Enhancing Scalability of Quantum Eigenvalue Transformation of Unitary Matrices for Ground State Preparation through Adaptive Finer Filtering

Hamiltonian simulation is a domain where quantum computers have the potential to outperform their classical counterparts. One of the main challenges of such quantum algorithms is increasing 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) combined with 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>10</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:math>. To demonstrate the key results in this work, we ran simulations with the transverse-field Ising Model on classical computers using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext mathvariant="monospace">Qiskit</mml:mtext></mml:mrow></mml:math>. 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.