{"id":"5100c2b0-cc3d-40bf-888e-834f33a920a3","arxiv_id":"1908.03902","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A quantum algorithm and circuits for constructing one-particle Green's functions of molecules via probabilistic state preparation and statistical sampling, demonstrated in classical simulations for LiH and H2O.","lead":"This paper proposes quantum circuits that let a quantum computer compute the one-particle Green's function, the spectrum of adding or removing an electron from a molecule, by statistical sampling with at most two auxiliary qubits. It is a concrete recipe for extracting quasiparticle spectra, the quantities measured in photoemission experiments, on future quantum hardware.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The UCC trial state is not an eigenstate, so even with ideal QPE the assembled Lehmann GF is not the physical Green's function of the prepared molecular state.","rationale":"The reader correctly identifies the ideal-QPE and known-spectra assumption as a serious gap for real hardware, but I find the non-eigenstate premise more load-bearing because it affects the formal validity of the construction even with perfect QPE and exact energies. The circuits and probability identities in Sec. II B are internally consistent; the sampling histograms do equal the squared overlaps shown in eqs. (25)-(28), and the classical simulations demonstrate convergence of those overlaps. The problem is that eqs. (3)-(4) only represent the physical one-particle Green's function when the input N-electron state is an eigenstate of H. UCC trial states are not eigenstates, so the assembled Lehmann expression is a biased spectral object rather than the true GF of the prepared state. The paper acknowledges the related failure of the Galitskii-Migdal energy identity, but the numerical 'UCC GF' is still presented as a molecular GF and compared directly with FCI spectra. This does not invalidate the general algorithm if exact ground-state preparation is assumed, and the limitation is disclosed, so the appropriate verdict remains CONDITIONAL. I rank the eigenstate issue ahead of QPE noise because it is a correctness issue in the ideal limit, not only an implementation issue; however, the two are related, and the reader's verdict already captures the conditional status.","tokens_in":17860,"tokens_out":13790,"duration_ms":146990,"concrete_test":"For the LiH STO-3G optimized U1 state used in Sec. IV A, diagonalize H in the N-electron sector and compute the projection of |Psi_UCC> onto the FCI ground state. Then reconstruct the exact spectral function from the full Lehmann sum over all N and N+-1 eigenstates and compare it with A_UCC(omega) built from eqs. (3)-(4) using E_N_gs = E_UCC and the same broadening delta = 0.02 a.u. If the difference exceeds the statistical sampling error at N_meas = 32000, the UCC demonstration is not a valid construction of the physical GF; if the difference is negligible for this weak-correlation case, the concern is quantitative rather than structural.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing limitation is not finite QPE precision but the eigenstate premise behind eqs. (3)-(4). The pole expansion G(e)_mm'(z) = sum_lambda B(e)_lambda mm' / (z + E_N_gs - E^{N+1}_lambda) is derived by inserting eigenstates of H and using H|Psi_N_gs> = E_N_gs|Psi_N_gs>. If |Psi_N_gs> is only a UCC trial state, this final identity fails: the exact Lehmann representation of <Psi_UCC| a_m (z + E_ref - H)^{-1} a^dagger_m' |Psi_UCC> contains an extra sum over all N-electron eigenstates |E_k> with overlap <E_k|Psi_UCC>, and there is no canonical E_ref that makes eqs. (3)-(4) exact. The circuit identities (25)-(28) do measure |<E_lambda^{N+1}| a^dagger_m |Psi_UCC>|^2 correctly under ideal QPE, but assigning those sampled values to poles at z + E_UCC - E_lambda is an additional, unvalidated approximation. The paper concedes this in the Galitskii-Migdal paragraph (EGM[G_trial] differs from <H>), and for weakly correlated LiH/H2O the bias may be small. But the central claim is stronger: it presents the sampled histograms as molecular GFs, whereas what is demonstrated is Monte Carlo estimation of overlap squares for a non-eigenstate reference. This affects the meaning of the algorithm, not just hardware precision.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a scheme for constructing the one-particle Green's function (GF) of a molecular system on a quantum computer via statistical sampling. The key idea is to encode creation and annihilation operators as linear combinations of unitaries, so that circuits C_m and C_{mm'} probabilistically prepare electron-added and electron-removed states. After the ancilla measurement, quantum phase estimation (QPE) projects onto (N±1)-electron energy eigenstates, and the histogram frequencies directly give the diagonal and off-diagonal transition matrix elements B^{(e/h)}_{λmm'} that enter the Lehmann representation. The authors demonstrate the sampling procedure in classical simulations for LiH and H2O using unitary coupled-cluster (UCC) trial states, comparing the sampled spectra with spectra 'exact within UCC' and with FCI spectra, and they analyze convergence of Galitskii-Migdal correlation energies as the number of measurements grows.","tokens_in":18178,"tokens_out":12949,"duration_ms":120101,"significance":"If the claims are taken as stated, the paper provides a simple and explicit circuit-level method for the previously missing ingredient in quantum-computing approaches to spectral functions: the measurement of the numerators of the Lehmann representation. The algebraic derivations in Eqs. (8)-(20) are internally consistent, and the probability counting in Eqs. (25)-(28) correctly recovers the transition matrix elements. The classical sampling simulations are honest Monte Carlo experiments with no fitted parameters, and the pseudocodes make the protocol reproducible. The main weakness, discussed below, is that the object computed for a UCC reference state is not the exact Green's function of the prepared state, which limits the physical interpretation of the demonstrated spectra.","major_comments":[{"comment":"The Lehmann representation in Eqs. (3) and (4) is derived under the assumption that |Ψ_N_gs⟩ is an eigenstate of H with energy E_N_gs. The UCC trial states used in Secs. IV.A.1 and IV.B.1 are not eigenstates of H. For a non-eigenstate |Ψ_UCC⟩, the resolvent expression ⟨Ψ_UCC| a_m (z+E_ref-H)^{-1} a†_{m'} |Ψ_UCC⟩ contains an additional sum over all N-electron eigenstates, and no choice of E_ref makes Eqs. (3)-(4) exact. The assembled object is therefore not the one-particle Green's function of the system, but rather a model GF defined by the trial state and the trial energy. The paper acknowledges the associated discrepancy for the Galitskii-Migdal formula in Sec. II.C, but the abstract and Sec. IV present the sampled spectra as the GF of an interacting electronic system and compare them with FCI spectra without a clear statement of this approximation. The authors should revise the claims to distinguish the exact case (exact ground state prepared) from the approximate demonstration (UCC state), and ideally quantify the error of the trial-state GF relative to the exact resolvent GF of |Ψ_UCC⟩.","section":"II.B.1 and IV.A.2"}],"minor_comments":[{"comment":"The paper uses classical FCI eigenvalues for the (N±1)-electron states and assumes ideal QPE. Since the algorithm's output is the transition matrix elements, not the poles, the authors should state more prominently that the complete GF construction requires a separate accurate determination of the (N±1) spectra; the current text says this in Sec. III, but the abstract's phrasing could mislead readers.","section":"III"},{"comment":"The pseudocode in Procedures 2-4 instructs 'Find E among {E^{N±1}_λ}' after QPE. Under ideal precision this is fine, but with finite-precision QPE the eigenvalue estimate will not exactly coincide with a known eigenvalue; a brief discussion of binning and the associated pole-shift error would make the protocol more complete.","section":"II.B.4"},{"comment":"In Eq. (20), the appearance of the phase gate Z(π/4) and the factor e^{iπ/4} in the resulting state is not derived; spelling out the connection between the phase gate and the definitions in Eqs. (13)-(14) would improve readability.","section":"II.B.3"},{"comment":"There are minor typographical issues, e.g., 'ancillae' is used where 'ancillas' is standard, and 'H2O' is used without subscript in some places; these do not affect the technical content.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid proof-of-principle of a measurement scheme for Green's functions on quantum computers. The main issue is the mismatch between the strength of the claims in the abstract/conclusions and the actual object computed for UCC trial states. Once the authors clarify this distinction, the paper is likely acceptable for publication. I would suggest that the referee report focuses on the eigenstate issue, which is the only substantive technical concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper proposes a practical circuit scheme for building the one-particle Green's function of a molecule on a quantum computer. What's genuinely new is the probabilistic state-preparation trick: writing fermionic creation/annihilation operators as combinations of unitaries, then using an ancilla to create and detect the electron-added/removed states, with a two-ancilla extension for off-diagonal weights. The circuit algebra and the probability counting in eqs. (25)–(28) are consistent, and the off-diagonal recovery via the auxiliary operators works; I checked the phases. The classical simulations for LiH and H2O behave as expected, with sampling error scaling as N^{-1/2}.\n\nThe paper is also honest about its limits. It assumes ideal QPE and that the (N±1) spectra are already known (it takes them from FCI). It also states, in the Galitskii–Migdal section, that a GF assembled from a non-eigenstate trial state is not the exact GF. The stress-test note is right that this is the load-bearing caveat: with a UCC trial state, the circuits do measure the correct overlap products, but the Lehmann poles at E_UCC – E_λ^{N±1} are only exact if the reference is an eigenstate. For weakly correlated LiH the error is small, and the paper shows that. But the narrative sometimes slips from 'GF of the trial state under the eigenstate approximation' to 'molecular GF,' and that distinction deserves to be front and center. It's a presentation issue, not a flaw in the core algorithm, provided the input is taken to be an exact ground state.\n\nOther soft spots are minor: the sampling cost for satellite peaks grows quickly (H2O needs tens of thousands of measurements), the circuits are shown for the JW/BK encodings, and the ansatz truncation is crude. These are acknowledged. Self-citations are heavy but relevant.\n\nVerdict: this is a worthwhile algorithm paper for anyone working on quantum-chemistry simulations beyond ground-state energies. The math holds up, the new circuit is real, and the limitations are disclosed. I'd send it to peer review; the main revision request would be to separate the algorithm's claim from the UCC demonstration more cleanly, and to move the eigenstate caveat to the front. Accept after minor revision.","headline":"Solid, honest algorithm paper for molecular Green's functions on a quantum computer; the eigenstate caveat is real but disclosed.","tokens_in":18705,"tokens_out":17926,"would_cite":true,"duration_ms":152482,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68","81V55"],"pacs":["03.67.Ac","31.15.-p"],"model":"deepseek-v4-flash","headline":"This paper claims that a quantum computer can construct the one-particle Green's function of an interacting molecular system through statistical sampling: specially built circuits probabilistically prepare electron-added and…","keywords":["Green's function","quantum phase estimation","statistical sampling","unitary coupled-cluster","probabilistic state preparation","quasiparticle spectra","Galitskii–Migdal formula","molecular quantum chemistry"],"falsifier":"Construct the Green's function for LiH on a device with a deliberately coarse quantum phase estimation register and compare the pole positions of the reconstructed spectrum with the exact FCI spectrum: any shift beyond the known statistical sampling error would falsify the assumption of perfect QPE precision.","tokens_in":17628,"feed_emoji":"⚛️","tokens_out":6009,"duration_ms":58287,"temperature":0.7,"pith_summary":"The paper proposes a way to build the one-particle Green's function of an interacting molecule on a quantum computer without ever preparing a specific ionized state on demand. The non-unitarity of electron creation and annihilation operators normally blocks such preparations, but the paper shows that each operator can be written as a combination of two unitary operations, so a circuit with one or two ancilla qubits prepares the added and removed states probabilistically. After quantum phase estimation on the prepared state, the measured energy histogram equals the Green's function's transition amplitudes. Simulated runs on LiH and H2O using unitary coupled-cluster ground states reproduce the quasiparticle spectra and give correlation energies via the Galitskii–Migdal formula that converge with more measurements. If valid, the scheme yields molecular Green's functions—and with them photoelectron spectra and total energies—directly from qubit measurements.","feed_headline":"Counting qubit outcomes builds molecular Green's functions","feed_subtitle":"Two-ancilla circuits turn electron addition and removal into a histogram whose bins are the transition amplitudes.","key_machinery":"The load-bearing object is the probabilistic state-preparation circuit, built on the Majorana-like decomposition of the fermionic creation and annihilation operators into two unitaries: $a_m^\\dagger = (U_{0m} - U_{1m})/2$ and $a_m = (U_{0m} + U_{1m})/2$, with $U_{0m}$ and $U_{1m}$ the combinations of Pauli operations from the Jordan–Wigner or Bravyi–Kitaev representation. Circuit $C_m$ uses one ancilla and controlled $U_{0m}$, $U_{1m}$ to entangle the ancilla with the register; post-selecting on the ancilla outcome yields the normalized electron-added or electron-removed state. Circuit $C_{mm'}$ uses two ancillas and the phase-shifted auxiliary operators $a^{\\pm}_{mm'} = (a_m \\pm e^{-i\\pi/4} a_{m'})/2$ to reach the off-diagonal amplitudes. These circuits convert the non-unitary action of $a_m^\\dagger$ and $a_m$ into a unitary operation on a larger Hilbert space, which is exactly what a quantum circuit can implement. The subsequent QPE histogram then reads out the transition matrix elements as probabilities.","core_discovery":"On the paper's own terms, the central discovery is that the Lehmann representation of the one-particle Green's function can be evaluated by sampling rather than by computing overlaps. Because $a_m^\\dagger \\pm a_m$ are unitary by the fermionic anticommutation relations, one writes $a_m^\\dagger = (U_{0m} - U_{1m})/2$ and $a_m = (U_{0m} + U_{1m})/2$, and a controlled circuit $C_m$ entangles an ancilla with the register so that measuring the ancilla leaves $|0\\rangle \\otimes a_m|\\psi\\rangle$ or $|1\\rangle \\otimes a_m^\\dagger|\\psi\\rangle$. Feeding the result into quantum phase estimation turns the histogram of measured (ancilla, energy) pairs into the diagonal transition elements $B^{(e)}_{\\lambda mm}$ and $B^{(h)}_{\\lambda mm}$ (eqs. 25–26); the two-ancilla circuit $C_{mm'}$ similarly yields the off-diagonal elements through auxiliary operators $a^{\\pm}_{mm'}$. The full Green's function then follows by inserting these $B$'s and the known $(N \\pm 1)$-electron energies into the Lehmann sums (eqs. 3–4).","pith_inferences":["A natural extension is to replace the ideal QPE with finite-resolution phase estimation; the resulting pole shifts would turn the ideal-precision assumption into a concrete resource estimate for the required energy-register accuracy.","The same probabilistic-preparation trick might be adapted to other non-unitary operations in quantum simulation, such as projecting onto symmetry sectors or implementing imaginary-time evolution steps.","For strongly correlated molecules, where satellite weights are large, the number of measurements needed to resolve them may scale unfavorably; the paper's LiH and H2O data already hint at this, and a formal scaling bound would be a testable follow-up."],"forward_implications":["The same sampling scheme can be applied to any ground state that a quantum computer can prepare, not just UCC states, since the circuits only need the state as input.","The full Green's function, including quasiparticle and satellite peaks, can be reconstructed on a quantum computer using at most two ancilla qubits, regardless of system size.","Total energies can be extracted from the sampled Green's function via the Galitskii–Migdal formula, converging to the expected value as the number of measurements grows.","The scheme converts the problem of computing transition matrix elements into one of counting histogram bins, a purely classical post-processing task."],"supporting_citations":[{"why":"Supplies the Lehmann representation of the one-particle Green's function and the Galitskii–Migdal formula used for total energies.","marker":"[41]"},{"why":"Gives the many-body definition of Green's functions and spectral representations that the scheme evaluates.","marker":"[42]"},{"why":"Provides the Jordan–Wigner and Bravyi–Kitaev mappings from which the unitary operators $U_{0m}$ and $U_{1m}$ are read off.","marker":"[4]"},{"why":"Introduces the unitary coupled-cluster ansatz whose optimized state serves as the input ground state in the simulations.","marker":"[39]"},{"why":"Names the Majorana-fermion viewpoint justifying that $a_m^\\dagger \\pm a_m$ are unitary, the property the circuits exploit.","marker":"[49]"},{"why":"Defines the quantum phase estimation procedure whose output histogram carries the transition amplitudes.","marker":"[3]"}],"fun_headline_variants":["Two-ancilla circuits sample molecular Green's functions","Sampling qubit outcomes builds molecular Green's functions","Quantum scheme constructs Green's functions via ancilla sampling","Ancilla-controlled sampling yields molecular Green's functions","Counting ancilla bits produces molecular Green's functions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The scheme assumes the energy eigenvalues of the $(N \\pm 1)$-electron states are already known exactly and that quantum phase estimation returns them with no error; in addition, the input state is treated as an exact eigenstate of the Hamiltonian, which the UCC trial states used here are not.","fun_headline_variants_meta":{"raw":{"variants":["Two-ancilla circuits sample molecular Green's functions","Sampling qubit outcomes builds molecular Green's functions","Quantum scheme constructs Green's functions via ancilla sampling","Ancilla-controlled sampling yields molecular Green's functions","Counting ancilla bits produces molecular Green's functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000188,"raw_usage":{"total_tokens":1336,"prompt_tokens":950,"completion_tokens":386,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":566,"completion_tokens_details":{"reasoning_tokens":311}},"tokens_in":566,"tokens_out":386,"duration_ms":4400,"temperature":1.0,"reasoning_tokens":311,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:59:25.598286+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct the Green's function for LiH on a device with a deliberately coarse quantum phase estimation register and compare the pole positions of the reconstructed spectrum with the exact FCI spectrum: any shift beyond the known statistical sampling error would falsify the assumption of perfect QPE precision.","supporting_citations":[{"cited_title":"(10) The action of the circuit to the whole system is easily confirmed to be unitary due to the anti-commutation re- lation between the electronic operators","cited_arxiv_id":null,"evidence_quote":"Provides the Jordan–Wigner and Bravyi–Kitaev mappings from which the unitary operators $U_{0m}$ and $U_{1m}$ are read off."},{"cited_title":"Nooijen and J","cited_arxiv_id":null,"evidence_quote":"Names the Majorana-fermion viewpoint justifying that $a_m^\\dagger \\pm a_m$ are unitary, the property the circuits exploit."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the quantum phase estimation procedure whose output histogram carries the transition amplitudes."}],"review_version":1}