Sample-based quantum diagonalization as parallel fragment solver for the localized active space self-consistent field method

TL;DR

This work introduces LASSQD, a hybrid quantum–classical framework that uses sample-based quantum diagonalization (SQD) as a fragment solver within the localized active space self-consistent field (LASSCF) method. By incorporating carryover SQD and the LUCJ circuit, LASSQD enables solving larger fragment active spaces while preserving meaningful orbital optimization, and LASSQD-PDFT adds external dynamical correlation. Demonstrations on bimetallic and trimetallic iron systems show LASSQD approaches LASSCF accuracy with a fraction of the Hilbert space and competitive results against classical SCI, highlighting viability on near-term quantum hardware. The work also demonstrates the method's capacity to tackle fragment sizes beyond classical LASSCF reach, including iron-porphyrin, suggesting a practical pathway toward scalable multireference treatments in complex transition-metal systems. Overall, LASSQD provides a flexible, quantum-assisted route to treat strong correlation in large, challenging systems and outlines clear avenues for hardware- and algorithmic improvements.

Abstract

Accurately and efficiently describing strongly correlated electronic systems is a central challenge in quantum computational chemistry, with classical and quantum computers. The localized active space self-consistent field method (LASSCF) uses a product of fragment active spaces as a variational space, with the Schrödinger equation solved exactly in each fragment and the fragment active-space orbitals defined in a self-consistent manner. LASSCF is accurate for systems with strong intra-fragment and weak inter-fragment correlation, and its computational cost is combinatorial with respect to the size of the individual fragment active spaces, rather than their product. However, exactly solving the Schrödinger equation in each fragment remains a substantial bottleneck. Here, we address the possibility of solving the fragment active space Schrödinger equation with approximate methods, particularly sample-based quantum diagonalization (SQD). SQD is a technique that uses a quantum computer to sample configurations from a chemically motivated quantum circuit and a classical computer to mitigate errors and solve the Schrödinger equation in a subspace of the configuration space. We apply the proposed method, LASSQD, to the [Fe(H$_2$O)$_4$]$_2$bpym$^{4+}$ compound and the [Fe$^{\mathrm{III}}$Fe$^{\mathrm{III}}$Fe$^{\mathrm{II}}$($μ$$_3$-O)-(HCOO)$_6$] complex for calculating the intermediate-spin ground state energies. We observe that LASSQD can tackle fragment sizes intractable by LASSCF, achieves within 1kcal/mol agreement to LASSCF, and delivers results that are competitive with alternative classical methods to solve the Schrödinger equation, and thus can be used as a starting point for a perturbative treatment (LASSQD-PDFT) to recover correlation external to the active space.

Figures (6)

• Figure 1: a): Bimetallic compound [Fe(H$_2$O)$_4$]$_2$bpym $^{4+}$, fragmentation illustrated in dotted box. Each fragment contains 5 3$d$ orbitals and 6 electrons. The first fragment has 4 spin-up electrons and 2 spin-down electrons, while the second fragment has 2 spin-up electrons and 4 spin-down electrons. b): Qubits used on ibm_sherbrooke are indicated by red (purple) circles, which correspond to $\alpha$ ($\beta$) spin orbitals. Orange encodes auxiliary qubits that mediate the density-density interactions among orbitals of opposite spin in the LUCJ ansatz. c): absolute energy plots of LASSCF and LASSQD energies from hardware across consecutive macro cycles. d): energy difference from each macro cycles of classical LASSCF and LASSQD hardware calculations in log scale with the converged LASSCF energy ($E_\mathrm{LASSCF}$ in the legend) as the reference.
• Figure 2: a) Energy difference with error bars between LASSQD QASM simulations and converged LASSCF energy ($E_\mathrm{LASSCF}$ in the legend) as the reference. 1-3 kcal/mol deviations from LASSCF energy is plotted as an acceptable range of error. The dashed vertical line indicates the cutoff for statistical analysis. b): Energy difference with error bars between LASSQD QASM1 simulation and LASSCF in log scale. The light blue box shows the ensemble's mean with 1 standard error after the burn-in cutoff.
• Figure 3: a): Bimetallic compound [Fe(H$_2$O)$_4$]$_2$bpym$^{4+}$, fragmentation illustrated in dotted box. Each fragment contains 10 orbitals (5 3$d$ orbitals and 5 4$d$ orbitals) and 6 electrons. The first fragment has 4 spin-up electrons and 2 spin-down electrons, while the second fragment has 2 spin-up electrons and 4 spin-down electrons. b): Qubits used on ibm_torino are indicated by red (purple) circles, which correspond to $\alpha$ ($\beta$) spin orbitals. Orange encodes auxiliary qubits that mediate the density-density interactions among orbitals of opposite spin in the LUCJ ansatz. c): LASSQD energy difference in log scale to the reference LASSCF energy. The dashed grey line is 1 kcal/mol deviation from LASSCF. The legend indicates the backend type (QASM or hardware), and the carryover threshold in powers of 10. d): The maximum subspace dimension as the percentage of FCI space plotted across each macro cycle for LASSQD hardware calculation. “d sampled” denotes the raw SQD dimension and “d final” denotes the final subspace after carryover
• Figure 4: a): Triiron oxo-centered complex [Fe$^{\mathrm{III}}$Fe$^{\mathrm{III}}$Fe$^{\mathrm{II}}$($\mu$$_3$-O)-(HCOO)$_6$], with dashed box illustrating fragmentation. Each fragment has 10 orbitals, and (5$\alpha$, 1$\beta$),(4$\alpha$, 1$\beta$),(4$\alpha$, 1$\beta$) electrons respectively. b): LASSQD QASM simulations without carryover, shaded grey area indicates 1-3 kcal/mol accuracy with respect to LASSCF. c): LASSQD QASM simulations with carryover, $\epsilon=$$1 \cdot 10^{-5}$, $1 \cdot 10^{-8}$.
• Figure 5: Independent QASM LASSQD runs and classical LASSCI calculations with varying cutoffs. y-axis shows the absolute energies of LASSQD and LASSCI difference from LASSCF while x-axis indicates the percentage of max FCI space used for that energy value.