $T$ gate without
probabilistic $S$ gate correction. Consequently, we present a quantum circuit
that minimizes resource utilization for time-optimal quantum computation,
demonstrating efficient time-optimal quantum computation. Additionally, we
describe an efficient form involving initialization, CNOTs, and measurements,
laying the foundation for the development of an efficient compiler for
fault-tolerant quantum computation.
