From Random Determinants to the Ground State

TL;DR

TrimCI introduces a prior-knowledge-free framework that discovers accurate quantum ground states directly from random Slater determinants by iteratively expanding and trimming a core set connected by the Hamiltonian graph. The method yields compact, explicit ground-state wavefunctions and achieves state-of-the-art accuracy with orders-of-magnitude fewer determinants than traditional approaches, across both molecular and lattice models, including challenging [4Fe–4S] and nitrogenase clusters and large 8×8 Hubbard systems. Empirically, TrimCI exhibits systematic convergence, reveals a core-shell structure in the determinant space, and supports powerful extrapolations, outperforming AFQMC in several regimes. The work also provides a quantitative view of many-body complexity via amplitude statistics and suggests strong potential for integration with quantum algorithms and existing classical workflows, offering a scalable path to high-accuracy many-body computations with practical observables.

Abstract

Accurate quantum many-body calculations often depend on reliable reference states or good human-designed ansätze, yet these sources of knowledge can become unreliable in hard problems like strongly correlated systems. We introduce the Trimmed Configuration Interaction (TrimCI) method, a prior-knowledge-free algorithm that builds accurate ground states directly from random Slater determinants. TrimCI iteratively expands the variational space and trims away unimportant states, allowing a random initial core to self-refine into an accurate approximation of exact ground state. Across challenging benchmarks, TrimCI achieves state-of-the-art accuracy with strikingly efficiency gains of several orders of magnitude. For [4Fe-4S] cluster, it matches recent quantum computing results with $10^6$-fold fewer determinants and CPU-hours. For the nitrogenase P-cluster, it matches selected-CI accuracy using $10^5$-fold fewer determinants. For $8\times8$ Hubbard model, it recovers over $99\%$ of the ground-state energy using only $10^{-28}$ of the Hilbert space. In some regimes, TrimCI attains orders-of-magnitude higher accuracy than AFQMC method. These results demonstrate that high-accuracy many-body ground states can be discovered directly from random determinants, establishing TrimCI as a prior-knowledge-free, accurate and highly efficient framework for quantum many-body systems. The compact explicit wavefunctions it produces further enable direct and rapid evaluation of observables.

Figures (5)

• Figure 1: Overview and efficiency of the Trimmed Configuration Interaction (TrimCI) algorithm.(a) Schematic workflow. Starting from a random initial core of determinants, TrimCI alternates between expansion, which adds connected determinants via Hamiltonian couplings, and trimming, which removes unimportant ones, ultimately converging to a compact final core containing the dominant configurations of the ground state. One schematic iteration is illustrated. (b) Example of a TrimCI iteration using real data. Multidimensional scaling (MDS) visualizations project the high-dimensional determinant space onto two dimensions, with marker size indicating the magnitude of the wavefunction coefficients. The plots show the evolution from an initial core to an expanded pool and finally to a refined new core, illustrating how TrimCI autonomously identifies the most important determinants within the Hilbert space. (c) Application to the [Fe$_8$S$_7$] P-cluster. TrimCI achieves the same accuracy as selected-CI while using over $10^5$-fold fewer determinants, as visualized by the polar efficiency spiral, where the radius encodes the energy deviation $\log_{10}(\Delta E + 1)$ relative to a reference point and the angle represents the logarithmic determinant cost $2\pi \log_{10}(N_{\mathrm{dets}})$. (d) Application to the two-dimensional Hubbard model. TrimCI efficiently discovers the correlated ground-state basin, whereas selected-CI remains trapped near the Hartree--Fock minimum, highlighting TrimCI's ability to escape local basins and identify globally optimal determinant subsets.
• Figure 2: Comparative performance of TrimCI and Selected-CI across molecular benchmarks.(a,d,g) Energy convergence with respect to the number of determinants for Cr$_2$ (36o, 48e), [4Fe–4S] cluster (36o, 54e), and [Fe$_8$S$_7$] P-cluster (73o, 114e). TrimCI attains comparable variational energies with several orders of magnitude fewer determinants than Selected-CI . (b,e,h) Two-dimensional embeddings of the determinant space obtained via multidimensional scaling (MDS). Each point represents a Slater determinant, with marker size proportional to its coefficient magnitude. TrimCI states (gold) form compact, well-localized manifolds concentrated near the physically relevant region of the Hilbert space, whereas Selected-CI states (violet) spread more diffusely around the Hartree–Fock reference (red star) or wrong basins. (c,f,i) Weighted Hamming-distance distributions measured relative to the top determinant with the largest amplitude. TrimCI exhibits narrower distance distributions, indicating that it more efficiently identifies the entanglement core of the many-body wavefunction.
• Figure 3: Ground state energy of [4Fe–4S] cluster with TrimCI compared to SQD. TrimCI achieves the same energy ($E = -326.9127$ Ha) as the SQD method, which used $10^{10}$ determinants and $10^7$ CPU-hours, with only $1.3 \times 10^4$ determinants and 5 CPU-hours. This corresponds to over $7 \times 10^5$-fold efficiency in reaching the same accuracy. The TrimCI variational and perturbatively corrected energies show smooth convergence, and a linear extrapolation estimates the ultimate energy to be $E \approx -327.263$ Ha.
• Figure 4: TrimCI performance on the two-dimensional Hubbard model. (a) Relative energy error versus number of determinants for the $4\times4$ lattice with periodic boundary condition ($U=2$, half filling), compared to exact FCI results. Dashed blue lines show power-law fits; orange markers include perturbative corrections. (b) Linear extrapolation of total energy $E_{\mathrm{tot}} = E_{\mathrm{var}} + E_{\mathrm{per}}$ against $- E_{\mathrm{per}}$, converging to the exact FCI reference within a relative error of $4\times10^{-7}$; AFQMC reference and uncertainty are shown in gray. (c) Scaling for the $8\times8$ lattice with periodic boundary condition ($U=2$, half-filling); TrimCI achieves $97$–$99\%$ accuracy with $10^4$–$10^8$ determinants in a space of $10^{36}$ dets. (d) Energy extrapolation for the $8\times8$ lattice, where the extrapolated energy deviates from the AFQMC reference by only $0.011\%$.
• Figure 5: Evolution and structure of the TrimCI wavefunction for the 4×4 Hubbard model. (a1–a5) First independent run (Run 1) showing the self-organization of random determinants into a compact, high-fidelity core set. Starting from random Slater determinants, successive expansion–trimming iterations gradually reveal the correlated core of the Hilbert space. Each subpanel reports the number of determinants (#dets) and the energy accuracy relative to the exact ground state. The Hartree–Fock configuration (red star) is shown as a structural reference and notably has zero importance in the ground state. (b1–b5) Second independent run (Run 2) initialized with a different random set of determinants, exhibits nearly identical convergence dynamics and arrives at the same core set (up to spin exchange of electrons) in the early stage, demonstrating TrimCI’s robustness in locating ground state basin. (c) Final many-body wavefunction after convergence, containing $1.7\times10^{6}$ determinants and achieving $>99.99\%$ accuracy, revealing a well-organized geometry in the 2D MDS embedding (rotated by $90^\circ$ for clarity). Strikingly, comparing to panels (a3) and (b3), it shows that TrimCI identifies the most important determinants, yielding an almost perfect core set, already at a very early stage. (d) Density map of determinant amplitudes (total weights per cell, logarithmic color scale) shows concentric shells and arc-like correlations, indicating a hierarchical, core–shell structure in the amplitude distribution. (e) Complementary cumulative distribution $1-F(r)$ of sorted coefficients follows a power-law $r^{-\alpha}$ with $\alpha=0.53$ ($R^2=0.990$).