Hamiltonian simulation is a domain where quantum computers have the potential to outperform their classical counterparts due to their inherent quantum behavior. One of the main challenges of such quantum algorithms is up-scaling the system size, which is necessary to achieve meaningful quantum advantage. In this work, we present an approach to improve the scalability of eigenspace filtering for the ground state preparation of a given Hamiltonian. Our method aims to tackle limitations introduced by a small spectral gap and high degeneracy of low energy states. It is based on an adaptive sequence of eigenspace filtering through Quantum Eigenvalue Transformation of Unitary Matrices (QETU) followed by spectrum profiling. By combining our proposed algorithm with state-of-the-art phase estimation methods, we achieved good approximations for the ground state energy with local, two-qubit gate depolarizing probability up to $10^{-4}$. To demonstrate the key results in this work, we ran simulations with the transverse-field Ising Model on classical computers using Qiskit. We compare the performance of our approach with the static implementation of QETU and show that we can consistently achieve three to four orders of magnitude improvement in the absolute error rate.

翻译：哈密顿量模拟是量子计算机因其固有的量子行为而有望超越经典计算机的领域之一。此类量子算法的主要挑战在于扩大系统规模，这是实现有意义的量子优势所必需的。在本工作中，我们提出了一种改进给定哈密顿量基态制备中本征空间滤波可扩展性的方法。我们的方法旨在应对由小能隙和低能态高简并性带来的限制。该方法基于通过酉矩阵量子本征值变换（QETU）的自适应本征空间滤波序列，并结合谱分析。通过将我们提出的算法与最先进的相位估计方法相结合，我们在局域双量子比特门去极化概率高达$10^{-4}$的情况下，实现了对基态能量的良好近似。为展示本工作的关键结果，我们使用Qiskit在经典计算机上对横场伊辛模型进行了模拟。我们将所提方法与静态QETU实现的性能进行了比较，结果表明我们能够在绝对误差率上持续实现三到四个数量级的改进。