Quantum Discrete Adiabatic Linear Solver based on Block Encoding and Eigenvalue Separator

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.