  The noisy binary linear problem (NBLP) is known as a computationally hard
problem, and therefore, it offers primitives for post-quantum cryptography. An
efficient quantum NBLP algorithm that exhibits a polynomial quantum sample and
time complexities has recently been proposed. However, the algorithm requires a
large number of samples to be loaded in a highly entangled state and it is
unclear whether such a precondition on the quantum speedup can be obtained
efficiently. Here, we present a complete analysis of the quantum solvability of
the NBLP by considering the entire algorithm process, namely from the
preparation of the quantum sample to the main computation. By assuming that the
algorithm runs on "fault-tolerant" quantum circuitry, we introduce a reasonable
measure of the computational time cost. The measure is defined in terms of the
overall number of T gate layers, referred to as T-depth complexity. We show
that the cost of solving the NBLP can be polynomial in the problem size, at the
expense of an exponentially increasing logical qubits.
