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arXiv:2610.21907

Selected Stabilizer Subspaces for Compact Many-Body Wave Functions

Maria-Andreea Filip, George H. Booth

quant-phSubmitted Oct 15, 2026• Announced Oct 16, 2026• 0 citations
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Abstract

Selected configuration interaction (SCI) obtains accurate many-body solutions by variationally diagonalising the Hamiltonian in a sparse, iteratively selected set of configurations, but can lose compactness when the target state has broad configurational support. We introduce selected stabilizer interaction (SSI), a fully classical, SCI-inspired method that instead iteratively constructs a nonorthogonal subspace spanned by a systematically enlargeable set of selected stabilizer states. A single stabilizer, generated by Clifford operations, can encode a structured superposition of (possibly exponentially) many configurations, shifting the relevant compression from configurational to stabilizer sparsity. We benchmark ground-state energies, fidelities, superposition degrees and required expansion lengths against exact diagonalization and equivalent SCI schemes for the transverse-field Ising, XXZ Heisenberg, and Hubbard models, finding particularly strong compression from the stabilizer representation in the spin models, and increasing advantage with system size for these systems. The results establish a proof of principle for heuristic selected stabilizer diagonalization, while also identifying particular challenges in symmetry handling and effective exploration of the stabilizer space.

Comments
20 pages, 8 figures, 1 table

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Selected configuration interaction (SCI) obtains accurate many-body solutions by variationally diagonalising the Hamiltonian in a sparse, iteratively selected set of configurations, but can lose compactness when the target state has broad configurational support. We introduce selected stabilizer interaction (SSI), a fully classical, SCI-inspired method that instead iteratively constructs a nonorthogonal subspace spanned by a systematically enlargeable set of selected stabilizer states. A single stabilizer, generated by Clifford operations, can encode a structured superposition of (possibly exponentially) many configurations, shifting the relevant compression from configurational to stabilizer sparsity. We benchmark ground-state energies, fidelities, superposition degrees and required expansion lengths against exact diagonalization and equivalent SCI schemes for the transverse-field Ising, XXZ Heisenberg, and Hubbard models, finding particularly strong compression from the stabilizer representation in the spin models, and increasing advantage with system size for these systems. The results establish a proof of principle for heuristic selected stabilizer diagonalization, while also identifying particular challenges in symmetry handling and effective exploration of the stabilizer space.
High Resolution Image Download MS PowerPoint Slide Selected Configuration Interaction (SCI) is a method in molecular electronic structure theory that allows for the construction of configuration subspaces adapted to the particular system under study. This adaptability is achieved by guiding subspace construction with respect to a selection criterion designed to identify and retain the most important configurations for an accurate description of the system. Quantum-SCI (QSCI) introduces quantum resources to inform the construction of these subspaces, motivated by the classical hardness of state sampling and system dynamics. As with conventional SCI, the CI routine is still performed classically and is therefore not corrupted by hardware noise; only the subspace quality is affected by deficiencies in the quantum hardware. Previous QSCI approaches take the measurements produced by a physically motivated circuit construction, often with a recovery step to mitigate against errors, and form a subspace from the resulting configurations. We propose an alternative approach that is more aligned with conventional selection criteria but attempts to inject classically inaccessible information into the selection step with the aim of enabling new subspace expansion pathways. The approach presented in this work uses the population statistics of a time-evolved quantum state to predict likely single and double excitations away from existing configurations to bias the subspace expansion procedure. Importantly, this occupancy-guided expansion complements rather than replaces the direct inclusion of valid configurations sampled from the time-evolved quantum state, so determinants containing higher-order excitations can still enter the variational space in a single iteration. We also include multireference perturbation theory to capture missed correlations outside the configuration subspace. This is demonstrated on hardware by using 42 qubits of an IQM superconducting device to compute the potential energy curve of SiH 4 in a 6-31G basis set as the Si–H bonds are stretched. We benchmark against the best-in-class Heatbath CI algorithm to assess the compactness of the resulting wave function.
A procedure is discussed that searches for the best description of the eigenstates of a Hamiltonian of a finite quantum many-body system in terms of a selected set of physically relevant configurations. The procedure resorts to iterative sequences of diagonalizations in spaces of very reduced size. Each diagonalization provides an energy-based importance measure that governs the selection of the configurations to be included in the states. The procedure is strictly variational and preserves the symmetries of the Hamiltonian throughout the iterative process. We report on some test applications to the Na8{}_{8}8​ metal cluster. A series of calculations is performed on a Hartree-Fock basis for a number of orbitals ranging from 5 to 20. In the case of 5 (10) orbitals, we compare the low-lying spectra generated with our procedure to complete (truncated) configuration interaction calculations. The procedure accurately reproduces the results in the complete space. In the most complex cases of 15 and 20 orbitals (the latter involving a many-particle space of 1011{}^{11}11 states), we construct the ground state and study its structure. The analysis highlights the efficiency of the method in generating the most appropriate model space for the description of individual eigenstates of the Hamiltonian.
Selected configuration interaction (SCI) methods are currently enjoying a resurgence due to several recent developments which improve either the overall computational efficiency or the compactness of the resulting SCI vector. These recent advances have made it possible to get full CI (FCI) quality results for much larger orbital active spaces compared to conventional approaches. However, due to the starting assumption that the FCI vector has only a small number of significant Slater determinants, SCI becomes intractable for systems with strong correlation. This paper introduces a method for developing SCI algorithms in a way which exploits local molecular structure to significantly reduce the number of SCI variables. The proposed method is defined by first grouping the orbitals into clusters over which we can define many-particle cluster states. We then directly perform the SCI algorithm in a basis of tensor products of cluster states instead of Slater determinants. While the approach is general for arbitrarily defined cluster states, we find significantly improved performance by defining cluster states through a Tucker decomposition of the global (and sparse) SCI vector. To demonstrate the potential of this method, called tensor product selected configuration interaction (TPSCI), we present numerical results for a diverse set of examples: (1) modified Hubbard model with different inter- and intracluster hopping terms, (2) less obviously clusterable cases of bond breaking in N 2 and F 2, and (3) ground state energies of large planar π-conjugated systems with active spaces of up to 42 electrons in 42 orbitals. These numerical results show that TPSCI can be used to significantly reduce the number of SCI variables in the variational space, thus paving a path for extending these deterministic and variational SCI approaches to a wider range of physical systems.
Subspace diagonalization techniques based on quantum sampling, such as quantum selected configuration interaction (QSCI) and sample-based quantum diagonalization (SQD), have recently emerged as promising quantum-centric approaches for approximating ground-state energies of many-body systems. However, their performance is fundamentally limited by an intrinsic trade-off between sampling efficiency and the sparsity of the ground-state wavefunction, which becomes particularly severe in strongly correlated systems. Here, we introduce a filter-assisted SQD protocol that engineers wavefunction sparsity via a quantum filter, i.e., a unitary transformation of the Hamiltonian designed to concentrate the ground-state weight onto a small number of computational basis states. Using the Gini coefficient as a robust sparsity measure, we establish a quantitative relationship between wavefunction sparsity and the resource requirements of SQD, providing theoretical bounds on the required subspace dimension and sampling cost. To realize the quantum filter, we employ a tensor-network-based circuit-encoding algorithm that maps target states to quantum circuits with controllable fidelity. We benchmark our approach on the quantum Ising model with transverse and longitudinal fields using both numerical simulations and quantum hardware experiments. Our results demonstrate that, compared with standard SQD, the proposed protocol significantly enhances wavefunction sparsity, reduces ground-state energy estimation errors by orders of magnitude, and substantially lowers sampling overhead. These findings establish filter-assisted subspace diagonalization as a powerful and scalable framework for quantum many-body calculations in the strongly correlated regime.
Selected configuration interaction (SCI) methods are effective for treating strongly correlated electronic systems, yet their scalability has long been limited by implementations that replicate the configuration interaction (CI) vector across processes, leading to severe memory bottlenecks. Here, we present a fully distributed diagonalization framework tailored for extremely large selected determinant spaces, directly addressing this major scalability bottleneck of modern SCI methods. The method is grounded in a tensor-product bitstring (TPB) representation, in which determinants are organized through a TPB structure constructed from selected α- and β-bitstrings, and is referred to as tensor-product bitstring SCI (TBSCI). An efficient TBSCI eigensolver is developed based on a novel bitstring-based Hamiltonian evaluation algorithm together with a suite of MPI communication strategies designed to improve parallel efficiency. Large-scale full configuration interaction (FCI) benchmarks, employed as communication-intensive stress tests, demonstrate that the implemented TBSCI eigensolver continues to reduce the wall time for distributed diagonalization of 2.6 × 1012 determinants, reaching 54 000 nodes (more than 2.5 × 106 cores) on supercomputer Fugaku. Beyond scalability, we investigate the structural compactness of the TPB representation and show that selecting α- and β-bitstrings according to their collective weights in a reference SCI wavefunction yields TPB-based wavefunctions approaching the FCI limit while using only a small fraction of determinants. These results establish TBSCI as a scalable SCI methodology and provide evidence for the intrinsic compactness of the TPB representation.
Accurate ground-state energy calculations remain a central challenge in quantum chemistry due to the exponential scaling of the many-body Hilbert space. Variational Monte Carlo and variational quantum eigensolvers offer promising ansatz optimization approaches but face limitations in convergence as well as hardware constraints. We introduce a particular Selected Configuration Interaction (SCI) algorithm that uses auto-regressive neural networks (ARNNs) to guide subspace expansion for ground-state search. Leveraging the unique properties of ARNNs, our algorithm efficiently constructs compact variational subspaces from learned ground-state statistics, which in turn accelerates convergence to the ground-state energy. Benchmarks on molecular systems demonstrate that ARNN-guided subspace expansion combines the strengths of neural-network representations and classical subspace methods, providing a scalable framework for classical and hybrid quantum-classical algorithms.