Quantum-Selected Configuration Interaction

• Quantum-Selected Configuration Interaction (QSCI) is a quantum–classical hybrid approach that uses quantum sampling to select a compact, chemically relevant subspace for classical diagonalization.
• It leverages adaptive state construction, time-evolution, and compressed selection to drastically reduce the number of determinants needed while ensuring chemical accuracy.
• QSCI integrates with high-performance computing and traditional post-processing to deliver superior performance in strongly correlated and resource-limited regimes.

Quantum-Selected Configuration Interaction (QSCI) is a family of quantum–classical hybrid algorithms that enable efficient electronic structure calculations by using quantum devices to select a compact, physically meaningful subspace of electronic configurations for subsequent high-accuracy classical diagonalization. In QSCI, quantum resources are directed toward the classically prohibitive task of configuration sampling, while noise-robust classical post-processing handles the variational solution of the Schrödinger equation. Recent research has extended QSCI to optimize stochastic Hamiltonian evolution, adaptive state preparation, configuration compression, high-performance distributed diagonalization, and integration with both quantum Monte Carlo and coupled-cluster methods. Theoretical analyses and hardware demonstrations show that strategically sampled and post-processed CI expansions can yield chemical accuracy with orders of magnitude fewer determinants than classical heuristics, and can match or outperform established algorithms under strong correlation or resource-limited regime.

1. Fundamental Principles and Formulation

QSCI algorithms operate by constructing an approximate eigenstate (ground or excited) on quantum hardware, then sampling the electronic configurations (Slater determinants) in the computational basis. These configurations—typically those most prevalent in the output distribution—define a “selected” subspace $\mathcal{S}_R$:

$\mathcal{S}_R = \{\,|x\rangle\, :\, x \text{ is among the $R$ most frequently observed outcomes}\,\}$

A classical computer builds the effective Hamiltonian restricted to $\mathcal{S}_R$, with elements $H_{xy} = \langle x | \hat{H} | y \rangle$. Diagonalization in this subspace yields approximate eigenvalues and eigenvectors, improving upon the initial quantum-state guess. The resulting QSCI wavefunction is

$$|\psi^{(0)}_{\mathrm{out}}\rangle = \sum_{x \in \mathcal{S}_R} c_x |x\rangle,$$

where the $c_x$ coefficients are free of additive quantum measurement noise and determined entirely by the deterministic classical diagonalization step (Kanno et al., 2023).

Excited-state energies are obtained either by combining multiple input quantum states (single diagonalization scheme) or by sequentially “deflating” previously found eigenstates with explicit penalty terms (Kanno et al., 2023). The accuracy of QSCI depends crucially on how well the sampled configuration subspace spans the true eigenstate. Post-selection based on conserved quantum numbers (e.g., $N_e$, $S_z$) is routinely employed to mitigate sampling errors due to hardware noise or bit-flip events.

2. Methods for Subspace Selection: Adaptive, Time-Evolution, and Hamiltonian Simulation Approaches

Depending on the intended system and available quantum resources, QSCI can leverage multiple modes of input-state preparation and configuration amplification:

• Adaptive Input State Construction (ADAPT-QSCI): Builds the input state $|\Phi_k\rangle$ iteratively by selecting operators $P_t$ that most significantly lower the variational energy, updating the state as $$|\Phi_{k+1}\rangle = e^{i\theta_k^* P_t} |\Phi_k\rangle$$. Operator selection is based on the energy gradient $h_j = \langle c_k | i[\hat{H}, P_j] | c_k \rangle$. This results in shallow circuits with high overlap to the true ground state, reduced measurement requirements, and resilience to noise (Nakagawa et al., 2023).
• Time-Evolved QSCI (TE-QSCI, HSB-QSCI): Uses the real-time evolution operator $|\psi(t)\rangle = e^{-iHt}|\psi_I\rangle$ to systematically generate higher-order excitations from an initial state. Sampling from the time-evolved state naturally populates configurations that capture increasing correlation orders. This approach is optimization-free (no variational circuit parameter adjustment) and can use Trotterized circuits of limited depth suitable for near-term devices (Sugisaki et al., 10 Dec 2024, Mikkelsen et al., 18 Dec 2024, Weaving et al., 2 Sep 2025). Stochastic Hamiltonian evolution can be further employed to “filter” and select configurations most relevant in the ground-state expansion, with configuration sampling probabilities modulated by orbital occupancies (Weaving et al., 2 Sep 2025).
• Variational and Compressed Selection: Qubit-efficient strategies (e.g., VQ-SCI) encode only the dominant configurations, reducing the required register size from $O(M)$ to $O(\log_2 D)$ where $D$ is the number of configurations retained (Yoffe et al., 2023). Lossy-QSCI frameworks employ chemical bias and random linear encoding (RLE) with a neural-network Fermionic Expectation Decoder to compress the sampled subspace, achieving further resource savings and high fidelity reconstruction (Chen et al., 23 May 2025).

These flexible input strategies allow QSCI to achieve a balance between circuit depth, sampling overhead, and subspace compactness, accommodating hardware limitations and problem-specific requirements.

3. Algorithms for Classical Post-Processing and Integration with HPC

Once the selected configuration subspace is defined, the Hamiltonian projected onto this subspace is diagonalized classically. Algorithms such as the determinant-driven CI diagonalization in Quantum Package (Garniron et al., 2019), often using stochastic or hybrid stochastic/deterministic selection (as in stochastic CIPSI), enable scaling to tens of millions of determinants on supercomputers. High-performance computing (HPC) integration has been advanced by distributed CI vector storage and tensor-product SCI algorithms, allowing full-CI or SCI calculations on trillion-determinant spaces without exhausting single-node memory (Xu et al., 13 Mar 2025).

For scalable calculations, the selected subspace can be further tailored by:

• Hamiltonian truncation through locality constraints (restricting operator strings to $k$-local terms for shallow circuits and reduced classical storage) (Sugisaki et al., 10 Dec 2024).
• Error mitigation through self-consistent configuration recovery (SCCR), ensuring sampling respects electronic symmetry sectors in the presence of noise (Sugisaki et al., 10 Dec 2024, Weaving et al., 2 Sep 2025).
• On-the-fly Hamiltonian construction, exploiting spin adaptation and factorization to minimize the storage and computational cost associated with very large determinant sets (Xu et al., 13 Mar 2025).

These hybrid quantum-HPC strategies make QSCI-compatible frameworks competitive for strongly correlated systems previously inaccessible to classical methods.

4. Applications and Numerical Results: Molecules, Materials, and Benchmarking

QSCI has been demonstrated on diverse molecular and materials systems, including:

• Small Molecules and Excitations: In water and formaldehyde, QSCI combined with fixed-node diffusion Monte Carlo retrieves vertical excitation energies with errors as low as 0.03–0.07 eV across transition types and determinant set sizes (Scemama et al., 2018). Compact multideterminant expansions constructed with modest basis sets are shown to achieve chemical accuracy ($<1$ kcal/mol).
• Large Active Spaces and Strong Correlation: Ground and excited states in oligoacenes, phenylene-dinitrene, and hexa-1,2,3,4,5-pentaene were resolved using HSB-QSCI or TE-QSCI with only 0.8–1.2% of the CAS-CI space, capturing over 99.75% of correlation energy (errors $<0.15$ kcal/mol) (Sugisaki et al., 10 Dec 2024).
• Hardware Demonstrations: A QSCI calculation of the SiH$_4$ potential energy curve was carried out on a 42-qubit IQM superconducting device, validating run-time stochastic Hamiltonian evolution and sampling, and demonstrating compactness and noise resilience (Weaving et al., 2 Sep 2025).
• Quantum Monte Carlo Integration: QSCI-derived CI expansions serve as high-quality trial wavefunctions in phaseless Auxiliary-Field Quantum Monte Carlo (ph-AFQMC), substantially improving energy recovery and reducing sampling redundancy (Mahajan et al., 2022, Yoshida et al., 28 Feb 2025, Danilov et al., 7 Mar 2025). For N$_2$, QSCI-AFQMC delivered energy differences within chemical accuracy relative to full configuration interaction.
• Coupled-Cluster Tailoring: QSCI-active space wavefunctions are embedded into tailored coupled-cluster theory (QSCI-TCC), fixing active-space amplitudes and optimizing the rest. QSCI-TCC provides accurate dissociation curves for challenging bond-breaking in H$_2$O and N$_2$, maintaining accuracy where conventional CCSD or CCSD(T) fail (Erhart et al., 20 Jun 2025).
• Quasiparticle Band Structures: For crystalline silicon, QSCI sampling followed by quantum subspace expansion (QSE) enabled computation of accurate valence and conduction bands using only partially optimized VQE states (Ohgoe et al., 1 Apr 2025).

5. Comparative Analysis, Limitations, and Sampling Bottlenecks

While QSCI can, in principle, emulate or outperform classical SCI heuristics for some systems by producing more compact wavefunctions, severe sampling inefficiencies can arise when sampling from the ground-state (or approximate) quantum distributions:

• Sampling Redundancy: QSCI frequently “resamples” already known determinants, leading to a sublinear increase in unique configurations as the sample count grows. For example, in N$_2$ a million samples yielded only $1.5 \times 10^4$ unique determinants out of a possible $7 \times 10^8$, requiring astronomical shot counts for high-precision estimation (Reinholdt et al., 13 Jan 2025).
• Trade-off in Determinant Discovery: Adjusting the sampling probability can bias towards high-weight determinants or increase exploration, but cannot simultaneously achieve compactness and completeness. Neither uniform nor strongly weighted sampling achieves the optimal balance found in classical heuristics like HCI or CIPSI (Reinholdt et al., 13 Jan 2025).
• Resource Overheads: In QSCI with redundant sampling, the needed number of determinants for chemical accuracy may far exceed that of classical Sci approaches, increasing the overhead in both quantum sampling and classical diagonalization (Reinholdt et al., 13 Jan 2025).
• Noise and Qubit Limitations: Qubit noise restricts the size of reliably selected configuration subspaces, making it difficult to capture dynamic correlation in larger molecules unless combined with perturbation theory or MRPT (Shirai et al., 28 Mar 2025).

Nevertheless, recent evidence indicates that with intelligent configuration sampling and time-evolution approaches, QSCI can yield more compact CI expansions than competitive heuristics in some regimes (e.g., SiH$_4$, where subspaces were 200 times smaller than HCI for comparable accuracy) (Weaving et al., 2 Sep 2025).

6. Extensions, Hybridization, and Future Prospects

The QSCI paradigm is being rapidly extended in several directions:

• Integration with Multireference Perturbation Theory: MRPT corrections with QSCI reference states (e.g., GMC-QDPT on top of QSCI-selected parents plus single/double/triple excitations) systematically improve dynamic correlation and excitation energies, as shown for naphthalene and tetracene (Shirai et al., 28 Mar 2025, Weaving et al., 2 Sep 2025).
• Compression and Neural Decoding: Lossy-QSCI leverages number-conserving compressed binary encodings in concert with a neural-network decoder for efficient sampling and robust configuration recovery (Chen et al., 23 May 2025).
• HPC Scalability: Distributed storage, tensor-product bit string selection, and on-the-fly Hamiltonian evaluation overcome classical memory bottlenecks, enabling trillion-determinant FCI calculations and peta-flop HPC integration (Xu et al., 13 Mar 2025).
• Hybrid Quantum–Classical QMC: Trial wavefunctions from QSCI can guide nonperturbative quantum Monte Carlo, with energy-variance extrapolation further improving ground-state energy estimates (Yoshida et al., 28 Feb 2025, Danilov et al., 7 Mar 2025).

Continued advances in quantum hardware, improved quantum subspace sampling algorithms, and the development of adaptive, noise-mitigated preparation strategies (ADAPT-QSCI, stochastic time evolution) are expected to expand QSCI’s application domain. Key future directions include scaling to larger active spaces, integration with other quantum algorithms (QSE, Krylov subspace, QPE), and the use of machine learning to further optimize configuration sampling schemes (Chen et al., 23 May 2025, Weaving et al., 2 Sep 2025).

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In summary, QSCI is an extensible, unifying framework for combining quantum sampling and classical configuration interaction, facilitating tractable, accurate electronic structure calculations in both molecular and material contexts, and bridging the capabilities of quantum and classical high-performance computing for strongly correlated quantum systems (Scemama et al., 2018, Kanno et al., 2023, Sugisaki et al., 10 Dec 2024, Reinholdt et al., 13 Jan 2025, Weaving et al., 2 Sep 2025).