Quantum subroutine problem and the robustness of quantum complexity classes

This paper positively solves the quantum subroutine problem for fully quantum oracles. The quantum subroutine problem asks whether a quantum computer with an efficiently computable oracle can be efficiently simulated by a non-oracle quantum computer. We extends the earlier results obtained by Bennett, Bernstein, Brassard, and Vazirani, and by Aharonov, Kitaev, and Nisan to the case where the oracle evaluates a unitary operator and the computer is allowed to be in the superposition of a query state and a non-query state during computation. We also prove the robustness of EQP, BQP, and ZQP under the above general formulation, extending the earlier results on the robustness of BQP shown by Bennett et al.