用 Daubechies Wavelet 分子軌域在量子化學中進行量子計算

Abstract

本篇論文的目的是探索研究使用 Daubechies wavelet 基組在量子化學中來進行量子計算。量子化學的量子計算取決於分子哈密頓算符在一組基底上用分子軌道來表示。常見的編碼方法將一個分子軌道編碼為一個量子位元，因此在模擬分子系統時所需的量子位元數量對應於基組中使用分子軌道的數量。在 noisy intermediate-scale quantum (NISQ) 時代，僅有有限的量子資源可用。目前，使用小基組的大多數量子計算所預測分子性質的準確度仍遠遠不及其相應的實驗數據 （未達到化學準確度）。為了獲得準確的結果， 需要使用很大的建構於傳統高斯基底函數之分子軌道基組，這也對應到大量量子位元的使用。因此，需要量子計算方法來克服這一有限量子資源的挑戰，但仍能實現與實驗數據有相當的準確結果。在這工作中，我們提出一個有前景的節省量子位元的量子計算方法，並進行全面性的研究藉由通過對大量中性閉殼雙原子的簡諧振動頻率進行基準測試，其結果與實驗數據非常一致。為此，我們使用從密度泛函理論導出的分子軌道以考慮電子相關性來建構準確的哈密頓算符，且在 Daubechies wavelet基組中展開以在實空間網格點上精確表示，並進一步選擇了一個最佳緊緻的活行空間，以便只需要少量的量子位元即可得到準確的結果。為了驗證這種方法的有效行，我們首先將選定的分子軌道生成之哈密頓算符轉換為量子位元哈密頓算符，然後使用精確對角化方法計算結果，其被視為量子計算可達到的最佳解，以與實驗數據進行比較。此外，在使用構建的量子位元哈密頓算符進行 variational quantum eigensolver 演算法計算，我們展示了 chemistry-inspired UCCSD ansatz 的變分量子電路可以達到與精確對角化方法相同的準確度，除了那些 Mayer 鍵級數大於2的系統。對於這些系統，我們進一步展示了啟發式的 hardware-efficient RealAmplitudes ansatz，即使電路深度明顯較短，能夠有一個相對於UCCSD ansatz顯著的改善，證實了在NISQ時代下可以準確計算簡諧振動頻率。

The aim of this thesis is to explore the use of the Daubechies wavelet basis set in quantum computation for quantum chemistry. Quantum computation of quantum chemistry depends on a representation of the molecular Hamiltonian by the molecular orbitals (MOs) in a basis set. Common encoding methods encode a MO into a qubit, and thus the number of qubits needed to simulate a molecular system corresponds to the number of MOs used in a basis set. During the noisy intermediate-scale quantum (NISQ) era, only limited quantum resources are available. Nowadays, the accuracy of the molecular properties predicted by most of the quantum computations using small basis sets is still far off (not within chemical accuracy) compared to their corresponding experimental data. To yield accurate results, a large MO basis set using the traditional Gaussian basis functions is required, consequently corresponding to the use of a large number of qubits. Therefore, quantum computational approaches to overcome this challenge of limited quantum resources but can still achieve accurate results comparable to the experimental data are desirable. In this thesis, we propose a promising qubit-efficient quantum computational approach and present a comprehensive investigation by benchmarking quantum computation of the harmonic vibrational frequencies of a large set of neutral closed-shell diatomic molecules with results in great agreement with their experimental data. To this end, we construct the accurate Hamiltonian using molecular orbitals, derived from density functional theory to account for the electron correlation and expanded in the Daubechies wavelet basis set to allow an accurate representation in real space grid points, where an optimized compact active space is further selected so that only a reduced small number of qubits is sufficient to yield an accurate result. To justify the approach, we benchmark the performance of the Hamiltonians spanned by the selected molecular orbitals by first transforming the molecular Hamiltonians into qubit Hamiltonians and then using the exact diagonalization method to calculate the results, regarded as the best results achievable by quantum computation to compare to the experimental data. Furthermore, using the variational quantum eigensolver algorithm with the constructed qubit Hamiltonians, we show that the variational quantum circuit with the chemistry-inspired UCCSD ansatz can achieve the same accuracy as the exact diagonalization method except for systems whose Mayer bond order indices are larger than 2. For those systems, we then demonstrate that the heuristic hardware-efficient RealAmplitudes ansatz, even with a substantially shorter circuit depth, can provide a significant improvement over the UCCSD ansatz, verifying that the harmonic vibrational frequencies could be calculated accurately by quantum computation in the NISQ era.