Cluster-Adaptive Sample-Based Quantum Diagonalization for Strongly Correlated Systems

Abstract

Strongly correlated electronic systems exhibit inherently multiconfigurational wave functions, making it difficult to construct compact variational subspaces that preserve the essential multireference character. Quantum computing has emerged as a promising route to alleviate these limitations, and sample-based quantum diagonalization (SQD) is a representative hybrid approach that uses quantum hardware as a determinant sampler followed by classical diagonalization in the projected subspace. To mitigate hardware noise, SQD employs a self-consistent particle-number recovery guided by a single global reference occupancy vector. However, in strongly correlated, multimodal regimes, this global reference can become mixture-averaged and bias recovery toward a mean pattern, diluting mode-specific occupation structure and degrading the determinant pool. Here, we introduce cluster-adaptive SQD (CSQD), which clusters measurement samples via unsupervised learning and performs particle-number recovery using cluster-specific, self-consistently updated reference occupancy vectors. Under a matched variational budget, we benchmarked CSQD against SQD for N2 dissociation in a (10e,26o) active space and the [2Fe-2S] cluster in a (30e,20o) active space. Our results indicate that CSQD offers an advantage over SQD in estimating the ground-state energy in the strongly correlated regime, lowering the variational estimate by up to 15.95 mHa for stretched N2 and up to 45.53 mHa for [2Fe-2S], with modest additional classical overhead.

Figures (4)

• Figure 1: CSQD workflow for ground-state energy estimation. The left (blue) block shows quantum sampling and preprocessing: a parameterized quantum state $|\tilde{\psi}\rangle$ is prepared and measured repeatedly to collect bitstring samples $x=(\mu_\alpha,\mu_\beta)$, whose spin strings are pooled into single-spin strings $\mathcal{X}_\mu=\{\mu\}$. These strings are partitioned into $K$ clusters via unsupervised clustering. The right (red) block depicts the self-consistent recovery loop for CSQD: for each cluster $k$, a cluster-specific reference occupancy vector $\mathbf{n}^{(k)}$ guides particle-number recovery. Recovered strings are subsampled into $B$ batches, each defining a determinant subspace. For each batch $b$, the Hamiltonian projected into the corresponding subspace is constructed and diagonalized to obtain the lowest-energy eigenpair $\bigl(E^{(b)},|\psi^{(b)}\rangle\bigr)$. The best batch $b^*=\arg\min_b E^{(b)}$ is selected, $\{\mathbf{n}^{(k)}\}_{k=1}^K$ is updated from $|\psi^{(b^*)}\rangle$, and the recovery--diagonalization loop is iterated until a stopping criterion is met, yielding final estimates $(E_0,|\psi_0\rangle)$.
• Figure 2: Dissociation of $\mathrm{N_2}$: Comparison of CSQD with SQD and classical references.(a) Total energies along the $\mathrm{N\!-\!N}$ bond-stretching coordinate $R$ for RHF, CCSD, HCI, SQD, and CSQD (BMM, $K=5$). HCI provides a high-accuracy classical benchmark within the chosen active space (CCSD is non-variational). (b) Upper panel: energy difference $\Delta E(R)=E_{\mathrm{CSQD}}(R)-E_{\mathrm{SQD}}(R)$ in mHa. Negative values indicate a lower variational energy obtained by CSQD. Lower panel: RHF determinant weight $|c_{\mathrm{RHF}}|^2$ (squared coefficient of the RHF determinant in the HCI expansion), which decreases upon bond stretching, reflecting the multireference character. At each $R$, SQD and CSQD are evaluated from the same quantum inputs and compared at a matched final projected-subspace dimension $\dim(\mathcal{V})=d_{\max}^2$ with $d_{\max}=2000$.
• Figure 3: $\boldsymbol{\mathrm{[2Fe\!-\!2S]}}$ iteration-wise energy trajectories and reference vector separation in CSQD.(a) Iteration-wise lowest variational energies for SQD and CSQD (BMM, $K=5$) at $d_{\max}=2000$. At each iteration, both methods were evaluated at the fixed projected-subspace dimension $\dim(\mathcal{V})=d_{\max}^2=4\times10^{6}$, and the plotted value is the lowest energy observed so far across all batches and iterations. Horizontal dashed lines indicate the CCSD and DMRG energies for the same effective active-space Hamiltonian, providing single-reference and high-accuracy classical baselines, respectively. (b) Heatmap of one-half the Manhattan distance ($L_1$) computed for all pairs among the SQD global reference occupation vector $\mathbf{n}_{\mathrm{SQD}}$ and the CSQD cluster-specific reference occupation vectors $\mathbf{n}^{(k)}_{\mathrm{CSQD}}$. The values can be interpreted as the effective number of occupation swaps—that is, the total occupation weight, in electrons, required to transform one reference pattern into another. Larger values indicate more strongly separated occupation patterns.
• Figure 4: Hardware mapping on IBM quantum processors. Device schematics showing the qubit layouts and role assignments used in this work: (a)$\mathrm{N_2}$ experiment on ibm_fez and (b)$\mathrm{[2Fe\!-\!2S]}$ cluster experiment on ibm_torino. Qubits are colored by role: $\alpha$ spin-orbitals (red), $\beta$ spin-orbitals (blue), and auxiliary qubits (green).