{"id":"e8880de5-5904-4f84-bb5a-48e18d253180","arxiv_id":"2507.19145","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"On IBM Eagle hardware and in simulations of several chip layouts, the unitary growing protocol beats the measurement-based merging protocol in GHZ-state fidelity, while merging produces shallower circuits at the cost of more measurements and two-qubit gates.","lead":"This paper compares two ways to create large entangled GHZ states on noisy quantum computers with limited qubit-to-qubit connections: a measurement-based merging method and a unitary growing method. It finds that the unitary method works better on today's IBM hardware, while the measurement method may win in the future because it needs fewer circuit layers.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sec. III B subgraph samples are not stated to be connected; if many are disconnected, the reported depth/gate trade-off is biased. The reader's merge-edge concern is largely resolvable by symmetry of GHZ states.","rationale":"The reader's weakest assumption focused on the merging protocol's recursive edge condition and the lack of a proof for arbitrary merge orders. That concern is less decisive than it appears: Appendix A proves merging for two GHZ states with an edge between an arbitrarily labeled qubit A2 of the second state and a qubit A1 of the first; all qubits of a GHZ state are symmetric, so the proof holds for any inter-state edge. Furthermore, any partition of a connected graph into disjoint stars yields a connected 'star graph', so a merge order along a spanning tree always exists. The real gap is the connectivity of sampled subgraphs. The algorithms as described operate on a connected graph, but Sec. III B does not enforce connectivity for subgraph samples, unlike Sec. III C. If disconnected samples are included, the reported metrics likely understate the resources needed for genuine full-size GHZ generation, specifically favoring the merging protocol's shallow-depth claim. This directly impacts the paper's central trade-off on IBM- and Google-like architectures. Therefore, the manuscript should be accepted only conditionally, pending either a justification that sampling was connectivity-preserving or a re-analysis on connected subgraphs. Our recommendation does not change the reader's CONDITIONAL verdict.","tokens_in":15677,"tokens_out":9569,"duration_ms":97036,"concrete_test":"Re-run the subgraph sampling exactly as described in Sec. III B: for each size N in {25,50,75,100,125} (IBM) and {50,100,200,300,400} (grid), sample 100 induced subgraphs of size N uniformly from the layout graph and compute the fraction that are connected. If a substantial fraction (e.g., >10%) are disconnected, re-evaluate the depth, measurement, and two-qubit-gate counts using only connected subgraphs and compare with Figures 4 and 5. If the qualitative trade-off (merging shallower, more measurements) persists, the concern is not load-bearing; if the advantage flips or shrinks, the central claim needs revision and the sampling method must be corrected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III B ('Performance on quantum hardware layouts') states: 'For each GHZ state size, we generated 100 subgraph samples of this size randomly from the initial layout graph.' No connectivity constraint is mentioned. Both protocols require a connected qubit-connectivity graph: the growing protocol uses BFS from a starting node and can only incorporate qubits in that node's connected component, and the merging protocol requires an edge between the current GHZ state and the next star at each merge (which follows from connectedness of the underlying graph). Uniformly sampled induced subgraphs of the sparse IBM Eagle (degrees ≤3) or a rectangular grid are very likely disconnected for sizes up to a few hundred qubits. If some instances are disconnected, the reported circuit depth and measurement counts cannot correspond to generation of a GHZ state over all sampled qubits; the protocol would either have to restrict to a connected component (reducing effective size) or use unphysical long-range CX gates. Since the comparison in Figures 4 and 5 depends on these averages, the central trade-off claim could be biased. Notably, Section III C explicitly modifies Erdős–Rényi sampling to enforce connectivity, demonstrating awareness of the issue, but no analogous step is described for subgraph sampling. This is an internal inconsistency and a concrete methodological gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript compares two topology-aware GHZ-state preparation protocols on fixed qubit-connectivity graphs: a measurement-based \"merging\" protocol, which constructs small star-shaped GHZ states and fuses them via mid-circuit measurements, and a unitary \"growing\" protocol, which expands a GHZ state by breadth-first search along graph edges. The authors benchmark both on an IBM Eagle r3 device using Hellinger fidelity and in simulation on IBM Eagle, rectangular (Willow-like), and Erdős–Rényi random layouts, reporting circuit depth, number of measurements, and number of two-qubit gates. The central claim is a trade-off: the growing protocol gives higher fidelity on current NISQ hardware because it avoids mid-circuit measurement and feedforward, while the merging protocol gives shallower circuits and may become preferable on more error-resilient devices. Appendix A provides a proof of the two-GHZ merging operation, and Appendix B studies alternative star-selection heuristics.","tokens_in":15907,"tokens_out":8674,"duration_ms":90491,"significance":"If the claims are correct, the paper provides a useful, hardware-topology-aware comparison and a concrete measurement-based construction whose shallower depth could benefit future devices. Strengths include the real-device execution, the 100-sample statistics with error bars, the comparison against the external growing protocol of Liao et al., and the explicit merge proof in Appendix A. The main value is the quantified trade-off among depth, measurements, and two-qubit-gate count across three graph families. The result is not a fundamental conceptual breakthrough but is a solid engineering-oriented contribution, provided the gaps identified below are addressed.","major_comments":[{"comment":"The subgraph sampling procedure is not stated to produce connected subgraphs. Both protocols require a connected qubit-connectivity graph: the growing protocol uses BFS from a single root and can only reach the root's connected component, and the merging protocol needs an edge between the existing GHZ state and the next star at each merge. Uniformly sampling induced subgraphs of a degree-3 Eagle layout or a rectangular grid will frequently yield disconnected instances, and for such instances the reported circuit depth, measurement count, and two-qubit-gate count cannot correspond to GHZ generation over the full sampled vertex set without introducing nonlocal CX gates. Section III C explicitly modifies the Erdős–Rényi sampling to enforce connectivity, but no analogous statement is given for Section III B. This is an internal inconsistency and a potential bias in the central trade-off figures; the authors must state whether subgraphs were connected and, if not, redo the sampling with a connected-subgraph method.","section":"Section III B; captions of Figures 4 and 5"},{"comment":"The main text states that after applying CX on the two chosen centers, the qubit is measured in the X-basis, with a possible corrective X gate on the remaining qubits of the second GHZ state. Appendix A proves a different operation: after the CX, the measured qubit is measured in the Z-basis, and the correction is X on the remaining qubits of the second GHZ state. These are not equivalent: a direct calculation shows that an X-basis measurement after the CX leaves the second star's remaining qubits in a product state factor, so no local correction produces a GHZ state. The protocol description in Section II A and the proof in Appendix A must be aligned, and the hardware implementation must be checked against the corrected description.","section":"Section II A, Step 3 versus Appendix A"},{"comment":"The merge proof only covers the case where the centers of the two GHZ states share an edge, but the recursive merge order for arbitrary connected layouts is not proved to provide such an edge at every step. Because a GHZ state is symmetric under qubit permutations, one can choose any qubit from the current GHZ state and any qubit from the next star as the effective centers, so a crossing edge is sufficient. The paper should state this explicitly and prove that the star-decomposition order always yields a crossing edge. Without this statement, the depth and measurement counts for the merging protocol are not rigorously established for all sampled layouts.","section":"Appendix A and Section II A, Step 3"},{"comment":"The Hellinger fidelity is a fidelity between classical probability distributions, not the quantum state fidelity between the ideal GHZ state and the actual density matrix. The paper repeatedly refers to these values as \"GHZ state fidelity\" and uses them to conclude that the growing protocol outperforms the merging protocol on current hardware. This metric cannot certify multipartite entanglement or coherence; for instance, a completely dephased state can have high output-distribution overlap. The authors should either use a genuine quantum fidelity estimator (for example, based on stabilizer measurements or parity checks) or explicitly and consistently qualify the quantity as a classical output-distribution fidelity and temper the fidelity-based conclusions accordingly.","section":"Section III A, Eq. (3)"},{"comment":"The paper acknowledges a consistent discrepancy between simulated and hardware fidelities but offers no explanation. Because the simulations are used to project near-term performance and to support the fidelity comparison, this unexplained gap weakens the quantitative fidelity predictions. At minimum, the authors should discuss possible causes such as calibration drift, incomplete noise modeling, or measurement-error contributions, and state whether the discrepancy affects the relative ordering of the two protocols.","section":"Section III A"}],"minor_comments":[{"comment":"The typeset formula for the Hellinger fidelity is garbled; please rewrite it with the Hellinger distance explicitly defined before giving the simplified form (sum of sqrt(pi qi)) squared.","section":"Section III A, Eq. (3)"},{"comment":"Step 4 should specify that the CX used to re-add a measured qubit must have the existing GHZ qubit as control and the re-initialized qubit as target; the current wording is ambiguous.","section":"Section II A, Step 4"},{"comment":"Circuit depth is reported before hardware-specific compilation; statements about \"shorter execution times\" should be qualified, since compiled depth and the latency of mid-circuit measurement and feedforward can alter the comparison on real hardware.","section":"Section III and abstract"},{"comment":"The caption contains a typo: \"Each circuit as been executed\" should be \"Each circuit has been executed\"; the legend in panel (a) is also hard to parse because average and maximum markers share similar symbols.","section":"Figure 3 caption"},{"comment":"The captions contain \"thegrowing protocol\"; this should be \"the growing protocol\".","section":"Figures 6 and 13 captions"}],"recommendation":"major_revision","confidential_remarks":"Please ask the authors to verify which basis was actually used for the mid-circuit measurement in their hardware and simulation code. If the X-basis description in Section II A is not a typo, the measured fidelity data for the merging protocol may not correspond to GHZ generation. Obtaining the exact connected-subgraph sampling procedure is also essential before the simulation results in Figures 4 and 5 can be interpreted. These issues are fixable, but they are load-bearing for the central claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's my take on 2507.19145. It is a solid incremental benchmarking paper. The two protocols are not new—merging comes from the authors' prior work, growing from Liao et al.—but the hardware-layout adaptation, the systematic star-selection comparison, and the IBM Brussels data are a legitimate extension. The central trade-off claim is plausible: growing wins on current hardware, merging is shallower and may win with better measurements and feedforward. I agree with the reader's conditional verdict, with one caveat: the connectivity gap in Section III B is weightier than the reader's soundness score suggests.\n\nThe stress-test note is right. Section III B says \"we generated 100 subgraph samples of this size randomly from the initial layout graph\" without any connectivity constraint. Both protocols require a connected graph. Section III C explicitly enforces connectivity for the Erdős–Rényi graphs, and the fact that this step is missing for the IBM and Willow subgraph samples is a real internal inconsistency. Uniformly sampled induced subgraphs of Eagle or a grid will often be disconnected, and for those instances the reported depth and two-qubit-gate counts cannot correspond to actual GHZ generation on the sampled qubits. This could bias the averages in Figures 4 and 5. The fix is easy—sample connected subgraphs or say you did—but it has to happen before those numbers are used.\n\nOther soft spots are minor. Hellinger fidelity is a classical distribution distance, not a quantum state fidelity; it is fine as a proxy but should be labeled honestly. Depth is pre-compilation, which is acceptable for protocol comparison but should be stated in every figure caption. The unexplained simulation-hardware fidelity gap needs a few sentences; readers will wonder whether the noise model is off. The star-selection conclusion (highest degree wins on IBM/Willow, not on random graphs) is drawn from the same simulations used to evaluate it, but the claim is descriptive, so the circularity is mild. And no code or data are released, which is a real loss for a benchmarking paper.\n\nThe reader's merge-edge concern is largely a non-issue once you notice that GHZ states are permutation-symmetric. Any qubit in one GHZ can serve as the control and any qubit in the other can be the measured one, so an edge between any pair of qubits in the two states suffices. The Appendix only proves the center-to-center case, and the paper should say this explicitly, but the protocol does not fall apart.\n\nWho is this for: people designing GHZ preparation circuits, NISQ compiler groups, and experimentalists benchmarking connectivity-aware protocols. It will be a useful citation. I would send it to a serious referee, with the connectivity issue as a required revision. If they fix that and release the sampling code, I'd be happy to rely on the numbers.\n\nBest.","headline":"A solid, incremental benchmarking paper whose central trade-off claim holds up, but the Section III B subgraph sampling has an unstated connectivity requirement that must be fixed before the quantitative comparisons can be trusted.","tokens_in":16427,"tokens_out":4337,"would_cite":true,"duration_ms":41246,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68","81P40"],"pacs":["03.67.Lx","03.67.Mn"],"model":"deepseek-v4-flash","headline":"The right way to make a GHZ state depends on the chip: unitary growing wins fidelity today, measurement-based merging wins circuit depth and may win later.","keywords":["GHZ states","mid-circuit measurement","qubit connectivity","circuit depth","NISQ devices","fidelity benchmarking","entanglement generation","star decomposition"],"falsifier":"One concrete check is to take any connected qubit-layout graph, apply the paper's star-selection and merge-ordering rules, and see whether every merge step connects two GHZ blocks through an edge between a qubit in each block; a single connected graph where a step lacks such an edge, or where the sampled subgraph is disconnected, would invalidate the reported depth and measurement counts for that layout. A second check is to run both protocols on a device whose mid-circuit measurement and feedforward errors are comparable to its two-qubit gate errors; if merging's fidelity does not at least match growing's at the same GHZ size, the projected crossover would be refuted.","tokens_in":15472,"feed_emoji":"⚛️","tokens_out":10880,"duration_ms":107048,"temperature":0.7,"pith_summary":"This paper tries to settle how to prepare Greenberger–Horne–Zeilinger (GHZ) states—fully entangled states of many qubits—on quantum hardware whose qubits can only talk to nearby neighbors. It compares two connectivity-aware protocols: a measurement-based \"merging\" protocol that builds small star-like GHZ states and fuses them with mid-circuit measurements, and a purely unitary \"growing\" protocol that expands a GHZ state outward in a breadth-first fashion. On a 127-qubit superconducting chip, growing gives higher fidelity, because it avoids mid-circuit measurements and classical feedforward; merging gives shallower circuits, at the cost of more measurements and more two-qubit gates. Simulations across hardware-like and random layouts show the trade-off depends on topology, and the paper argues merging will become the better choice as two-qubit gates and readout improve. A sympathetic reader would take the central message: protocol choice should be made with the device's connectivity and noise profile in hand, not in the abstract.","feed_headline":"Merging cuts GHZ depth; unitaries keep fidelity today","feed_subtitle":"Measurement-based merging gives shallower circuits but more overhead; the purely unitary approach still wins on today's noisy chips","key_machinery":"Two mechanisms carry the argument. The first is the GHZ-merging gate sequence: given two GHZ blocks whose central qubits are connected by an edge, apply a controlled-NOT between the two central qubits, measure one of them, and apply Pauli corrections conditioned on the outcome; Appendix A proves this produces a deterministic larger GHZ state. The second is star decomposition: partition the layout graph by repeatedly taking the highest-degree remaining node and its neighbors, prepare a small GHZ state on each star with Hadamard and controlled-NOT gates, then merge the blocks recursively, re-using measured qubits so every qubit ends in the final state. The growing protocol's counterpart mechanism is breadth-first expansion, where already-entangled qubits act as control qubits for controlled-NOT gates that add neighboring qubits. The tuning knobs—star size, scaling factor, and star-degree criterion—determine where a given layout sits on the depth-versus-measurement trade-off.","core_discovery":"The paper's central claim is that GHZ-state generation on a fixed qubit-connectivity graph is best treated as a protocol-selection problem: a measurement-based merging protocol and a unitary growing protocol occupy opposite ends of a depth-versus-overhead trade-off. On a 127-qubit superconducting chip, growing—a breadth-first expansion that uses only controlled-NOT and single-qubit gates—produced higher Hellinger fidelities than merging for every tested GHZ size, because merging's mid-circuit measurements and classical feedforward add noise and because it needs slightly more two-qubit gates. In simulations of hardware-like layouts with up to hundreds of qubits, merging produced shallower circuits, with the improvement growing with state size, at the cost of measurements and extra two-qubit gates from re-adding measured qubits. The paper also finds that the best star-selection rule depends on topology: highest-degree stars are best on regular hardware layouts, while on random graphs smaller stars reduce depth and larger stars reduce measurements, leaving a case-by-case trade-off. All fidelity results are without error mitigation, and the paper reports a systematic discrepancy between its noisy simulations and the hardware runs.","pith_inferences":["Beyond the paper: if mid-circuit measurement and feedforward latencies continue to fall faster than two-qubit gate error rates, the projected crossover could arrive before full fault tolerance, and the reported depth counts give a way to estimate when.","Beyond the paper: the star-selection step is a graph-cover optimization, so the paper's greedy heuristics are probably not optimal; an exact optimizer or learned policy could reduce measurement counts and shift the crossover point.","Beyond the paper: on regular grids the highest-degree rule's dominance comes partly from the small maximum degree, so a natural test is to repeat the comparison on higher-degree regular lattices, where the trade-off may reappear."],"forward_implications":["On current hardware, choose the unitary growing protocol when GHZ fidelity is the goal; its advantage comes from avoiding mid-circuit measurements and feedforward and using slightly fewer two-qubit gates.","Merging's depth advantage grows with GHZ size, so for large entangled states its shorter execution time becomes an increasingly strong argument, assuming measurements are reliable.","On hardware-like regular layouts, selecting stars by highest degree is the best merging strategy because it minimizes the number of merges and hence measurements without a depth penalty.","On random graphs representing distributed quantum systems, no single star-selection rule dominates: smaller stars reduce depth, larger stars reduce measurements, and the choice must be made per layout.","As two-qubit-gate fidelities and readout improve, the merging protocol is expected to surpass growing, with the crossover depending on those error rates."],"supporting_citations":[{"why":"This work supplies a prior hardware comparison in which a unitary GHZ construction beat a measurement-based one on a superconducting chip, partly because of slow classical feedforward, motivating the paper's fidelity hierarchy.","marker":"[14]"},{"why":"This work supplies the breadth-first unitary growing protocol that the paper adapts and benchmarks against its merging protocol.","marker":"[15]"},{"why":"This work reports ion-trap experiments where constant-depth measurement-based GHZ generation looked promising under limited connectivity, supporting the paper's forward-looking assessment.","marker":"[17]"},{"why":"This work introduces the original merging protocol for random graph architectures, which the paper reworks for physical chip layouts.","marker":"[22]"},{"why":"This work defines the 127-qubit heavy-hex topology used for both the hardware experiment and the fixed-layout simulations.","marker":"[23]"},{"why":"This work defines the rectangular grid layout used as the second fixed hardware-inspired topology in the simulations.","marker":"[24]"},{"why":"This work provides the Erdős–Rényi random graph model used to represent distributed quantum systems in the random-layout benchmarks.","marker":"[25, 26]"}],"fun_headline_variants":["GHZ depth vs fidelity: unitaries win today, merging for future","Unitaries beat merging for GHZ on today's noisy chips","Shallow GHZ via merging, but unitaries keep fidelity now","On IBM Eagle, unitaries win over merging for GHZ"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that every layout can be partitioned into stars and then merged recursively with the required edge available at each merge step; the proof in Appendix A covers only the case where the two central qubits of the blocks being merged are directly connected, and no proof shows that the star-selection and merging order always guarantees such a connection, or even connected subgraphs, on arbitrary sampled layouts.","fun_headline_variants_meta":{"raw":{"variants":["GHZ depth vs fidelity: unitaries win today, merging for future","Unitaries beat merging for GHZ on today's noisy chips","Shallow GHZ via merging, but unitaries keep fidelity now","On IBM Eagle, unitaries win over merging for GHZ"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000841,"raw_usage":{"total_tokens":3705,"prompt_tokens":1028,"completion_tokens":2677,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":644,"completion_tokens_details":{"reasoning_tokens":2603}},"tokens_in":644,"tokens_out":2677,"duration_ms":19810,"temperature":1.0,"reasoning_tokens":2603,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:59:50.572345+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete check is to take any connected qubit-layout graph, apply the paper's star-selection and merge-ordering rules, and see whether every merge step connects two GHZ blocks through an edge between a qubit in each block; a single connected graph where a step lacks such an edge, or where the sampled subgraph is disconnected, would invalidate the reported depth and measurement counts for that layout. A second check is to run both protocols on a device whose mid-circuit measurement and feedforward errors are comparable to its two-qubit gate errors; if merging's fidelity does not at least match growing's at the same GHZ size, the projected crossover would be refuted.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This work supplies the breadth-first unitary growing protocol that the paper adapts and benchmarks against its merging protocol."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This work reports ion-trap experiments where constant-depth measurement-based GHZ generation looked promising under limited connectivity, supporting the paper's forward-looking assessment."},{"cited_title":"Ball, First quantum computer to pack 100 qubits en- ters crowded race, Nature 599 (2021)","cited_arxiv_id":null,"evidence_quote":"This work defines the 127-qubit heavy-hex topology used for both the hardware experiment and the fixed-layout simulations."},{"cited_title":"Acharya, D","cited_arxiv_id":null,"evidence_quote":"This work defines the rectangular grid layout used as the second fixed hardware-inspired topology in the simulations."}],"review_version":2}