Faulty towers: recovering a functioning quantum random access memory in the presence of defective routers

Proposals for quantum random access memory (QRAM) generally have a binary-tree structure, and thus require hardware that is exponential in the depth of the QRAM. For solid-state based devices, a fabrication yield that is less than $100\mathrm{\%}$ implies that certain addresses at the bottom of the tree become inaccessible if a router in the unique path to that address is faulty. We discuss how to recover a functioning QRAM in the presence of faulty routers. We present the $\mathtt{\text{IterativeRepair}}$ algorithm, which constructs QRAMs layer by layer until the desired depth is reached. This algorithm utilizes ancilla flag qubits which reroute queries to faulty routers. We present a classical algorithm $\mathtt{\text{FlagQubitMinimization}}$ that attempts to minimize the required number of such ancilla. For a router failure rate of $1\mathrm{\%}$ and a QRAM of depth $n=13$, we expect that on average 430 addresses need repair: we require only 1.5 ancilla flag qubits on average to perform this rerouting.