Constant Depth Bucket Brigade Quantum RAM Circuits Without Introducing Ancillae

Bucket brigade quantum RAM (QRAM) circuits were proposed for their advantageous addressing of the memory. Another quality of these circuits is that queries, once the addresses are determined, can be parallelised. State-of-the-art error-corrected formulation of these circuits, however, had to be decomposed into the Clifford+T gate set, and the initial parallelism was lost in the process. By using advantageous Toffoli gate decompositions, and without introducing any additional ancilla qubits, we construct bucket brigade QRAM circuits with a constant depth. This depth is reduced from $\mathcal{O}(2^q)$ to $\mathcal{O}(q)$ in the worst case when the QRAM is queried for any of its $2^q$ memory cells being used. Incidentally, the presented construction has a T-count more than half smaller, compared to the existing Clifford+T formulations. The construction shows that, if quantum hardware would not be a scarce resource, exponential query speed ups are possible compared to state-of-the art quantum read-only memory (QROM) designs.