  In this paper, we propose a novel quantum compiler optimization, named
relaxed peephole optimization (RPO) for quantum computers. RPO leverages the
single-qubit state information that can be determined statically by the
compiler. We define that a qubit is in a basis state when, at a given point in
time, its state is either in the X-, Y-, or Z-basis. When basis qubits are used
as inputs to quantum gates, there exist opportunities for strength reduction,
which replaces quantum operations with equivalent but less expensive ones.
Compared to the existing peephole optimization for quantum programs, the
difference is that our proposed optimization does not require an identical
unitary matrix, thereby named `relaxed' peephole optimization. We also extend
our approach to optimize the quantum gates when some input qubits are in known
pure states. Both optimizations, namely the Quantum Basis-state Optimization
(QBO) and the Quantum Pure-state Optimization (QPO), are implemented in the
IBM's Qiskit transpiler. Our experimental results show that our proposed
optimization pass is fast and effective. The circuits optimized with our
compiler optimizations obtain up to 18.0% (11.7% on average) fewer CNOT gates
and up to 8.2% (7.1% on average) lower transpilation time than that of the most
aggressive optimization level in the Qiskit compiler. When running on real
quantum computers, the success rates of 3-qubit quantum phase estimation
algorithm improve by 2.30X due to the reduced gate counts.
