Classical computers struggle to simulate many-body quantum systems like molecules due to the factorial scaling of computational cost with the number of electrons. Quantum computers, leveraging superposition, entanglement, and quantum tunneling, offer a potential solution by reducing this scaling to polynomial. Two approaches exist: gate-based and adiabatic quantum computing. While gate-based computers are available, true adiabatic quantum computers are still under development. Current quantum annealers, like the D-Wave 2000Q, are naturally suited to adiabatic quantum computation but lack the non-stoquastic Hamiltonian technology for optimal scaling. Most near-term quantum chemistry efforts focus on gate-based computers (VQE and FQE), with limited work on quantum annealers. Existing methods for quantum annealers often involve qubit copying, classical optimization of many variables, or added constraints, limiting their efficiency. This paper introduces a generalized Quantum Annealer Eigensolver (QAE) to address these limitations, enabling the calculation of ground and excited electronic states for small molecules.

Previous attempts to solve electronic structure problems on D-Wave quantum annealers have been limited to small molecules (H₂, LiH) and involved either extensive qubit copying or the classical optimization of numerous parameters. These methods typically employed Jordan-Wigner or Bravyi-Kitaev transformations to map the second-quantized electronic Hamiltonian to a qubit Hamiltonian, often requiring extra constraints and sacrificing qubits to handle high-order terms. This paper builds upon previous work on the QAE, initially applied to vibrational quantum problems, and extends it to electronic structure calculations. Unlike previous methods, the QAE is Hamiltonian and basis-agnostic, uses an efficient power-of-two wave function encoding, and is hardware-dominant, requiring only one classical parameter.

The proposed method consists of two steps (Figure 1a). First, an electronic Hamiltonian matrix is constructed in the basis of Slater determinants using molecular orbitals optimized via Hartree-Fock methods. Full Configuration Interaction (FCI) and Complete Active Space Self-Consistent Field (CASSCF) matrices are generated using a modified Psi4 code. Second, the QAE algorithm diagonalizes this matrix to obtain eigenvalues (electronic energies) and eigenvectors (electronic wavefunctions). The QAE maps the eigenvalue problem to a Quadratic Unconstrained Binary Optimization (QUBO) problem solvable by D-Wave annealers. This mapping minimizes the Rayleigh-Ritz quotient (RRQ), with a classically optimized Lagrange multiplier (λ) enforcing vector normalization. The QUBO problem is then solved iteratively, generating a new guess vector for each iteration until a stopping condition is met (Figure 1b). Excited states are computed using eigenvalue shifts based on Brauer's theorem. The original QAE's normalization penalty (λ) scan is automated, improving speed and accuracy. The QAE utilizes the qbsolv software, which partitions large QUBO problems into smaller subQUBOs that can be solved classically or on the D-Wave annealer. These subQUBOs are minimized individually, and then refined by qbsolv's post-processing system, offering both classical and hardware modes of operation. The accuracy of the eigenvector representation is controlled by the number of qubits (K) used per eigenvector element, in a power-of-two scheme. The paper uses various basis sets and methods (FCI, CASSCF) for several small molecules (H₂, HF, H₂O, CH₂²⁺, BeH₂, H₃⁺, BH₃) and investigates their performance with the QAE.

The QAE successfully computes ground and excited state energies and wavefunctions for various small molecules. For small molecules and basis sets, chemical accuracy (within 1 kcal/mol) is achieved. The method's accuracy decreases with larger molecules and basis sets, reaching up to 7 kcal/mol error in this study. The difference between the classical and hardware modes is generally less than 1 kcal/mol, except for H₂O. Calculations for the H₂ molecule demonstrate basis set convergence, showing that larger basis sets generally lead to more accurate ground-state energies (Figure 2a). The potential energy curve for H₃⁺ was also computed, with similar accuracy trends as the ground and excited state calculations (Figure 2b). A convergence study on the number of qubits (K) per eigenvector element reveals that 10 qubits are sufficient to reach a plateau in accuracy for the molecules studied (Figure 2c). The analysis of excited states for H₂O shows increasing errors for higher excited states, possibly due to accumulated noise from spectral transformations and limitations in the QUBO solver. Overall, the results confirm the viability of the QAE approach for electronic structure calculations but highlight the need for improvements, particularly regarding the accuracy of the underlying QUBO solver (qbsolv).

The study demonstrates the feasibility of solving electronic structure problems using a quantum annealer via the QAE method. However, the observed errors, exceeding the target chemical accuracy, indicate room for improvement. The primary source of error is identified as the qbsolv software, which introduces classical noise in the QUBO solutions. Statistical analysis reveals a larger energy spread for larger matrices, suggesting that the accuracy of qbsolv diminishes with problem size. Furthermore, the hardware mode is significantly slower than the classical mode due to the overhead of minor embedding and the VFYC post-processing system. Strategies to mitigate these issues are suggested, including running qbsolv multiple times and optimizing its parameters, potentially integrating an exact QUBO solver or utilizing D-Wave's Hybrid Solver Service (HSS) as a more accurate alternative to qbsolv. The impact of hardware limitations and the role of quantum effects, such as tunneling, remain areas for future investigation.

The Quantum Annealer Eigensolver (QAE) provides a viable method for solving electronic structure problems on a quantum annealer, particularly for smaller molecules and basis sets, achieving chemical accuracy. While limitations concerning accuracy and speed remain, the QAE's flexible design, efficient qubit usage, and hardware-dominant nature make it a promising approach for future quantum computing hardware. The current limitations, primarily stemming from the QUBO solver (qbsolv), highlight the need for improved algorithms and quantum hardware. Future work should focus on enhancing the accuracy of the underlying QUBO solver, addressing the speed limitations of the hardware mode, and investigating the influence of quantum features in the computation.

The main limitation is the accuracy of the QUBO solver qbsolv, which introduces noise and limits the overall accuracy of the QAE, especially for larger systems. The hardware mode is substantially slower than the classical mode due to minor embedding and post-processing overhead. The current implementation is limited to relatively small molecules and basis sets, due to the scaling limitations of the QUBO solver and available quantum hardware. The study focuses on a limited set of molecules and methods, hence further investigation on a broader range of systems is required to fully evaluate the approach’s generality.