{"id":"1371d0ce-9a55-48d3-af79-7701d04eb62d","arxiv_id":"2506.24109","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"MTDMRG-X combines multi-target DMRG with DMRG-X to simultaneously target strongly hybridized excited eigenstates, demonstrated on 2D transmon arrays.","lead":"The paper introduces MTDMRG-X, a tensor-network algorithm that computes multiple excited quantum eigenstates at once by matching them to a set of reference states, without first finding all lower-energy states. It applies the method to a 2D array of superconducting qubits and couplers, showing how coupler excitations delocalize and how two-qubit couplings depend on nearby 'aggressor' qubits.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The MTDMRG-X update rule in Eq. (14) is ambiguous for exactly the strongly hybridized states the paper claims to target: equal-overlap bare states can be matched to the same eigenstate, and no exact-diagonalization benchmark rules out this failure.","rationale":"The reader identified the localization-in-the-bare-basis assumption and the absence of exact benchmarks as the weakest point. My concern sharpens this into a concrete algorithmic failure mode: even when the localization assumption holds, Eq. (14) is ill-defined for symmetric hybridization because the reference states have equal overlap with the two dressed states. This is not a mere numerical nuisance; it is the flagship example the paper uses to motivate MTDMRG-X (Sec. IV: 'psi_+/- ~ |01> +/- |10>'). Without a tie-breaking rule, the algorithm can assign both target indices to the same effective eigenstate, causing the multi-target SVD to lose one member of the subspace. The variance checks in the paper cannot detect this because a wrong superposition of nearly degenerate eigenstates has small variance. The proposed check is deliberately small: a two-transmon Hamiltonian is exactly diagonalizable, so the test directly isolates the update rule from other sources of error such as bond-dimension truncation or environment construction. If the test passes, the central claim is substantially supported; if it fails, the method as written does not deliver its advertised capability. I do not, however, regard the concern as a demonstrated contradiction: the paper's phrase 'matched by largest overlap' could be implemented as an assignment problem, but that implementation is not specified. The manuscript also supplies no code or data, so the tie-handling behavior cannot be checked post hoc. These facts justify keeping the reader's CONDITIONAL verdict: the paper is promising and self-consistent in outline, but its central claim needs this validation before it can be accepted as reliable. I therefore leave the verdict unchanged rather than escalating to rejection.","tokens_in":21736,"tokens_out":6615,"duration_ms":85074,"concrete_test":"Run MTDMRG-X on the minimal resonant model: two coupled transmons with H = omega_1 a^dag a + omega_2 b^dag b + g(a + a^dag)(b + b^dag), local dimension 3 or 4, with omega_1 = omega_2 and g chosen so that psi_+ = (|01> + |10>)/sqrt(2) and psi_- = (|01> - |10>)/sqrt(2) are exact eigenstates. Initialize the MTMPS with S = {|01>, |10>} as prescribed by Eq. (12), run the sequential-Lanczos matching described in Appendix A, and compute the fidelity of the two output states to psi_+ and psi_-. Repeat at detunings delta/(2g) = 0, 0.05, and 0.5. If at delta = 0 the method fails to return both states with fidelity above 0.99, or returns the same state twice, the central claim fails; if it succeeds across all detunings, the tie-handling concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that overlap-based tensor updates resolve strongly hybridized excited states. That premise is insecure at Eq. (14): for a resonant pair with eigenstates psi_+ = (|01> + |10>)/sqrt(2) and psi_- = (|01> - |10>)/sqrt(2), both bare reference states |01> and |10> have equal squared overlap (1/2) with both eigenstates. The argmax in Eq. (14) is therefore not unique for either bare state. If the two target labels are matched to the same eigenstate, the SVD of gamma_opt collapses the multi-target state and one dressed state is lost. The manuscript describes sequential matching but specifies no tie-breaking or assignment rule that prevents duplicate selection, and the claimed robustness to strong hybridization is exactly the regime where ties occur. The reported Hamiltonian variances (10^-9 to 10^-7 GHz^2 in Sec. V C 1; 10^-10 to 10^-8 in Appendix E) do not resolve this: inside a near-degenerate resonant subspace, a wrong member of the subspace has near-zero variance, so variance cannot certify that the intended pair was found. No exact-diagonalization comparison for MTDMRG-X in a strongly hybridized subspace is provided; Appendix C, Fig. 9 compares MTDMRG and DMRG-X on a single coupler state, not MTDMRG-X on a resonant pair. The algorithm's central advantage therefore rests on an unvalidated tie-handling assumption precisely where it matters most.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces MTDMRG-X, a multi-target extension of the DMRG-X algorithm, which simultaneously targets several excited eigenstates by selecting, at each two-site update, the eigenstates of the effective Hamiltonian with largest overlap with a set of reference bare states (Sec. IV, Eqs. (13)-(14)). The authors also describe a Lanczos-based variant, Lanczos-X, that builds the Krylov subspace around the reference states to reduce runtime for deeply excited states, and they apply DMRG-X and MTDMRG-X to a 5x5 transmon array with qubits and couplers. Numerically, they report Hamiltonian variance checks (10^-9-10^-7 GHz^2 for DMRG-X, 10^-10-10^-8 GHz^2 for MTDMRG-X), an analysis of single-excitation localization in the chip, and state-dependent exchange (g) and ZZ (zeta) couplings for a target qubit pair in the presence of an on-resonance aggressor pair.","tokens_in":22130,"tokens_out":6197,"duration_ms":76586,"significance":"If the algorithm performs as claimed, MTDMRG-X would be a useful tool for obtaining excited eigenstates of circuit-QED Hamiltonians without first computing all lower-energy states, including in regimes with strong hybridization. The manuscript is clearly written and provides a concrete algorithmic prescription, a practical Lanczos-X speedup with a runtime benchmark, honest convergence metrics (Hamiltonian variances), and physically motivated applications to localization and spectator-induced couplings in a realistic transmon architecture. However, the central claim of robustness to strong hybridization rests on an under-specified matching rule at Eq. (14) and lacks a direct exact-diagonalization benchmark in the strongly hybridized regime, so the numerical evidence does not yet establish the advertised capability.","major_comments":[{"comment":"The central update rule is under-specified precisely in the regime the paper claims to target. Eq. (14) defines, for each reference projection |P_k>, an argmax over the eigenstates of the effective Hamiltonian, but it does not require that different k select distinct eigenstates. In the advertised resonant case psi_+ = (|01> + |10>)/sqrt(2) and psi_- = (|01> - |10>)/sqrt(2), both reference projections have identical squared overlap 1/2 with both eigenstates, so the argmax is not unique. The sequential-Lanczos description in Appendix A 1 does not state a tie-breaking rule or explicitly exclude already-matched eigenstates from subsequent matching; if two reference states are assigned to the same effective eigenstate, the SVD of gamma_opt collapses the multi-target state and one dressed state is lost. Please specify the assignment rule (e.g., greedy matching with exclusion of used eigenstates, or a global assignment such as the Hungarian algorithm) and demonstrate on a resonant pair that the intended distinct states are recovered.","section":"Sec. IV, Eq. (14)"},{"comment":"There is no direct validation of MTDMRG-X against exact diagonalization in a strongly hybridized subspace. The 5x5 results in Fig. 6 report Hamiltonian variances of 10^-10-10^-8 GHz^2, but as Appendix B shows, variance is small for any near-eigenstate; inside a near-degenerate resonant subspace, a wrong linear combination can also have near-zero variance, so the variance check does not certify that the intended pair of dressed states was found. Appendix C, Fig. 9, compares MTDMRG with DMRG-X on a single coupler excitation, not MTDMRG-X on a resonant pair. Please add an exact-diagonalization benchmark for a small system (e.g., two transmons at resonance, or a single resonant pair embedded in a small lattice) reporting fidelities, energies, and assignment of the recovered states to the exact psi_+ and psi_-. This is the minimal evidence needed to support the central claim of simultaneous resolution of strongly hybridized states.","section":"Sec. V C 2 and Appendix E"}],"minor_comments":[{"comment":"The localization measure P_(x,y),k is the same overlap used as the DMRG-X targeting objective, so the finding that qubit eigenstates are localized is partly built into the method. The distance-decay profiles and the qubit-vs-coupler difference are not forced by the construction and are informative, but the text should explicitly acknowledge this partial circularity and state that the localization magnitude itself is not an independent verification of the method.","section":"Sec. V C 1, Eq. (20)"},{"comment":"The reported variance ranges are given as 10^-10-10^-8 (GHz); the units should be GHz^2, matching Eq. (7) and the main-text statement in Sec. V C 1.","section":"Appendix E"},{"comment":"The dashed gray line is described as an 'approximate bound consistent with the requested DMRG accuracy' with values below it shown only to display the trend. Please define this bound numerically (which variance or infidelity it corresponds to) so that readers do not interpret values below it as quantitatively meaningful.","section":"Fig. 5b"},{"comment":"The thresholds theta (used to define the reference set S) and th (used in the sequential matching) are never specified or varied. Please state the values used in the numerical experiments and comment on the sensitivity of the results to these thresholds.","section":"Sec. IV, Eq. (15) and Appendix A 1"},{"comment":"The notation lambda in the argmax is confusing: Eq. (6) uses lambda to label eigenstates of H_eff, and Eq. (14) reuses lambda as the variable being optimized. Please rename the eigenstate index (e.g., to nu) for clarity.","section":"Sec. IV, Eq. (14)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a solid methods-plus-applications paper, and the core idea is plausible, but the missing ED benchmark and the ambiguous matching rule in Eq. (14) are load-bearing for the advertised capability. Both can be fixed within the manuscript's scope: add a small-system exact-diagonalization validation, specify the assignment/tie-breaking rule, and check the resonant-pair case explicitly. No concerns about citation or scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the useful part: MTDMRG-X is a natural and clearly written extension of DMRG-X to multiple target states, and the Lanczos-X Krylov variant (Appendix A 2) is a real algorithmic addition that appears to give near-constant runtime in excitation number (Fig. 7). The circuit-QED application—extracting state-dependent exchange and ZZ couplings in a 2D transmon array—is well motivated, and the variance checks (10^-10 to 10^-8 GHz^2) show the states are self-consistent. The paper also honestly flags that its results are for a typical parameter set, not a real device.\n\nThe soft spots are in the central claim. The stress-test concern about Eq. (14) is valid: for a resonant pair, both bare states have equal overlap with both eigenstates, so the argmax is not unique. The sequential matching in Appendix A 1 may implicitly avoid duplicates by moving to the next eigenstate once one is matched, but the paper never states a rule that prevents two bare states from being assigned to the same eigenstate, and the cumulative-overlap stopping condition (\"remaining unaccounted-for overlap\") could, in the equal-overlap case, stop after one eigenstate and match both bare states to it. If that happens, the SVD of gamma_opt collapses one dressed state. This is exactly the strong-hybridization regime the algorithm claims to handle.\n\nThe validation gap makes this more than a theoretical quibble: Appendix C (Fig. 9) compares MTDMRG and DMRG-X on a single coupler state, not MTDMRG-X on a resonant pair. No exact-diagonalization benchmark of the new method is provided. Hamiltonian variance alone cannot certify that the intended member of a near-degenerate subspace was found, since any state inside that subspace has near-zero variance. The localization analysis in Sec. V C 1 is also partially circular: Eq. (20) is the same overlap that DMRG-X maximizes, so finding qubit states localized is partly built into the targeting objective. That doesn't invalidate the distance-decay profiles, but it softens the claim. No code or data is shipped, and the coupling numbers in Fig. 6 have no error bars.\n\nOn balance, this is a clear, honest paper with a plausible new method and a practically relevant demonstration. The missing ED benchmark and the tie ambiguity are addressable. I would send it to peer review, but I'd ask the authors for a small-system exact diagonalization comparison in a resonantly hybridized subspace and a precise assignment rule in the equal-overlap case.","headline":"Plausible new DMRG variant for multiple excited states, but the strong-hybridization claim needs an exact-diagonalization benchmark and a tie-breaking rule.","tokens_in":22686,"tokens_out":4060,"would_cite":false,"duration_ms":44309,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"New algorithm finds many excited states at once, skipping lower ones","keywords":["density matrix renormalization group","matrix product states","excited eigenstates","circuit quantum electrodynamics","transmon arrays","eigenstate localization","state-dependent couplings","multi-target DMRG"],"falsifier":"Run MTDMRG-X on a small transmon array (for example, a 3x3 qubit grid) with parameters that produce a nearly degenerate resonant subspace, and compare each converged MPS against the corresponding exact-diagonalization eigenstate by computing their squared overlap; if any returned state has infidelity above the threshold implied by the reported variances, or the states are returned in the wrong order within the subspace, the overlap-based assignment has converged to the wrong eigenstates.","tokens_in":21546,"feed_emoji":"⚛️","tokens_out":8037,"duration_ms":79220,"temperature":0.7,"pith_summary":"The paper introduces MTDMRG-X, a density matrix renormalization group algorithm that simultaneously targets several excited eigenstates of a many-body Hamiltonian by maximizing, at each local update, the overlap between effective-Hamiltonian eigenstates and a set of reference bare states. The motivation is practical: exact diagonalization of a transmon-array Hamiltonian becomes intractable beyond tens of qubits, and existing DMRG methods for excited states either require computing all lower-energy states first or handle only one localized state at a time. The authors show numerically that DMRG-X alone can find single-excitation eigenstates on a 5x5 transmon chip with 40 couplers, and that MTDMRG-X can resolve strongly hybridized resonant subspaces to estimate state-dependent exchange and ZZ couplings between qubit pairs. If the algorithm works as claimed, it gives circuit-QED designers a way to compute dressed frequencies, crosstalk, and spectator-induced error channels for order-100 devices without full diagonalization.","feed_headline":"New algorithm finds many excited states at once, skipping lower ones","feed_subtitle":"MTDMRG-X resolves strongly hybridized transmon states directly, revealing crosstalk and ZZ shifts on 5x5 arrays.","key_machinery":"The load-bearing mechanism is the overlap-based local update rule of Eq. (14): in each two-site DMRG sweep step, the eigenstate |ψ^(λ)> of the effective two-site Hamiltonian is chosen by argmax |<P_k|ψ^(λ)>|, where |P_k> is the projection of the current MPS onto the k-th reference bare state, rather than by minimizing energy. This rule is combined with the multi-target MPS ansatz (Eq. 9), which carries m states on an extra index k, so that m effective-Hamiltonian eigenstates are selected and decomposed together by singular value decomposition. The algorithm is initialized with the reference bare states themselves (Eq. 12), and the Lanczos-X variant seeds the Krylov subspace with the reference-state projections, giving O($χ^{3}$) updates that appear independent of excitation number in the tested regime.","core_discovery":"The central claim is that replacing the energy-minimization step of two-site DMRG with an overlap-maximization step, and carrying this out for multiple reference states at once, yields the simultaneous convergence of a set of strongly hybridized excited eigenstates without first computing lower-energy states. Concretely, for a set S of m reference bare product states, the update builds projections {|P_k>} of the current variational MPS onto those states, then selects the eigenstate of the effective Hamiltonian with the largest overlap with each |P_k> and assembles them into a multi-target two-site state (Eqs. 13-14). The authors demonstrate the method on a 5x5 transmon array with qubits and couplers: DMRG-X produces qubit-like single-excitation states with high localization and coupler-like states with noticeable delocalization, and MTDMRG-X resolves four-state resonant subspaces to extract state-dependent exchange coupling g and ZZ coupling ζ. They also introduce Lanczos-X, a Krylov-subspace variant built around the reference states, which reduces runtime when the target states lie deep in the effective spectrum.","pith_inferences":["One testable extension is to scan coupler-frequency disorder and recompute the localization metric ς(d) for coupler-like eigenstates, which would show whether the observed coupler delocalization can be suppressed by increasing frequency spread among couplers.","The overlap-based update rule is not limited to transmon arrays; it could be applied to other bosonic circuit elements with quasi-localized eigenstates, such as fluxonium devices, as a way to compute dressed spectra without full diagonalization.","A stress test for the method is to compare MTDMRG-X results against exact diagonalization on small arrays (for example, 3x3) at several detunings; confirming that the returned states match true eigenstates by fidelity would establish the reliability of variance-based convergence metrics in near-degenerate subspaces.","The overlap-based update rule could also be used to benchmark Schrieffer-Wolff perturbation theory in the nonperturbative regime, supplying numerically exact dressed couplings against which truncated expansions can be tested."],"forward_implications":["Order-100 transmon arrays, such as a 5x5 chip with 25 qubits and 40 couplers, can have their dressed single-excitation states computed with DMRG-X at bond dimension χ=80, reaching Hamiltonian variances of 10^-9 to 10^-7 GHz^2.","Excited states deep in the spectrum can be targeted directly and in parallel per state or per subspace, avoiding the m^2 cost and error accumulation of orthogonalization against all lower-energy states.","Strongly resonant subspaces with near-equal bare-state projections, such as ψ± ≈ |01> ± |10>, can be resolved simultaneously by MTDMRG-X instead of by sequential projection out of previously found states.","State-dependent coupling analysis becomes numerically accessible: exchange and ZZ couplings of a target qubit pair can vary with the excitation state of a nearby aggressor pair, with ZZ corrections reaching about 10 MHz at distance 1 for two aggressor excitations in the studied parameters."],"supporting_citations":[{"why":"Provides the standard two-site DMRG sweep procedure and effective-Hamiltonian construction that MTDMRG-X modifies in step ii).","marker":"[13]"},{"why":"Supplies the overlap-based DMRG-X update rule, which MTDMRG-X generalizes to multiple reference states.","marker":"[16]"},{"why":"Supplies the multi-target MPS ansatz and multi-target update procedure that MTDMRG-X adopts.","marker":"[15]"},{"why":"Introduces the multi-target DMRG idea of simultaneously optimizing several eigenstates, which the MTMPS ansatz builds on.","marker":"[26]"},{"why":"Lanczos iteration is the base of the sequential Lanczos and Lanczos-X eigenstate searches used in the two-site update.","marker":"[42]"},{"why":"Maps out the localized versus chaotic regime of transmon processors, motivating the need to target dressed computational-basis eigenstates directly.","marker":"[4]"},{"why":"Demonstrates prior tensor-network modeling of superconducting circuits with DMRG, the approach that this paper extends to excited states and hybridized subspaces.","marker":"[25]"}],"fun_headline_variants":["MTDMRG-X finds many transmon excited states at once, skipping lower ones","Multi-target DMRG-X resolves hybridized transmon states directly","Skip lower states: MTDMRG-X computes many excited transmon eigenstates at once","MTDMRG-X targets multiple hybridized states in one run for 5x5 qubit array","New DMRG-X variant handles strong hybridization in transmon arrays directly"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes that each target eigenstate has high overlap with at least one of the supplied reference bare states—product states of single-site levels—so that maximizing local overlap tracks the intended global eigenstate.","fun_headline_variants_meta":{"raw":{"variants":["MTDMRG-X finds many transmon excited states at once, skipping lower ones","Multi-target DMRG-X resolves hybridized transmon states directly","Skip lower states: MTDMRG-X computes many excited transmon eigenstates at once","MTDMRG-X targets multiple hybridized states in one run for 5x5 qubit array","New DMRG-X variant handles strong hybridization in transmon arrays directly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000577,"raw_usage":{"total_tokens":2726,"prompt_tokens":954,"completion_tokens":1772,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":1663}},"tokens_in":570,"tokens_out":1772,"duration_ms":13328,"temperature":1.0,"reasoning_tokens":1663,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:24:14.759847+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run MTDMRG-X on a small transmon array (for example, a 3x3 qubit grid) with parameters that produce a nearly degenerate resonant subspace, and compare each converged MPS against the corresponding exact-diagonalization eigenstate by computing their squared overlap; if any returned state has infidelity above the threshold implied by the reported variances, or the states are returned in the wrong order within the subspace, the overlap-based assignment has converged to the wrong eigenstates.","supporting_citations":[{"cited_title":"Berke, E","cited_arxiv_id":null,"evidence_quote":"Provides the standard two-site DMRG sweep procedure and effective-Hamiltonian construction that MTDMRG-X modifies in step ii)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the overlap-based DMRG-X update rule, which MTDMRG-X generalizes to multiple reference states."},{"cited_title":"Bluvstein, H","cited_arxiv_id":null,"evidence_quote":"Supplies the multi-target MPS ansatz and multi-target update procedure that MTDMRG-X adopts."},{"cited_title":"Gyenis, A","cited_arxiv_id":null,"evidence_quote":"Lanczos iteration is the base of the sequential Lanczos and Lanczos-X eigenstate searches used in the two-site update."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Maps out the localized versus chaotic regime of transmon processors, motivating the need to target dressed computational-basis eigenstates directly."}],"review_version":1}