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

TL;DR

This paper analyzes the quantum solvability of the noisy binary linear problem (NBLP), demonstrating that it can be solved with polynomial T-depth complexity, but requires exponentially many logical qubits, highlighting the trade-off in quantum resources.

Contribution

The paper provides a comprehensive analysis of the quantum algorithm for NBLP, focusing on T-depth complexity and resource trade-offs, extending previous work on quantum speedup.

Findings

NBLP can be solved with polynomial T-depth complexity.

The solution requires exponentially increasing logical qubits.

The analysis covers from quantum sample preparation to main computation.

Abstract

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.