Linear system solvers are widely used in scientific computing, with the primary goal of solving linear system problems. Classical iterative algorithms typically rely on the conjugate gradient method. The rise of quantum computing has spurred interest in quantum linear system problems (QLSP), particularly following the introduction of the HHL algorithm by Harrow et al. in 2009, which demonstrated the potential for exponential speedup compared to classical algorithms. However, the performance of the HHL algorithm is constrained by its dependence on the square of the condition number. To address this limitation, alternative approaches based on adiabatic quantum computing (AQC) have been proposed, which exhibit complexity scaling linearly with the condition number. AQC solves QLSP by smoothly varying the parameters of the Hamiltonian. However, this method suffers from high Hamiltonian simulation complexity. In response, this work designs new Hamiltonians and proposes a quantum discrete adiabatic linear solver based on block encoding and eigenvalue separation techniques (BEES-QDALS). This approach bypasses Hamiltonian simulation through a first-order approximation and leverages block encoding to achieve equivalent non-unitary operations on qubits. By comparing the fidelity of the original algorithm and BEES-QDALS when solving QLSP with a fixed number of steps, and the number of steps required to reach a target fidelity, it is found that BEES-QDALS significantly outperforms the original algorithm. Specifically, BEES-QDALS achieves higher fidelity with the same number of steps and requires fewer steps to reach the same target fidelity.

翻译：线性系统求解器在科学计算中应用广泛，其主要目标是解决线性系统问题。经典迭代算法通常依赖于共轭梯度法。量子计算的兴起激发了人们对量子线性系统问题的兴趣，特别是在Harrow等人于2009年提出HHL算法之后，该算法展现了相较于经典算法实现指数级加速的潜力。然而，HHL算法的性能受限于其对条件数平方的依赖。为克服这一局限，基于绝热量子计算的替代方法被提出，其复杂度随条件数线性增长。绝热量子计算通过平滑改变哈密顿量参数来求解量子线性系统问题，但该方法存在哈密顿量模拟复杂度高的问题。为此，本研究设计了新的哈密顿量，并提出一种基于块编码与特征值分离技术的量子离散绝热线性求解器。该方法通过一阶近似绕过了哈密顿量模拟，并利用块编码在量子比特上实现等效的非酉操作。通过比较原始算法与BEES-QDALS在固定步数下求解量子线性系统问题时的保真度，以及达到目标保真度所需的步数，发现BEES-QDALS显著优于原始算法。具体而言，BEES-QDALS在相同步数下获得更高保真度，且达到相同目标保真度所需步数更少。