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REVIEW 2 major objections 5 minor 70 references

A hybrid approach to extending selected configuration interaction and full configuration interaction quantum Monte Carlo

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper shows that fixing the initiator space in i-FCIQMC to a selected space from SCI lets the simulation sample the first-order interacting space with a walker population only a small multiple of |V|, and that rejected spawnings…

desk verdict Solid, honest algorithmic paper that deserves serious refereeing; the PT2 estimator's derivation has a minor hidden approximation, but the benchmarks support its use. read the letter →

arxiv 1908.04158 v2 pith:7LV57KN4 submitted 2019-08-12 physics.chem-ph cond-mat.str-el

Nick S. Blunt This is my paper · ORCID
classification physics.chem-phcond-mat.str-el
keywords i-FCIQMC(SCI)selectedconfigurationinteractionheat-bathCIPT2correctionpopulationplateaufermionsignproblemfirst-orderinteractingspaceorbitaloptimization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

i-FCIQMC(SCI) combines selected configuration interaction with initiator full configuration interaction quantum Monte Carlo by taking the selected space V from a prior SCI calculation as the fixed initiator space. Because initiators may spawn to any connected determinant, the simulation effectively performs a variational calculation in V plus its first-order interacting space (FOIS), even though the FOIS may be orders of magnitude larger than V. The paper shows that the population plateau, the sign-problem measure that sets the minimum walker memory, is typically only a small multiple of |V|. Rejected spawnings further supply a PT2 correction in the second-order interacting space, which brings weakly correlated systems to near-exact energies. A sympathetic reader would care because this yields SCI+PT2-like accuracy with rigorously variational energies and available reduced density matrices, at a memory cost set by V rather than by the full space sampled.

What carries the argument

The load-bearing object is the fixed initiator space V taken from a prior SCI calculation (here heat-bath CI), together with the initiator spawning rule that walkers in V may spawn to any connected determinant while non-initiators may spawn only to occupied determinants or to V. This construction makes the sampled space V plus its first-order interacting space without ever storing the FOIS vector. The other central object is the population plateau: its height measures how severe the fermion sign problem is, and the paper shows this plateau is a small multiple of |V| under semi-stochastic propagation and optimized orbitals. Finally, the PT2 estimator is assembled from spawnings rejected by the initiator rule, using an equilibrium relation C_i = -S_i/(Δτ(E-H_ii)) to cancel allowed-spawning terms, so the correction is obtained almost for free from data already present in the simulation.

What would settle it

Take a small active space where the H_B matrix in V+FOIS can be diagonalized exactly, run i-FCIQMC(SCI), and record for each non-rejected determinant the difference C_i + S_i/(Δτ(E-H_ii)) over equilibrated iterations; if this difference does not average to zero within statistical error while the variational energy is converged, the PT2 estimator is biased.

Watch

Extended reading notes

Core claim

The central claim is that the initiator approximation in i-FCIQMC can be improved by fixing the initiator space to equal the selected space V of a prior SCI calculation instead of letting population thresholds pick initiators dynamically. In this hybrid method, the Hamiltonian acting on the sampled wave function interpolates between H_A (diagonal in the first-order interacting space) and H_B (the full V+FOIS Hamiltonian), so that as the walker population grows, the wave function converges toward the ground state of V+FOIS. The central numerical discovery is that the walker population needed to sample the FOIS accurately is, in every system studied, only a small factor of |V| itself, even when the FOIS is larger by several orders of magnitude; from those walkers, rejected spawning events give a near-free PT2 correction that removes most of the remaining error in weakly correlated systems. This makes i-FCIQMC(SCI)+PT2 a complementary alternative to SCI+PT2: more accurate for a common selected/initiator space, albeit slower, and able to supply variational energy estimates and reduced density matrices in the larger space.

Load-bearing premise

In Section IID, the PT2 derivation assumes that at equilibrium the amplitude C_i and total spawning S_i on every non-rejected determinant satisfy C_i = -S_i/(Δτ(E-H_ii)), so the allowed-spawning terms in Eq. (27) cancel on average; the paper gives no error bound for this cancellation, so a systematic failure would bias the PT2 energies even when variational energies are accurate.

Editorial extensions

If this is right

  • For any system where an SCI calculation can produce a selected space V, i-FCIQMC(SCI) can recover most of the correlation energy in V plus its first-order interacting space with memory proportional to a small multiple of |V|, not of the much larger FOIS.
  • The same selected space gives variational energies comparable to, and sometimes lower than, SCI+PT2 energies, with a rigorous variational bound and with exact reduced density matrices available for the V+FOIS wave function.
  • The rejected-spawn PT2 estimator brings weakly correlated systems to near-exact total energies at selected-space sizes where SCI+PT2 still has residual error.
  • The method extends to large active spaces and to basis sets up to quadruple-zeta, with plateau height growing only modestly with basis cardinal number when optimized orbitals and the semi-stochastic adaptation are used.
  • Using a fixed initiator space systematically lowers variational energies relative to standard i-FCIQMC for the same walker population, showing that less severe Hamiltonian truncation is possible while keeping the sign problem manageable.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: the plateau-to-|V| scaling suggests a design rule: choose the initiator space as the top of a deterministic importance ranking, and the stochastic sampler will populate its connections at a cost set by the ranked space rather than by the connection space; this could generalize to other projector QMC formulations.
  • Inference: if the replica-trick statistical error on the PT2 correction is reduced by a single-replica estimator, the method's practical advantage over SCI+PT2 would grow, since the current bottleneck is noise on the correction, not the variational energy.
  • Inference: since the paper tests only heat-bath CI as the SCI generator, comparing CIPSI, ASCI, or other selection criteria would show whether the favorable plateau scaling is a property of the fixed-initiator construction itself or of the particular importance metric.
  • Inference: the interpolation between H_A and H_B implies a sharp testable prediction: variational energies should improve monotonically with walker population toward the H_B ground state, and deviations from that trend would signal that the initiator rules are not the only source of bias.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proposes i-FCIQMC(SCI), a hybrid of selected CI and initiator FCIQMC in which the initiator space is fixed to be the SCI-selected space V rather than chosen by a population threshold. Initiators in V may spawn freely into the first-order interacting space (FOIS), so the simulation samples a variational energy in V ⊕ FOIS, while rejected spawns are used to construct a second-order perturbative correction. The manuscript derives the PT2 estimator, studies population plateaus as a function of basis set and orbital choice, and compares energies against standard i-FCIQMC and SHCI/SCI+PT2 for formamide, hexacene, butadiene, BPEA, and the water dimer. The central conclusion is that the walker population required to sample the FOIS is only a small multiple of |V|, and that i-FCIQMC(SCI)+PT2 provides near-exact energies for weakly correlated systems at memory cost comparable to SCI+PT2.

Significance. Should the claims hold, the method is a practically useful synthesis of SCI and FCIQMC: it offers a variational energy in V ⊕ FOIS without storing that space explicitly, a nearly-free PT2 correction built from rejected spawns, and a reduced need for extrapolation relative to SCI+PT2. The paper is strong on transparency: the PT2 estimator is derived from the Guo-Sharma effective-Hamiltonian expression, the interpolation between the two truncated Hamiltonians HA and HB is clearly formulated, and the benchmarks use independent DMRG or extrapolated SHCI references. No parameters are fitted to benchmark energies; the only tunable inputs are the SCI threshold and the walker population. The main weakness is the unproven average cancellation in the derivation of Eq. (30), which is the basis of all +PT2 results.

major comments (2)
  1. [Section II D, Eqs. (27)–(30)] The cancellation of the allowed-spawning terms is asserted ‘on average’ via the equilibrium relation (28), but in i-FCIQMC(SCI) the coefficients are produced by a Hamiltonian truncation that depends on the instantaneous set of occupied determinants, so (28) is not an exact property of the full-H eigenvector and the error in the cancellation is uncontrolled. Since Eq. (30) is the basis of every ‘+PT2’ result in Section III, please add a direct numerical verification, for example comparing the full expression (26) with the rejected-only estimator (30) from the same simulation and showing that the difference is within statistical noise, or provide a rigorous bound on the neglected terms. The empirical agreement with DMRG does not isolate this step, because the PT2 contribution is small in the strongly correlated cases where it is least likely to mask a bias.
  2. [Section III B, Table I] The conclusion that the walker population needed to sample the FOIS is ‘only a small factor of the size of V itself’ is not uniformly supported by the data: Table I reports plateau/NV = 34.7 for the water dimer with HF orbitals and no semi-stochastic adaptation. Please qualify the claim to the regime in which optimized orbitals and/or the semi-stochastic adaptation are used, or explicitly justify why a factor of roughly 35 is considered small in the context of the much larger FOIS.
minor comments (5)
  1. [Section III A] The system list contains a typo: ‘actone’ should be ‘acetone’.
  2. [Figure 1 caption] The label ‘W alker pop.’ contains an extraneous space and should read ‘Walker pop.’.
  3. [Section III D] The sentence ‘these results are all exact within statistical errors’ would be more precise as ‘these results are all consistent with the benchmark within statistical errors.’
  4. [Section II D, Eq. (27)] The symbol S_i is used for two different quantities in Eq. (27): for rejected determinants it is the amplitude of rejected spawns, while for allowed determinants it is the accepted spawned amplitude. Please clarify this distinction in the text.
  5. [Section III B, Table I] The statement that the increase in plateau height with basis set cardinal number is ‘only a small factor’ is not immediately evident from the table; for HF orbitals with the semi-stochastic adaptation the ratio changes from 1.7 (cc-pVDZ) to 5.3 (cc-pVTZ). Reporting the ratios explicitly would help the reader assess the claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained; the PT2 estimator is a standard projector-perturbation expression using the sampled variational energy, not a fit to benchmark energies.

full rationale

The paper's central derivation chain is self-contained and does not reduce any claimed prediction to its own inputs. The variational energy estimator in Eq. (15) follows directly from the definition of the spawned-array S_i in Eq. (4) and replica sampling; no benchmark energy or fitted parameter enters. The PT2 correction in Eqs. (16)-(26) is the exact second-order energy for the zeroth-order Hamiltonian H0 = P E0 P + Q H_d Q, with E0 taken as the sampled Evar, and the algebra from Eq. (25) to Eq. (26) is an identity using Eq. (15). The passage from Eq. (27) to Eq. (30) uses the equilibrium relation C_i = -S_i/(Delta_tau (E-H_ii)) for determinants whose incoming spawns are not rejected, and the paper explicitly states this holds only on average and only where Eqs. (2) or (5) apply. Whether this cancellation is accurate is a statistical and numerical correctness question, not circularity: the final estimator is not equal to its input by construction and is tested against independent benchmarks (SHCI and DMRG). The only self-citations (Refs. 40, 43, 51) concern implementation details or a prior statement of the same PT2 estimator, and the paper re-derives the estimator from Guo et al. rather than relying on a self-citation as evidence. The SCI-selected space V is an input, and the comparison with SCI+PT2 uses the same V, which is a fair shared-input comparison rather than a fitted advantage. No pattern of self-definitional prediction, fitted-input renaming, or uniqueness imported from authors is present.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The method introduces no new physical entities and no constants fitted to benchmark energies. The main external inputs are the SCI selection threshold and the walker population, which are convergence parameters chosen per system. The formal derivation relies on standard FCIQMC convergence and perturbation theory, plus the unproven equilibrium relation used for the PT2 estimator.

free parameters (2)
  • SHCI selection threshold epsilon = varied per system; e.g., 1e-4 to 2e-3 Ha for formamide, 6e-5 for BPEA
    Controls the size of V and hence the initiator space. It is chosen by the user, not fitted to output energies.
  • Walker population N_w = roughly 1.5 to 10 times |V|; e.g., 1.8e4 for formamide, up to 1e7 for butadiene
    Chosen to exceed the plateau and to reduce statistical error on the PT2 correction. No target-energy fitting is involved.
assumptions (4)
  • standard math FCIQMC's projector P = 1 - delta_tau (H - E_S) converges to the ground state of the truncated Hamiltonian for sufficiently small delta_tau.
    Used in Sec. II A, Eq. (1). This is a standard result of projector quantum Monte Carlo, not proved in the paper.
  • domain assumption In equilibrium, the relation C_i = -S_i / (delta_tau (E - H_ii)) holds on average for determinants not subject to initiator rejection.
    Invoked in Sec. II D to cancel the allowed-spawning contributions in Eq. (27), leading to the PT2 estimator Eq. (30). Only numerical validation is given, with no proven error bound.
  • domain assumption The SCI selected space V from SHCI contains the most important determinants, so using V as the fixed initiator space gives a good approximation for the available memory.
    This is the basis of the hybrid method in Sec. II C. It is the standard selected-CI assumption and is inherited from SHCI.
  • domain assumption The instantaneous occupation of FOIS determinants is sufficient to make the i-FCIQMC(SCI) effective Hamiltonian close to HB, the full Hamiltonian in V plus the FOIS.
    Sec. II C argues that the method interpolates between HA and HB as the walker population grows. The closeness is demonstrated empirically in Table II and Figs. 2-6, but not proven.

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Cite this review

Pith. "Pith review of A hybrid approach to extending selected configuration interaction and full configuration interaction quantum Monte Carlo." pith.science (2026). https://pith.science/paper/7LV57KN4

@misc{pith2026190804158,
  author       = {Pith},
  title        = {Pith review of: A hybrid approach to extending selected configuration interaction and full configuration interaction quantum Monte Carlo},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7LV57KN4}},
  note         = {Machine review of arXiv:1908.04158}
}
read the original abstract

We present an approach to combining selected configuration interaction (SCI) and initiator full configuration interaction quantum Monte Carlo (i-FCIQMC). In the current i-FCIQMC scheme, the space of initiators is chosen dynamically by a population threshold. Here, we instead choose initiators as the selected space (V) from a prior SCI calculation, allowing substantially larger initiator spaces for a given walker population. While SCI+PT2 adds a perturbative correction in the first-order interacting space (FOIS) beyond V, the approach presented here allows a variational calculation in the same space, and a perturbative correction in the second-order interacting space. The use of a fixed initiator space reintroduces population plateaus into FCIQMC, but it is shown that the plateau height is typically only a small multiple of the size of V. Thus, for a comparable fundamental memory cost to SCI+PT2, a substantially larger space can be sampled. The resulting method can be seen as a complementary approach to SCI+PT2, which is more accurate but slower for a common selected/initiator space. More generally, our results show that approaches exist to significantly improve initiator energies in i-FCIQMC, while still ameliorating the fermion sign problem relative to the original FCIQMC method.

Figures

Figures reproduced from arXiv: 1908.04158 by the authors.

Figure 1
Figure 1. FIG. 1. Example population plateaus in i-FCIQMC(SCI) sim [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Energies for hexacene in the (26e [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 5
Figure 5. FIG. 5. Results for butadiene with the ANO-L [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figures from the paper (1 more)
Figure 6
Figure 6. Figure 6: FIG. 6. Results for BPEA. The active space is the (30e [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

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