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The noisy binary linear problem (NBLP) is known as a computationally hardproblem, and therefore, it offers primitives for post-quantum cryptography. Anefficient quantum NBLP algorithm that exhibits a polynomial quantum sample andtime complexities has recently been proposed. However, the algorithm requires alarge number of samples to be loaded in a highly entangled state and it isunclear whether such a precondition on the quantum speedup can be obtainedefficiently. Here, we present a complete analysis of the quantum solvability ofthe NBLP by considering the entire algorithm process, namely from thepreparation of the quantum sample to the main computation. By assuming that thealgorithm runs on "fault-tolerant" quantum circuitry, we introduce a reasonablemeasure of the computational time cost. The measure is defined in terms of theoverall number of T gate layers, referred to as T-depth complexity. We showthat the cost of solving the NBLP can be polynomial in the problem size, at theexpense of an exponentially increasing logical qubits.

Polynomial T-depth Quantum Solvability of Noisy Binary Linear Problem: From Quantum-Sample Preparation to Main Computation