Quantum Selected Heat-Bath CI (QSHCI)

• Quantum Selected Heat-Bath CI (QSHCI) is a hybrid method that fuses quantum subspace sampling with heat-bath configuration interaction and multireference perturbation theory for efficient wavefunction construction.
• It employs quantum-derived excitation scores to prioritize determinants while using classical heat-bath screening and perturbative corrections to refine energy estimates.
• Demonstrated on systems like SiH4 and N2, QSHCI achieves comparable accuracy to traditional HCI methods with significantly reduced variational space sizes.

Searching arXiv for QSHCI and closely related QSCI/HCI papers to ground the article with current sources. Quantum Selected Heat-Bath CI (QSHCI) denotes a hybrid selected-configuration-interaction paradigm that couples quantum-generated configuration statistics to the selection, screening, and post-variational machinery associated with Heat-Bath Configuration Interaction (HCI). In current usage, the term refers to at least two closely related constructions: a principled hybrid in which Quantum-Selected Configuration Interaction (QSCI) statistics are fused with HCI’s integral-driven selection and multireference perturbation theory, and an augmented CI-matrix-based QSCI variant in which the classical heat-bath threshold is replaced by a quantum-informed acceptance rule derived from sampled configuration probabilities (Weaving et al., 2 Sep 2025, Graves et al., 13 Mar 2026). Taken together, these works suggest that QSHCI is best understood as a class of quantum–classical SCI+PT methods whose defining feature is that a quantum device is used for subspace discovery, while final energies remain classical.

1. Terminological and methodological setting

QSHCI emerged against the background of HCI and SHCI, two classical selected-CI plus perturbation-theory methods. HCI is a deterministic analog of efficient heat-bath sampling for determinants; it iteratively builds a compact variational determinant space and then evaluates a screened Epstein–Nesbet-type second-order perturbative correction (Holmes et al., 2016). SHCI extends HCI by replacing the memory-intensive deterministic perturbative step with a semistochastic algorithm, thereby eliminating the severe memory bottleneck of the original method while preserving sub-millihartree accuracy targets on large active spaces (Sharma et al., 2016).

The original HCI and SHCI papers do not define QSHCI explicitly. In the HCI literature, “heat-bath” refers to the use of precomputed, sorted excitation lists that expose only large Hamiltonian couplings, and the user-facing controls are the variational threshold $\epsilon_1$ and the perturbative threshold $\epsilon_2$ (Holmes et al., 2016). In SHCI, semistochastic PT2 retains the same HCI selection structure but evaluates the perturbative correction by independent sampling with the Alias method and a control-variate-like deterministic–stochastic split (Sharma et al., 2016).

The later QSCI literature changed the role of the quantum processor. Rather than using the quantum device for final expectation-value estimation, QSCI uses it to sample determinants or configuration state functions (CSFs) from a dynamically generated state, after which the Hamiltonian is diagonalized classically in the sampled subspace (Weaving et al., 2 Sep 2025, Graves et al., 13 Mar 2026). QSHCI appears at the intersection of these lines of work: it imports quantum-guided subspace discovery into the HCI/SHCI selected-CI framework.

2. Classical antecedents: HCI and SHCI

In HCI, the variational wavefunction is

$|\Psi_V\rangle = \sum_{i\in V} c_i |D_i\rangle,$

where $V$ is the current variational determinant set. An external determinant $a\notin V$ is added if

$\max_{i\in V} |H_{ai} c_i| > \epsilon_1,$

or equivalently if, for some $i\in V$, $|H_{ai} c_i|>\epsilon_1$ (Holmes et al., 2016, Sharma et al., 2016). The efficiency comes from heat-bath precomputed lists: for each occupied-orbital pair, the relevant double-excitation couplings are sorted by magnitude, so traversal can stop as soon as the threshold is no longer met.

After diagonalizing the Hamiltonian in the selected space, HCI evaluates a screened Epstein–Nesbet-like second-order correction,

$$\Delta E^{(2)} \approx \sum_{k\notin V} \frac{\left(\sum_{i\in V}^{(|H_{ki}c_i|>\epsilon_2)} H_{ki} c_i\right)^2}{E^{(0)}-H_{kk}},$$

with $\epsilon_2$ controlling the truncation of the perturbative sum (Holmes et al., 2016). SHCI preserves this structure but replaces the deterministic accumulation of all external partial sums by stochastic or semistochastic sampling. Its sampling distribution is

$\epsilon_2$0

and the semistochastic combination

$\epsilon_2$1

removes the PT2 memory bottleneck while keeping the perturbative stage embarrassingly parallel (Sharma et al., 2016).

These classical ingredients are central to QSHCI because the hybrid methods inherit the HCI notion that subspace quality is controlled by large Hamiltonian couplings and by the quality of the variational coefficients $\epsilon_2$2. What changes in QSHCI is how candidate configurations are prioritized before the heat-bath filtering or instead of a fixed heat-bath threshold.

3. Quantum-selected configuration interaction as the precursor to QSHCI

QSCI uses a quantum processor only for selecting an exceptionally compact set of important configurations before all energies are obtained on classical HPC (Weaving et al., 2 Sep 2025). In the 2025 SiH$\epsilon_2$3 study, the quantum hardware explores short-time dynamics

$\epsilon_2$4

typically from a Hartree–Fock reference, and reports statistics that guide CI-space construction. The measured observables include single-orbital occupancies

$\epsilon_2$5

two-orbital number correlators

$\epsilon_2$6

and optionally off-diagonal one-body and two-body quantities aggregated over a time grid (Weaving et al., 2 Sep 2025).

These time-averaged statistics define scores for candidate excitations. For singles,

$\epsilon_2$7

and for doubles,

$\epsilon_2$8

with diagonal proxies based on occupancies and integral magnitudes used when direct reconstruction is too costly (Weaving et al., 2 Sep 2025). Determinants generated by the largest-scoring excitations are added to the candidate set, then the Hamiltonian is assembled and diagonalized classically, followed by an Epstein–Nesbet-style multireference perturbative correction.

The central empirical result was obtained on a 42-qubit IQM superconducting device for the SiH$\epsilon_2$9 potential energy curve in a 6-31G basis under Si–H bond stretching. In the strong-correlation regime, the QSCI-selected variational space was more than 200 times smaller than a comparable HCI variational space while yielding energies comparable to HCI and comparable non-parallelity errors along the stretch (Weaving et al., 2 Sep 2025). The paper describes this as the first evidence that QSCI can produce more compact representations of the ground-state wavefunction than conventional heuristics.

That result is the immediate motivation for QSHCI. If quantum short-time dynamics identify a subset of determinants carrying the dominant correlation weight, then heat-bath screening and perturbative completion can be applied after, or jointly with, that quantum-guided discovery stage.

4. Two explicit formulations of QSHCI

The literature now contains two technically distinct QSHCI constructions.

The first is the hybrid proposed from the QSCI compactness results. In that formulation, a candidate determinant $|\Psi_V\rangle = \sum_{i\in V} c_i |D_i\rangle,$0 connected to $|\Psi_V\rangle = \sum_{i\in V} c_i |D_i\rangle,$1 receives a combined score

$|\Psi_V\rangle = \sum_{i\in V} c_i |D_i\rangle,$2

where the quantum term is derived from measured dynamical statistics,

$|\Psi_V\rangle = \sum_{i\in V} c_i |D_i\rangle,$3

A determinant is included if

$|\Psi_V\rangle = \sum_{i\in V} c_i |D_i\rangle,$4

while simultaneously enforcing a minimal heat-bath condition

$|\Psi_V\rangle = \sum_{i\in V} c_i |D_i\rangle,$5

The same proposal retains HCI-style semistochastic perturbation theory, optionally sampling the external determinants with probability proportional to the hybrid score (Weaving et al., 2 Sep 2025). This construction is explicitly framed as a blend of QSCI’s dynamical selection statistics with HCI’s integral-weighted screening and perturbative rigor.

The second formulation is the CI-matrix (CIM) QSHCI algorithm introduced in 2026. Here the basis states are CSFs $|\Psi_V\rangle = \sum_{i\in V} c_i |D_i\rangle,$6, with CI-matrix elements

$|\Psi_V\rangle = \sum_{i\in V} c_i |D_i\rangle,$7

and the molecular wavefunction

$|\Psi_V\rangle = \sum_{i\in V} c_i |D_i\rangle,$8

Classical HCI selection in this language uses

$|\Psi_V\rangle = \sum_{i\in V} c_i |D_i\rangle,$9

but QSHCI replaces the fixed threshold by a quantum-informed rule: $V$0 where $V$1 is the normalized sampling probability for configuration $V$2 obtained from quantum measurements and $V$3 is a variance factor (Graves et al., 13 Mar 2026). This is the most explicit operational definition of QSHCI currently available.

The CIM-QSHCI framework also changes the qubit encoding. Instead of second quantization in Fock space, it uses a first-quantized CI-matrix encoding with exactly

$V$4

data qubits, plus one extra qubit for single-bit flip error mitigation when parity encoding is used (Graves et al., 13 Mar 2026). The Hamiltonian is decomposed into Pauli strings via the fast Walsh–Hadamard transform,

$V$5

and stochastic approximate Trotterization adapted from qDRIFT is used to generate the sampling distribution. A notable feature is the single-bit flip mitigation scheme, which detects and corrects non-physical bitstrings with an overhead of one qubit (Graves et al., 13 Mar 2026).

These formulations are not identical. One is a second-quantized, determinant-based hybrid that fuses QSCI statistics with HCI and MRPT; the other is a CIM-based CSF algorithm with a configuration-dependent quantum-informed heat-bath threshold. A plausible implication is that “QSHCI” currently names a methodological direction rather than a single canonical algorithm.

5. Resource models, workflow, and observed performance

In the QSCI-derived hybrid, the workflow begins with integral construction, active-space selection, qubit mapping, and reference-state preparation; proceeds through quantum sampling of short-time dynamics using Trotterization or randomized Trotter/qDRIFT; aggregates orbital occupancies and correlators; selects single and double excitations above thresholds $V$6 and $V$7; and finally performs classical CI diagonalization followed by MRPT(2) (Weaving et al., 2 Sep 2025). The quantum device is used only for selection, and the final molecular energies are evaluated by solving an interaction matrix on HPC, so they are not corrupted by hardware noise (Weaving et al., 2 Sep 2025).

The SiH$V$8 demonstration used 42 qubits of an IQM superconducting device and showed that, in a strongly correlated regime, a QSCI-selected space more than 200 times smaller than a standard HCI variational space achieved comparable ground-state energies and non-parallelity errors along the Si–H bond stretch (Weaving et al., 2 Sep 2025). In that study, the quantum sampling scheme predicted likely single and double excitations to generate connected configurations based on the orbital occupancies of a time-evolved quantum state (Weaving et al., 2 Sep 2025).

In the CIM-QSHCI work, benchmark systems included N$V$9 and naphthalene. The algorithm achieved similar accuracy as SQD methods but with significantly less quantum resources, yet CIM-QSCI and SQD could not match classical HCI for the same task; this motivated QSHCI as the augmented variant that replaces classical heat-bath sampling with quantum sampling to achieve performance comparable to HCI (Graves et al., 13 Mar 2026). For N$a\notin V$0, QSHCI used subspaces of 1.5%–10.5% of the full space on emulator and 2%–8% on hardware, with errors comparable to CIM-QSCI at 80% subspace but with much smaller subspaces (Graves et al., 13 Mar 2026). With $a\notin V$1 and HCI $a\notin V$2, QSHCI and HCI had similar error curves; with $a\notin V$3, HCI at $a\notin V$4 had lower errors despite similar subspace sizes, indicating more efficient determinant selection in HCI (Graves et al., 13 Mar 2026).

A further line of work, although framed around QSCI rather than CIM-QSHCI, directly strengthens the case for QSHCI hybrids. In “Generative Circuit Design for Quantum-Selected Configuration Interaction,” a GQE-based framework optimized low-depth, hardware-aware circuits so that the sampled determinant subspaces were unusually compact (Kemmoku et al., 10 Apr 2026). On N$a\notin V$5 in active spaces up to 32 qubits, the optimized circuits reached chemical precision with an average reduction of 98% in required two-qubit gate count relative to a single-step first-order Trotterized approximation and 83% relative to the qDRIFT approximation (Kemmoku et al., 10 Apr 2026). In stretched-bond, strongly correlated regimes, global refinement reached chemical precision with subspaces approximately 50% smaller than those required by HCI (Kemmoku et al., 10 Apr 2026). The paper explicitly presents these findings as motivating and instantiating a QSHCI workflow in which learned quantum sampling seeds a compact initial subspace and HCI-style refinement adds missed locally important determinants (Kemmoku et al., 10 Apr 2026).

6. Interpretation, misconceptions, and open problems

A common misconception is that QSHCI is a fully quantum energy-estimation method. The available formulations say otherwise. In the QSCI-derived hybrid, the quantum processor is used only for selecting configurations before classical CI and perturbative energy evaluation on HPC (Weaving et al., 2 Sep 2025). In the CIM framework, quantum evolution is used to estimate sampling probabilities $a\notin V$6, while subspace growth and diagonalization remain classical (Graves et al., 13 Mar 2026). The hybridization is therefore centered on subspace construction, not on replacing classical eigensolvers.

Another misconception is that QSHCI is already a standardized algorithm. The evidence does not support that view. The term is absent from the foundational HCI and SHCI papers (Holmes et al., 2016, Sharma et al., 2016), and later work uses it in at least two distinct ways: as a quantum-statistics-plus-heat-bath hybrid in second quantization and as a quantum-informed heat-bath acceptance rule in the CIM representation (Weaving et al., 2 Sep 2025, Graves et al., 13 Mar 2026). This suggests an active phase of methodological definition rather than terminological closure.

Several limitations recur across the literature. The QSCI-based approach depends on the quality of the reference state; if $a\notin V$7 is too far from the true ground state, short-time dynamics may miss important channels, though multi-reference seeding can mitigate this (Weaving et al., 2 Sep 2025). Hardware noise and finite-shot sampling can blur correlators or broaden the empirical configuration distribution $a\notin V$8; selection is more forgiving than direct energy estimation, but noisy sampling can still admit less efficient determinants (Weaving et al., 2 Sep 2025, Graves et al., 13 Mar 2026). In the CIM-QSHCI framework, the preprocessing cost for constructing the CI matrix and performing the FWHT is

$a\notin V$9

which is identified as a current drawback (Graves et al., 13 Mar 2026). The single-bit flip mitigation handles only single-bit errors and does not address more general noise channels (Graves et al., 13 Mar 2026).

Open directions are correspondingly clear. The CIM work identifies efficient CI-matrix access models and improved stochastic-Trotter implementations as natural routes to reducing classical preprocessing overhead (Graves et al., 13 Mar 2026). The QSCI–GQE work points toward learned circuit policies that expose compact determinant sets with much lower gate counts than time-evolution baselines, suggesting that future QSHCI algorithms may combine policy-learned global exploration with HCI-style local completion and SHCI-style perturbative tails (Kemmoku et al., 10 Apr 2026). A plausible implication is that the central research question is no longer whether quantum-selected subspaces can be compact, but how best to combine that compactness with the deterministic robustness and perturbative efficiency of HCI and SHCI.