{"id":"5d1937f8-b303-489b-9b69-77d9d72ac693","arxiv_id":"2411.14096","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A hybrid fermionic/bosonic qubit encoding splits molecular orbitals into fully resolved spin-orbitals and spin-paired hard-core bosons, reducing quantum resources with a tunable energy error.","lead":"This paper introduces a qubit encoding for quantum chemistry that treats some molecular orbitals as fermions and others as paired-electron 'hard-core bosons', and derives the interaction terms between the two subspaces. The hybrid scheme cuts qubit count, circuit depth and measurement terms, at a tunable cost in wavefunction flexibility.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The exactness of the hybrid decomposition rests on the unproven completeness and sign conventions of Eqs. 13-17; a normal-ordering reduction of the density couplings suggests Eq. 17's coefficient is not the projected Hamiltonian's, so the central claim is not yet established.","rationale":"The paper's distinctive contribution is the interaction Hamiltonian H_I, since H_F and H_B are standard. The reader's CONDITIONAL verdict is appropriate: the construction is plausible and the numerical program is coherent, but the exactness claim is not backed by a derivation or public artifacts. My stress-test narrows the condition: the issue is not only the absence of an exhaustiveness proof; a concrete normal-ordering reduction suggests that the printed summary Eq. (17) does not follow from Eqs. (14)-(15) and does not equal the projected fermionic Hamiltonian. If the proposed matrix-element test confirms this, the statement that Eqs. (13)-(17) exactly represent the projected Hamiltonian fails as written, and corrected coefficients or a corrected derivation are required. The numerical hybrid-FCI benchmarks may still be meaningful if they were computed by direct projection of the full Hamiltonian rather than from Eq. (17), but the paper does not say which is the case, and no code or data are shipped. The resource-reduction scalings would likely survive a coefficient correction, so the concern is about correctness of the central equivalence rather than about the overall framework. This is a standard bookkeeping hazard in fermionic Wick reductions and is not a reflection on the authors' intent.","tokens_in":12371,"tokens_out":46543,"duration_ms":417603,"concrete_test":"Implement the projection exactly: build the full second-quantized Hamiltonian of Eq. (1) for a small molecule (e.g., H2/6-31G or LiH/STO-3G), enumerate all determinants in which every B orbital has occupation 0 or 2, and form the projected matrix ⟨Φ|P H P|Ψ⟩. Independently construct H_hybrid from Eqs. (13)-(17) with the stated b†_i = a†_{i↓}a†_{i↑} convention, and compare all matrix elements on the allowed subspace. Any nonzero difference isolates a missing or mis-signed interaction term. A symbolic version for one B orbital and two F orbitals, obtained by normal-ordering Eq. (1), settles whether the coefficient in Eq. (17) should be 2g^{il}_{ki} - g^{il}_{ik} rather than the printed combination.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that H_ee = H_F + H_B + H_I with Eqs. 13-17 exactly representing the projection of Eq. (1) onto the subspace where each B orbital has occupation 0 or 2. The paper states, before Eq. (13), that 'only the terms that lead to the allowed occupations ... should be considered,' but gives no exhaustive enumeration and no sign proof. More concretely, a direct normal-ordering reduction of the density couplings indicates a sign/count problem. For one B orbital i and F orbitals k,l, the four orderings of a†_k, a†_i and a_l, a_i appearing in Eq. (1) reduce, after the 1/2 prefactor and with the stated b†_i = a†_{i↓}a†_{i↑} convention, to (2g^{il}_{ki} - g^{il}_{ik}) N_i a†_{kσ} a_{lσ} on the allowed subspace, using n_{i↑}+n_{i↓}=2N_i and b†_i b_i = n_{i↑}n_{i↓}=N_i. Eq. (17) as printed contains what appears to be a duplicated index term, and combining Eqs. (14a)-(15b) as written gives a coefficient that is not equal to this reduced value; it is twice it when the standard integral symmetries are applied, or has the exchange sign reversed. Every later resource count, circuit compilation, and the stated exactness of Eq. (10) inherit H_I, so this gap is load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a hybrid second-quantized encoding that splits molecular spatial orbitals into a fermionic set F, treated with standard fermionic encodings such as Jordan-Wigner, and a bosonic set B, treated as hard-core bosons with occupation restricted to zero or two. The central claim is that the electronic Hamiltonian can be exactly decomposed as H_ee = H_F + H_B + H_I, with H_I comprising pairwise interactions between the two subspaces (Eq. 10). The authors provide explicit interaction terms (Eqs. 13-17), analyze Pauli-term and measurement-group reductions (Fig. 2), give circuit compilations for paired and semi-paired excitations (Figs. 3-5), and benchmark the encoding against classical FCI and CCSD(T) for small molecules (Figs. 6-8), also presenting an ADAPT-based heuristic for choosing which orbitals to treat as fermionic.","tokens_in":12679,"tokens_out":46299,"duration_ms":427049,"significance":"If the decomposition in Eq. (10) is exact on the allowed subspace, the paper offers a practically useful framework: qubit counts reduce from 2N_MO to 2|F|+|B|, Hamiltonian terms reduce from O(N^4) to O(|F|^4+|B|^2+|F|^2|B|), and the circuit compilations in Sec. IV show depth reductions by more than a factor of two for representative paired excitations. The paper is also concrete: it uses a specific integral convention, reports Pauli-term counts for a realistic molecule, and makes its numerical benchmarks reproducible in principle through the tequila ecosystem. The observation that the hybrid encoding outperforms active-space truncation at the same resolved-orbital fraction is consistent with the fact that the former Hilbert space contains the latter, so the numerical comparison is informative but not surprising. However, the exactness of H_I is the load-bearing premise, and the derivation of Eqs. (14a)-(17) is not sufficiently supported; in fact, a direct normal-ordering reduction yields a different coefficient, which invalidates the stated exactness and requires that all numerical claims be re-examined.","major_comments":[{"comment":"The central claim that Eqs. (13)-(17) exactly represent the projection of Eq. (1) onto the allowed subspace is not established by exhaustive enumeration, and a direct calculation indicates a concrete error. For a bosonic orbital i and fermionic orbitals k,l, the four index assignments in Eq. (1) that preserve the 0-or-2 occupation restriction are (i,j,k,l) = (i,k,l,i), (k,i,l,i), (i,k,i,l), and (k,i,i,l). Reducing these with b†_i = a†_{i↓}a†_{i↑} and the subspace condition n_{i↑}=n_{i↓}, and including the 1/2 prefactor, gives, per spin channel, a projected density-coupling coefficient of (2g^{il}_{ki} - g^{il}_{ik}) b†_i b_i a†_{kσ}a_{lσ}. Combining Eqs. (14a)-(15b) as written and applying the standard eightfold symmetry of real-orbital two-electron integrals yields (2g^{li}_{ik}+2g^{il}_{ki}-g^{li}_{ki}-g^{il}_{ik}) = 2(2g^{il}_{ki}-g^{il}_{ik}), i.e., twice the correct projected value; if integral symmetries are not applied, the exchange sign is reversed. Since every resource count, circuit compilation, and numerical hybrid-FCI energy inherits H_I, this discrepancy is load-bearing for the paper's exactness claim and for the quantitative results.","section":"III, Eqs. (16b)-(17)"},{"comment":"The paper defines N_i = n_{i↑}+n_{i↓} in Eq. (16b) and immediately states that N_i can be rewritten as b†_i b_i, but these operators are not equal: on the allowed subspace b†_i b_i = n_{i↑}n_{i↓} equals 0 or 1, while n_{i↑}+n_{i↓} equals 0 or 2. This factor-of-two ambiguity appears to propagate directly into Eq. (17), where the occupation operator multiplies the interaction coefficients. The manuscript must adopt one consistent definition—preferably b†_i b_i for the pair occupation—and re-derive Eqs. (14a)-(17) under it.","section":"III, Eq. (16b)"},{"comment":"The numerical section compares 'hybrid-FCI' energies with full FCI and active-space FCI, but it never validates that the Hamiltonian used in the hybrid-FCI calculation actually matches the projection of Eq. (1) onto the allowed subspace. Given the derivation gap in H_I, the paper should include, for at least one small molecule, an explicit check of the spectrum or ground-state energy of H_F+H_B+H_I against a full fermionic FCI calculation restricted to the same hybrid subspace, in addition to the comparisons against energies in the full Hilbert space. This check should be performed after the interaction coefficients in Eqs. (14a)-(17) are corrected.","section":"V.A, Fig. 6"}],"minor_comments":[{"comment":"The hard-core-boson Hamiltonian in Eq. (8) has ambiguous operator ordering: writing b_i b†_j and b†_i b_i b†_j b_j without specifying the normal ordering of the hard-core bosons is confusing, since b_i b†_j and b†_i b_j differ by a Kronecker-delta term. Please specify the intended ordering and how it follows from the projection of Eq. (1).","section":"II.C, Eq. (8)"},{"comment":"The integral in Eq. (15b) is printed as g^{ij}_{ik}; this seems to be a typo, as the indices do not match the pattern of the surrounding terms. It should presumably be g^{il}_{ik} or g^{li}_{ik}, and the correct expression must be fixed before the signs in Eq. (17) can be checked.","section":"III, Eq. (15b)"},{"comment":"Equation (17) contains 'N†_i' where the occupation operator is meant; it should be N_i or b†_i b_i. Also, N_i is introduced after Eq. (17) in the text, making the equation hard to parse; please reorder or add a forward reference.","section":"III, Eq. (17)"},{"comment":"There are several typographical and wording issues: 'anty-symmetry' in Sec. III, 'Hamiltomian' in Sec. III.A, 'deepness' in Sec. V.A, 'encoeded' in the caption of Fig. 4, and 'loosing' in the Introduction. A careful proofread would improve clarity.","section":"Throughout"},{"comment":"The definition of the 'fraction x' used in Fig. 6 is not precisely stated: it should specify whether x is the fraction of orbitals treated as fermionic, the fraction of resolved spin-orbitals, or the fraction of the active space, and how the active-space comparison is matched to the same resolved-orbital fraction. The caption should define the symbols used in the legend.","section":"V.A, Fig. 6"}],"recommendation":"major_revision","confidential_remarks":"The factor-of-two issue in Eq. (17) is serious and affects the central exactness claim, but it appears to be a concrete, correctable error rather than a fundamental flaw in the hybrid-encoding idea. If the authors re-derive the interaction terms carefully and confirm the corrected Hamiltonian against full FCI in the restricted subspace, the paper could become a useful contribution. If the coefficient discrepancy persists after correction, the numerical results in Figs. 6-8 would need to be recomputed, and the paper's claims would be substantially weakened."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper proposes a hybrid fermionic/bosonic encoding for molecular Hamiltonians: some spatial orbitals are treated with JW fermions, others as hard-core bosons restricted to zero or double occupation. The split is intuitive, and the resource analysis is concrete: qubit counts, Pauli-term counts, and circuit compilations are all quantified on butadiene and small molecules. The numerical comparison against active-space truncation is honest, and the adaptive-encoding section is a nice practical touch.\n\nThe main thing to know: the central decomposition H_ee = H_F + H_B + H_I (Eq. 10) is the load-bearing claim, and the interaction terms in Eqs. 13-17 are asserted without a proof of completeness or sign correctness. The paper says 'only the terms that lead to the allowed occupations should be considered' but does not enumerate them. On reading, I think the stress-test note is right to worry: a normal-ordering reduction of the density couplings suggests Eq. 17's coefficient is off by a factor of two (or has the exchange sign reversed), and the printed 'N†_i' in Eq. 17 looks like a typo. There is also an ambiguous operator ordering in the HCB Hamiltonian in Eq. 8 and a factor-of-two slip in equating N_i with b†_i b_i. These are exactly the parts that need to be airtight because every resource count and circuit compilation inherits them.\n\nWhat the paper does well: the resource reduction trends are plausible even if the exact constant factors shift, the orbital-selection heuristics (PNOs, SPA-optimized orbitals) are sensible, and the numerical experiments are small but done with standard tools (pyscf, tequila, openfermion). The 'hybrid error beats active space' result is a subspace-inclusion consequence rather than an empirical surprise, but the paper does not oversell it.\n\nSoft spots in proportion: the interaction-term derivation is the big one. There is no proof that the listed terms exhaust all allowed couplings, no numerical verification of H_I against the exact projected Hamiltonian, and no shipped code or data to check the arithmetic. The circuit and measurement reductions are specific to the JW/HCB pairing, so they are illustrative rather than general.\n\nBottom line: this is a useful method paper with a load-bearing gap. It deserves a serious referee, but only if the authors can supply a rigorous derivation of H_I and verify it numerically on a few small systems. As it stands, the central exactness claim is not yet established. I would not cite it until that is fixed.\n\nRecommendation: send to peer review with a request for a major revision focusing on the derivation and verification of the interaction terms.","headline":"Useful hybrid fermionic/bosonic encoding idea, but the central interaction Hamiltonian is asserted without proof and likely has a sign/factor error in Eq. 17; worth a rigorous revision.","tokens_in":13202,"tokens_out":14866,"would_cite":false,"duration_ms":116045,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.67.Ac"],"model":"deepseek-v4-flash","headline":"By fixing a spin-ordering sign convention, the paper derives an exact three-term Hamiltonian that splits molecular orbitals into single-electron and electron-paired registers, cutting qubits, Hamiltonian terms, and circuit depth.","keywords":["hybrid qubit encoding","fermion-to-qubit mapping","hard-core bosons","Fock space splitting","quantum chemistry simulation","variational quantum eigensolver","paired-electron approximation","Pauli-string measurement grouping"],"falsifier":"On a small system (for example two fermionic and two bosonic spatial orbitals), build the full fermionic Hamiltonian of Eq. (1), enumerate every two-body term whose indices mix $F$ and $B$, and verify two things: (i) the subspace of allowed occupation states is invariant under the hybrid Hamiltonian built from Eqs. (10)–(17), and (ii) each matrix element of the hybrid Hamiltonian equals the corresponding matrix element of the full fermionic Hamiltonian on that subspace. Any term that maps an allowed state to an allowed state but is missing from Eqs. (13)–(17), or any sign discrepancy, would falsify the claim that the hybrid Hamiltonian is the exact projected Hamiltonian, since the hybrid-FCI energies of Section V are computed by zeroing all integrals not allowed by the encoding.","tokens_in":12136,"feed_emoji":"⚛️","tokens_out":14015,"duration_ms":118213,"temperature":0.7,"pith_summary":"The paper aims to establish that the electronic structure Hamiltonian can be rewritten in a hybrid form that splits the molecular orbital set into two groups: a fermionic group where each spin-orbital is individually resolved, and a bosonic group where each spatial orbital is restricted to zero or two electrons and encoded as a hard-core boson. The central claim is that this splitting is exact, meaning the full Hamiltonian decomposes into three terms, $H_{\\mathrm{ee}} = H_F + H_B + H_I$, with the interaction piece $H_I$ (Eqs. 13–17) collecting precisely those couplings that respect the occupation restriction on the bosonic group. If the claim is right, the payoff is concrete: qubits drop from $2N_{\\mathrm{MO}}$ to $2|F|+|B|$, Hamiltonian terms drop from $O(N^4)$ to $O(|F|^4+|B|^2+|F|^2|B|)$, and compiled two-electron excitation circuits shrink by more than a factor of two. Numerical results on CH$_4$ and ethene show the hybrid energy error is smaller than active-space truncation at the same fraction of restricted orbitals, which is why a sympathetic reader would care: near-term quantum chemistry is bottlenecked by qubit count, circuit depth, and measurements, and this encoding turns how many orbitals to resolve into a tunable knob instead of an all-or-nothing choice.","feed_headline":"Hybrid encoding halves qubits and circuit depth for quantum chemistry","feed_subtitle":"Pairing electrons as bosons on selected orbitals cuts resources and beats active-space truncation in tests.","key_machinery":"The load-bearing object is the three-term decomposition $H_{\\mathrm{ee}} = H_F + H_B + H_I$ of the electronic Hamiltonian, where $H_F$ is the usual fermionic Hamiltonian on the $F$ orbitals, $H_B$ is the hard-core-boson Hamiltonian of Eq. (8) on the $B$ orbitals, and $H_I$ is the interaction Hamiltonian built from Eqs. (13)–(17). Two identities carry the argument. First, the bosonic operator is defined with a fixed spin ordering, $b_i \\sim a_{i\\uparrow}a_{i\\downarrow}$, so that the exchange phase between two fermionic strings is no longer visible and must be reintroduced by the sign convention $s_{\\sigma_k,\\sigma_l} = \\sigma_k - \\sigma_l$ with $\\sigma \\in \\{\\pm\\tfrac12\\}$; this is what makes the interaction terms exactly equivalent to their fermionic counterparts. Second, because the two operator sets act on disjoint subspaces, the Jordan-Wigner parity string of Eq. (6) only needs to run over the fermionic qubits, which is what breaks the $O(N^4)$ term count and the linear qubit count. The interaction terms themselves are organized by the allowed transitions of Fig. 1: a boson swapping into two opposite-spin fermions (Eq. 13), Coulomb repulsion between a boson and two fermions of matched spin (Eq. 14), and exchange between a boson and one fermion (Eq. 15), compressed into Eq. (17) using the occupation operators $n_{i\\sigma}$ and $N_i$.","core_discovery":"On its own terms, this paper's discovery is a constructive recipe for partitioning Fock space: choose any subset of spatial orbitals to keep fully fermionic (indices in $F$) and encode the rest (indices in $B$) as hard-core bosons that are either empty or doubly occupied; then the electronic Hamiltonian restricted to the allowed occupation manifold is exactly $H_F + H_B + H_I$, not an uncontrolled truncation. The new content is the interaction Hamiltonian $H_I$: three families of terms, namely density-type (Eq. 13), Coulomb-type (Eq. 14), and exchange-type (Eq. 15) couplings between the subspaces, which compress into the compact population-operator form of Eq. 17. The paper argues these are all the couplings that map allowed states to allowed states, with the sign factor $s_{\\sigma_k,\\sigma_l} = \\sigma_k - \\sigma_l$ restoring the anticommutation phase that is hidden when a pair of fermionic operators is replaced by the bosonic operator $b^\\dagger_i$. The authors then demonstrate the consequences: a factor-of-two-plus reduction in compiled circuit depth for semi-paired excitations, monotonic decreases in Pauli-term counts and measurement groups as orbitals are bosonized (Butadiene, 22 orbitals), hybrid-FCI errors below active-space errors for CH$_4$ and ethene, an ADAPT-style procedure that automatically assigns orbitals to the two registers for BeH$_2$, and SPA-based circuit constructions for butadiene that recover a large part of the correlation energy at reduced depth.","pith_inferences":["The splitting is encoding-agnostic on the fermionic side: because $H_F$ and $H_I$ are written in second-quantized fermionic operators before any mapping, the same three-term structure should survive if the fermionic register uses a different standard fermionic encoding, with the bosonic register's simplicity and the sign convention unchanged.","The results suggest a practical orbital-selection rule that the paper leaves implicit: resolve orbitals in decreasing order of pair-natural occupation until the hybrid error falls below a chosen threshold; the CH$_4$ and ethene curves give a first quantitative calibration of that trade-off, but a systematic multi-molecule threshold study would be the natural follow-up.","The sign convention $s_{\\sigma_k,\\sigma_l} = \\sigma_k - \\sigma_l$ is a general phase-bookkeeping trick for any encoding that condenses fermion pairs into composite bosons, so the same machinery could extend to geminal-style or number-preserving pair ansatze beyond hard-core bosons.","Because the bosonic register's Pauli strings remain of the all-Z/XX/YY type that commute within groups, the hybrid scheme inherits the measurement-grouping advantage of pure hard-core bosons on the bosonic qubits; hardware with heterogeneous qubit connectivity could therefore assign the two registers to different topologies."],"forward_implications":["Qubit requirements drop from $2N_{\\mathrm{MO}}$ to $2|F|+|B|$ for $N_{\\mathrm{MO}}$ spatial orbitals, interpolating between full Jordan-Wigner (all fermionic) and full hard-core-boson (all paired) encodings.","The number of Pauli strings and of commuting measurement groups (using Sorted-Insertion grouping with both fully-commuting and qubit-wise-commuting conditions) falls monotonically as orbitals move from $F$ to $B$; for Butadiene with 22 orbitals the interaction part $H_I$ matters only near the fully bosonic regime.","Compilation of a two-electron semi-paired excitation in the hybrid encoding more than halves the circuit depth and resource count compared with full Jordan-Wigner, both for direct Pauli-string compilation and for the optimized decomposition; fully paired double excitations compile to near-trivially small circuits in the bosonic register.","Hybrid-FCI, defined by zeroing all integrals disallowed by the encoding, gives smaller energy errors than active-space truncation at the same percentage of resolved orbitals for CH$_4$ (16 pair-natural orbitals) and ethene (18), with further gains when the orbitals are re-optimized for the separable-pair approximation.","Orbital assignment can be automated: the ADAPT-based procedure selects as fermionic the orbitals that single or unpaired excitations act on, and for BeH$_2$ it reproduces the physical picture that dissociation requires the $\\sigma$ and $\\pi$ orbitals to be fermionic while the perpendicular $p_{x/y}$ orbitals stay bosonic."],"supporting_citations":[{"why":"Jordan-Wigner transformation, the fermionic encoding whose parity Z-strings the hybrid scheme restricts to the fermionic register.","marker":"[5]"},{"why":"the paired-electron hard-core-boson approximation that defines the bosonic register's zero-or-two occupation restriction.","marker":"[14]"},{"why":"source of the hard-core-boson Hamiltonian of Eq. (8) used for the bosonic subspace in the hybrid decomposition.","marker":"[23]"},{"why":"Sorted-Insertion grouping algorithm used to count commuting Pauli-string groups and quantify the measurement-resource reduction.","marker":"[30]"},{"why":"the optimized qubit-excitation decomposition used to compile the circuits whose depth is compared between Jordan-Wigner and hybrid encodings.","marker":"[36]"},{"why":"supplies the directly determined pair-natural-orbital bases for the CH4 and ethene hybrid-FCI error calculations.","marker":"[37]"},{"why":"the separable-pair-approximation wavefunction used for orbital re-optimization and as the base circuit for the applications.","marker":"[38]"},{"why":"the SPA+ circuit architecture whose fermionic core is extended with bosonic registers in the circuit-design demonstrations.","marker":"[42]"}],"fun_headline_variants":["Exact hybrid encoding splits Fock space into fermion and boson halves","Fermion-boson hybrid encoding reduces circuit depth and Pauli counts","New hybrid encoding: exact fermionic-bosonic split for quantum chemistry","Hybrid encoding: split Fock space into fermionic and bosonic registers"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire construction rests on the assumption that Eqs. (13)–(17) list every Hamiltonian coupling that maps an allowed state (zero or two electrons on each bosonic orbital) to another allowed state, with the correct signs: the paper states that only allowed-occupation terms should be kept, but it gives no exhaustive enumeration and no independent check of the sign conventions against a full fermionic calculation.","fun_headline_variants_meta":{"raw":{"variants":["Exact hybrid encoding splits Fock space into fermion and boson halves","Fermion-boson hybrid encoding reduces circuit depth and Pauli counts","New hybrid encoding: exact fermionic-bosonic split for quantum chemistry","Hybrid encoding: split Fock space into fermionic and bosonic registers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00057,"raw_usage":{"total_tokens":2698,"prompt_tokens":948,"completion_tokens":1750,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":564,"completion_tokens_details":{"reasoning_tokens":1668}},"tokens_in":564,"tokens_out":1750,"duration_ms":12814,"temperature":1.0,"reasoning_tokens":1668,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:32:58.266802+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a small system (for example two fermionic and two bosonic spatial orbitals), build the full fermionic Hamiltonian of Eq. (1), enumerate every two-body term whose indices mix $F$ and $B$, and verify two things: (i) the subspace of allowed occupation states is invariant under the hybrid Hamiltonian built from Eqs. (10)–(17), and (ii) each matrix element of the hybrid Hamiltonian equals the corresponding matrix element of the full fermionic Hamiltonian on that subspace. Any term that maps an allowed state to an allowed state but is missing from Eqs. (13)–(17), or any sign discrepancy, would falsify the claim that the hybrid Hamiltonian is the exact projected Hamiltonian, since the hybrid-FCI energies of Section V are computed by zeroing all integrals not allowed by the encoding.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the paired-electron hard-core-boson approximation that defines the bosonic register's zero-or-two occupation restriction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"source of the hard-core-boson Hamiltonian of Eq. (8) used for the bosonic subspace in the hybrid decomposition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the optimized qubit-excitation decomposition used to compile the circuits whose depth is compared between Jordan-Wigner and hybrid encodings."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the directly determined pair-natural-orbital bases for the CH4 and ethene hybrid-FCI error calculations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the SPA+ circuit architecture whose fermionic core is extended with bosonic registers in the circuit-design demonstrations."}],"review_version":1}