Quantum Random Access Stored-Program Machines

Random access machines (RAMs) and random access stored-program machines (RASPs) are models of computing that are closer to the architecture of real-world computers than Turing machines (TMs). They are also convenient in complexity analysis of algorithms. The relationships between RAMs, RASPs and TMs are well-studied. However, a clear relationships between their quantum counterparts are still missing in the literature. We fill in this gap by formally defining the models of quantum random access machines (QRAMs) and quantum random access stored-program machines (QRASPs) and clarifying the relationships between QRAMs, QRASPs and quantum Turing machines (QTMs). In particular, we prove: 1. A $T(n)$-time QRAM (resp. QRASP) can be simulated by an $O(T(n))$-time QRASP (resp. QRAM). 2. A $T(n)$-time QRAM under the logarithmic (resp. constant) cost criterion can be simulated by an $\tilde O(T(n)^4)$-time (resp. $\tilde O(T(n)^8)$-time) QTM. 3. A $T(n)$-time QTM can be simulated within error $\varepsilon > 0$ by an $O(T(n)^2 \operatorname{polylog}(T(n), 1/\varepsilon))$-time QRAM (under both the logarithmic and constant cost criterions). As a corollary, we have: $\textbf{P} \subseteq \textbf{EQRAMP} \subseteq \textbf{EQP} \subseteq \textbf{BQP} = \textbf{BQRAMP}$, where $\textbf{EQRAMP}$ and $\textbf{BQRAMP}$ stand for the sets of problems that can be solved by polynomial-time QRAMs with certainty and bounded-error, respectively.