Impact of molecular orbital localization on quantum computational resources for Hamiltonian simulation: A benchmark study of hydrogen chain systems

Technical Summary: Impact of Molecular Orbital Localization on Quantum Computational Resources for Hamiltonian Simulation

Problem Statement
While significant research has focused on optimizing quantum simulation algorithms (e.g., Trotterization, qubitization) and compression strategies (e.g., tensor hypercontraction, QCELS), relatively little attention has been paid to how the choice of molecular orbital basis affects quantum computational resources. In conventional quantum chemistry, canonical molecular orbitals (CMOs) derived from Hartree–Fock (HF) theory are delocalized, leading to "all-to-all" couplings in the second-quantized Hamiltonian. This delocalization necessitates a large number of quantum gates for simulation. Conversely, localized molecular orbitals (LMOs) are widely used in classical electronic structure methods to exploit spatial locality, where interactions between distant orbitals are negligible. The central question addressed by this study is whether orbital localization can similarly reduce the complexity and resource requirements of quantum Hamiltonian simulation, specifically for Trotter decomposition-based approaches.

Methodology
The authors conducted a benchmark study using one-dimensional linear hydrogen chain systems ($H_n$) with $n$ ranging from 8 to 100 atoms, at an interatomic distance of 1.0 Å. The study compared two orbital representations:

1. CMOs: Standard Hartree–Fock canonical orbitals.
2. LMOs: Pipek–Mezey localized orbitals, reordered according to their relative spatial positions.

The Hamiltonian was constructed using the Jordan–Wigner (JW) transformation. The authors investigated two distinct Hamiltonian truncation strategies to reduce the number of terms:

• Operator Coefficient-based Truncation ($c_{thre}$): Discarding terms where the absolute value of the coefficient is below a threshold.
• Operator Locality-based Truncation ($k_{max}$): Discarding terms where the operator locality $k$ (the number of non-identity Pauli operators in a Pauli string) exceeds a threshold. For two-electron terms, locality was defined based on the excitation of one $\alpha$ and one $\beta$ electron to avoid spin-bias.

Accuracy was evaluated by calculating the ground-state energy error ($\Delta E$) and fidelity ($F$) against untruncated reference calculations (Full-CI for small chains, DMRG for larger chains). Empirical formulas were fitted to the relationship between truncation thresholds and fidelity to estimate the thresholds required to achieve high fidelity ($F \geq 0.99$). These estimated thresholds were then used to project the quantum gate counts (Clifford, CNOT, and non-Clifford $R_z$ gates) for chains up to $H_{100}$.

Key Contributions and Results

1. Optimal Truncation Strategy per Basis:

   • For CMO-based expansions, coefficient-based truncation ($c_{thre}$) is effective. Even with loose thresholds, the ground-state energy error and fidelity remain close to ideal values because the delocalized nature of CMOs means that high-locality terms are often necessary to describe the wavefunction.
   • For LMO-based expansions, locality-based truncation ($k_{max}$) is superior. Coefficient-based truncation fails immediately for LMOs, causing rapid degradation in accuracy. However, locality-based truncation preserves high fidelity even at low $k_{max}$ values because spatially separated orbitals in the LMO basis have weak correlations.

2. Scaling of Truncation Thresholds:

   • To maintain $F \geq 0.99$, the coefficient threshold for CMOs scales as $c_{thre} \propto n^{-2.07}$.
   • For LMOs, the required locality threshold $k_{max}$ scales polylogarithmically with system size, specifically $k_{max} \propto (\log n)^{1.86}$.

3. Quantum Gate Resource Estimation:

   • CMO Basis: The number of quantum gates required for Hamiltonian simulation grows polynomially with system size, specifically $O(n^4)$ to $O(n^5)$. While truncation reduces the prefactor, it does not alter the polynomial scaling order.
   • LMO Basis: When combined with locality-based truncation, the gate count grows polylogarithmically, specifically $O((\log n)^{12})$ to $O((\log n)^{14})$.
   • CNOT Gates: In the CMO basis, many high-locality terms persist even after truncation, leading to a high CNOT count. In the LMO basis, the truncation restricts the Hamiltonian to low-locality terms, suppressing the growth of CNOT gates. For $n \geq 50$, the LMO-based approach requires fewer CNOT gates than the CMO-based approach, despite the LMO basis initially having more Hamiltonian terms due to broken point-group symmetry.

4. Encoding Comparison:

   • The study also compared Jordan–Wigner (JW) and Bravyi–Kitaev (BK) encodings. While BK reduces the CNOT scaling for CMOs from $O(n^5)$ to $O(n^4 \log n)$, the resulting gate counts remain comparable to the LMO-based approach. Additionally, BK increases the number of single-qubit Clifford gates due to the variable number of Pauli-X/Y terms in the strings.

Significance and Claims
The paper claims that the choice of orbital representation and the corresponding truncation strategy are critical factors in determining the feasibility of large-scale quantum chemical simulations. The authors conclude that for Trotter-based Hamiltonian simulation, utilizing LMOs combined with operator locality-based truncation offers a significant exponential advantage over traditional CMO-based approaches. Specifically, the transition from polynomial to polylogarithmic scaling in gate requirements suggests that LMOs could enable the simulation of much larger molecular systems on future fault-tolerant quantum computers.

The authors note that while the study focuses on 1D hydrogen chains where operator locality directly correlates with spatial distance, the findings provide useful guidelines for basis-design strategies. They suggest that for complex 3D molecules, distance-based truncation might be more appropriate than strict locality-based truncation, but the general principle of exploiting orbital localization remains a promising avenue for reducing computational costs. The study also highlights that state preparation remains a potential bottleneck and that further resource estimation for state preparation is necessary for a complete end-to-end cost assessment.