Resource-efficient quantum-selected configuration interaction for molecular properties

Technical Summary: Resource-efficient quantum-selected configuration interaction for molecular properties

Problem Statement
Quantum simulations of molecular electronic structures face significant scalability challenges on Noisy Intermediate-Scale Quantum (NISQ) devices. While Quantum-Selected Configuration Interaction (QSCI) offers a promising hybrid quantum-classical approach by identifying important determinantal basis functions through real-time evolution, its implementation using the full electronic Hamiltonian leads to rapidly increasing circuit complexity. Specifically, Hamiltonian simulation-based QSCI (HSB-QSCI) requires a number of Pauli terms scaling as $O(N^4)$ and two-qubit gates as $O(N^5)$ for an $N$-qubit system under the Jordan-Wigner transformation. This high resource demand limits the method's applicability to small systems and hinders the calculation of molecular properties beyond ground-state energies, such as permanent electric dipole moments (PDMs), particularly in relativistic regimes where Hamiltonian terms are even more numerous.

Methodology: eos-QSCI
The authors propose an effective Pauli operator sorting-based QSCI (eos-QSCI) framework designed to reduce Hamiltonian complexity prior to quantum simulation. The method proceeds in three distinct phases:

1. Phase I: Fermionic Screening: The second-quantized Hamiltonian is screened by examining the action of fermionic operator strings on a reference determinant (typically the Dirac-Hartree-Fock state). Operators that annihilate the reference state (e.g., those involving annihilation on virtual orbitals) are discarded. This imposes occupation-based selection rules, reducing the number of terms before mapping to qubits via the Jordan-Wigner transformation.
2. Phase II: Fidelity-Loss Based Selection: The retained Pauli operators are ranked by their importance using a reference-state fidelity loss metric. For each operator $P_\alpha$, the short-time evolution $|\Psi_\alpha(t)\rangle = e^{-ih_\alpha P_\alpha t}|\Phi_0\rangle$ is analyzed. The fidelity loss $\omega_\alpha(t)$, proportional to the variance of the operator with respect to the reference state, serves as a weight. Operators are sorted in descending order of $\omega_\alpha$, and a cumulative normalized weight is calculated. The Hamiltonian is truncated to retain only the top $K$ terms such that the cumulative weight exceeds a threshold (e.g., 0.99).
3. Phase III: Execution and Post-Processing: The truncated Hamiltonian is partitioned into sets of commuting observables (SCOs) to construct efficient time-evolution circuits. These circuits are optimized using pipeline routines (Qiskit and Pytket) and executed on quantum hardware. The sampled configurations define a reduced subspace Hamiltonian, which is diagonalized classically to obtain ground-state energies and wavefunctions. PDMs are subsequently evaluated from these wavefunctions.

Key Contributions

• Hamiltonian Reduction: The framework constructs a compact, effective Hamiltonian by identifying dominant fermionic excitation operators, substantially reducing the number of Pauli terms, circuit depth, and two-qubit gate counts compared to full-Hamiltonian HSB-QSCI.
• Relativistic Molecular Properties: The method extends QSCI to relativistic electronic-structure problems, successfully computing both ground-state energies and permanent electric dipole moments (PDMs) for Group IIIA monofluorides (BF, AlF, GaF, InF, TlF).
• Scalability Analysis: The authors demonstrate that the resource requirements of eos-QSCI scale more favorably than the full-Hamiltonian approach, achieving near-quadratic improvement in Hamiltonian-term scaling ($O(N^{2.3})$ vs. $O(N^4)$).

Results
The framework was validated through simulations and hardware execution on the IBM Marrakesh processor (Heron r2 architecture) for AlF and TlF systems with active spaces up to 20 qubits.

• Resource Efficiency: For a 20-qubit TlF system, the reduced Hamiltonian yielded a reduction of >98% in both circuit depth and two-qubit gate counts compared to the full-Hamiltonian approach. The number of Hamiltonian terms was reduced by 97.14% (AlF) and 98.00% (TlF).
• Accuracy:
  • Simulations: The computed ground-state energies deviated from Complete Active Space Configuration Interaction (CASCI) values by less than 1.6 mHa (chemical accuracy threshold) across all system sizes. Recovered correlation energies ranged from 90.02% to 98.46%. PDM deviations were within $1.49 \times 10^{-2}$ a.u. for AlF and $7.0 \times 10^{-3}$ a.u. for TlF.
  • Hardware Execution: On the IBM Marrakesh processor, the 20-qubit TlF calculation recovered 99.93% of the CASCI correlation energy. The resulting PDMs matched CASCI values within 99.99%. Even with hardware noise, energy deviations remained below 0.01% relative to CASCI.
• Determinant Space Reduction: The method concentrated sampling on dominant determinants, reducing the classical diagonalization cost. For the 20-qubit TlF system, only ~0.88% of the total CASCI determinant space was required to achieve high accuracy, reducing the diagonalization matrix size from $181 \times 181$ to $43 \times 43$.

Significance
The paper claims that the eos-QSCI framework significantly improves the practicality of HSB-QSCI on NISQ devices. By simultaneously reducing Hamiltonian complexity, quantum circuit resources, and classical post-processing costs, the approach enables resource-efficient simulations of relativistic molecular systems that were previously intractable due to circuit depth constraints. The successful hardware execution on 20 qubits demonstrates the scalability of this approach for calculating both energies and molecular properties like PDMs, which are critical for applications in quantum technologies and fundamental physics.