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REVIEW 5 major objections 5 minor 1 cited by

Shallow-depth GHZ state generation on NISQ devices

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2507.19145 v1 pith:2DOYHBO7 submitted 2025-07-25 quant-ph

S. Siddardha Chelluri , Stephan Schuster , Sumeet , Riccardo Roma This is my paper · ORCID
classification quant-ph MSC 81P6881P40 PACS 03.67.Lx03.67.Mn
keywords GHZstatesmid-circuitmeasurementqubitconnectivitycircuitdepthNISQdevicesfidelitybenchmarkingentanglementgenerationstardecomposition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

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.

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 (5)
  1. [Section III B; captions of Figures 4 and 5] 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.
  2. [Section II A, Step 3 versus Appendix A] 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.
  3. [Appendix A and Section II A, Step 3] 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.
  4. [Section III A, Eq. (3)] 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.
  5. [Section III A] 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.
minor comments (5)
  1. [Section III A, Eq. (3)] 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.
  2. [Section II A, Step 4] 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.
  3. [Section III and abstract] 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.
  4. [Figure 3 caption] 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.
  5. [Figures 6 and 13 captions] The captions contain "thegrowing protocol"; this should be "the growing protocol".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: protocols are self-contained, benchmarks are external, and self-citations are attributional rather than load-bearing.

full rationale

The paper's central comparison is grounded in external evidence: the growing protocol is taken from Liao et al. [15], and the hardware fidelities come from independent executions on IBM Brussels. The merging protocol is not defined in terms of the results it claims; its operation is specified in Section II A and proved algebraically in Appendix A, so the citation to the authors' prior work [22] is historical attribution, not a load-bearing derivation. The modified Erdos-Renyi connectivity model is described fully in Section III C ('each node is initially connected to at least one other node to ensure connectivity, and additional edges are then added according to the standard G(N,p) process'), so the self-citation there is likewise not doing logical work. The choice of the highest-degree star-selection strategy is an empirical summary of the same simulations used to display it, which is a mild selection-on-test-set caveat rather than a construction-level circularity: no predicted quantity is defined from a fitted parameter, and no equation is equivalent to its input by construction. The disconnected-subgraph concern in Section III B is a correctness or methodology risk, not a circularity, and is not counted here.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claims rest on the connectivity and mergeability of the layout graph, on pre-compilation circuit depth as a proxy for hardware cost, on Hellinger fidelity as a proxy for GHZ quality, and on random graphs as models of distributed architectures. No new physical entities are introduced. The scanned star-selection and graph parameters are design choices, not fitted constants.

free parameters (3)
  • star selection scaling factor = 0.7, 1.0, 1.3 (scanned, not fitted)
    Hand-chosen factor relating target star degree to average graph degree; the paper scans these values and concludes the best choice is topology-dependent.
  • absolute target star size = 2, 3, 4 (IBM); 3, 5, 7 (rectangular); scanned
    Hand-chosen star sizes evaluated in Appendix B to compare selection criteria.
  • Erdos-Renyi edge probability p = 0.1, 0.5, 1.0 (scanned)
    Graph generation parameter; results show circuit depth and measurement count depend on it.
assumptions (5)
  • domain assumption The layout graph is connected and every star-decomposition admits a sequence of merges along edges between qubits in the two GHZ states.
    Assumed in Section II A Step 3 and Appendix A; not proven for arbitrary star partitions, and random subgraph sampling does not state connectivity constraints.
  • domain assumption Pre-compilation circuit depth is a meaningful proxy for execution time and noise when comparing protocols.
    Stated in Section III: circuit depth is measured before hardware-specific compilation and is thus independent of any backend; this ignores the real overhead of mid-circuit measurements and classical feedforward.
  • domain assumption Hellinger fidelity between the ideal and measured computational-basis outcome distributions captures GHZ state quality.
    Used in Equation (3) and Section III A; it is a distribution distance, not quantum state fidelity, and does not directly measure coherence or entanglement.
  • domain assumption Erdos-Renyi random graphs with enforced connectivity model distributed quantum computing architectures.
    Section III C introduces this model; it is a modeling choice, not established as faithful to real distributed systems.
  • domain assumption Measured qubits can be reset to |0> and re-entangled with a CX gate without corrupting the remaining GHZ state.
    Step 4 of the merging protocol (Section II A) requires qubit reuse; mid-circuit reset and feedforward on current IBM hardware are error-prone and are a key reason the hardware fidelity of merging is lower.

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Cite this review

Pith. "Pith review of Shallow-depth GHZ state generation on NISQ devices." pith.science (2026). https://pith.science/paper/2DOYHBO7

@misc{pith2026250719145,
  author       = {Pith},
  title        = {Pith review of: Shallow-depth GHZ state generation on NISQ devices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2DOYHBO7}},
  note         = {Machine review of arXiv:2507.19145}
}
read the original abstract

In this work, we focus on GHZ state generation under the practical constraint of limited qubit connectivity, a hallmark of current NISQ hardware. We study the GHZ state preparation across different connectivity graphs inspired by IBM and Google chip architectures, as well as random graphs that reflect distributed quantum systems. Our approach is a measurement-based protocol designed to utilize qubit connectivity constraints for the generation of GHZ states on NISQ devices. We benchmark this against a tailored version of state-of-the-art unitary-based protocols, also incorporating physical connectivity limitations. To evaluate the performance of the protocols under realistic conditions, we conducted implementations on the IBM Eagle r3 chip. Additionally, to explore near-term scalability, we performed simulations across a range of graph sizes and connectivity configurations, assessing performance based on circuit depth, the number of two-qubit gates, and measurement overhead. We observe a trade-off between the two protocols across different figures of merit. For current state-of-the-art NISQ architectures, the unitary-based protocol is more suitable, as it avoids mid-circuit measurements and classical feedforward. However, the measurement-based protocol is expected to become more advantageous in the future with more error-resilient quantum devices, owing to its reduced circuit depth and consequently shorter execution times. In both settings, our proposed method provides an efficient means of leveraging the topology of qubit connections available on a given device.

Figures

Figures reproduced from arXiv: 2507.19145 by the authors.

Figure 1
Figure 1. FIG. 1. The figure illustrates star selection in the IBMQ device layout for the merging method. Coloured boxes indicate 4-, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The first three steps of the GHZ state growing method on the IBM layout as shown in this figure. The red nodes [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. GHZ state fidelity (cf. Equation (3)) over the number of qubits in the final GHZ state. Here we compare the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Random subgraph sampling from IBM’s 127-qubit Eagle layout [23]. Here we compare the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Random subgraph sampling from rectangular grid layout inspired by Google’s Willow chip [24]. Here we compare the [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Sampling from Erd˝os–R´enyi random graphs [25, 26]. Here we compare the [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Random subgraph sampling from IBM’s 127-qubit Eagle layout [23]. Here we compare our [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Random subgraph sampling from IBM’s 127-qubit Eagle layout [23]. Here we compare our [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Random subgraph sampling from IBM’s 127-qubit Eagle layout [23]. Here we compare our [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Random subgraph sampling from rectangular grid layout inspired by Google’s Willow chip [24]. Here we compare our [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Random subgraph sampling from rectangular grid layout inspired by Google’s Willow chip [24]. Here we compare [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Random subgraph sampling from rectangular grid layout inspired by Google’s Willow chip [24]. Here we compare [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Sampling from Erd˝os–R´enyi random graphs [25, 26]. Here we compare the [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]

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    Communicate the outcome x∈{ 0, 1} of the measurement to the nodes C1, C2, . . . , Cm

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    , Cm do nothing

    (a) If the outcome is x = 0, then C1, C2, . . . , Cm do nothing. (b) If the outcome is x = 1, then C1, C2, . . . , Cm apply the Pauli-X gate to their qubits. Let us now prove that after this protocol we deterministically have the GHZ state |GHZn+m+1⟩A1B1:nC1:m shared by A1, B1...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.