Reviewers' comments:

Reviewer #1 (Remarks to the Author):

I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Communications Physics initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.


Reviewer #2 (Remarks to the Author):

The manuscript "TrackHHL: The 1-Bit Quantum Filter for particle trajectory reconstruction" presents a newly developed quantum algorithm for particle track reconstruction on near-term quantum hardware. By reformulating the task from matrix inversion to signal-noise separation, the authors replace the deep circuits required for alternative approaches such as the Harrow-Hassidim-Lloyd (HHL) algorithm with circuits that may be applicable on current NISQ devices. They further reduce circuit depth by applying the Direct Structural Synthesis method instead of Trotterization.

With particle physics experiments increasing their luminosity, this work is motivated by the growing challenges faced by classical computers in performing accurate track reconstruction within strict time constraints. The manuscript is therefore timely and presents clear scientific methodology and original ideas.

However, while the conceptual reformulation is interesting, it remains unclear whether this constitutes a sufficiently broad advance in quantum algorithms or particle reconstruction to justify publication in Communications Physics. The current demonstrations are limited to highly idealized toy models, and no evidence is provided that the proposed spectral filtering mechanism remains effective under realistic detector conditions or scaling regimes relevant for HL-LHC experiments.

Thus, several points need to be addressed before this manuscript can be considered for publication in Communications Physics.

Major Comments:

- Discussion of classical alternatives:

The discussion of classical reconstruction methods should be significantly expanded. The authors state that classical algorithms may face insurmountable challenges in the future and use this as primary motivation for a quantum approach. However, quantitative detail is lacking. It would be essential to clarify how close current classical methods are to their practical limits, and on what timescale such limitations are expected to become critical. Moreover, a statement clarifying whether this algorithm is expected to outperform classical alternatives in terms of runtime is required. If this is indeed the case, the runtime of the complete process should be specified. In particular, the load complexity of O(N) is not discussed, although it has been shown to remove quantum speedups in certain cases (Phys. Rev. D 101, 094015 (2020), [arXiv:1908.08949]). Without this context, it is difficult to assess the potential future relevance of the proposed quantum algorithm.

Furthermore, if possible, incorporating classical benchmarks would significantly strengthen the manuscript. A comparison with state-of-the-art particle reconstruction methods (e.g., combinatorial track finding, graph-based methods, or machine-learning approaches) would help clarify the potential advantages and limitations of the proposed quantum algorithm and allow a more realistic assessment of its practical relevance.


- Limitations of the algorithm and realism of the model:

A clearer and more prominent discussion of the limitations of the algorithm should be included, particularly in the Introduction and Discussion sections. The method is demonstrated only on a simplified toy model consisting of straight tracks across up to three detector layers and currently limited to four particles. In realistic detector environments, additional complications such as multiple scattering, detector inefficiencies, non-Gaussian uncertainties, and combinatorial ambiguities will arise. The spectral filtering approach may become significantly more challenging under such conditions. While some of these limitations are mentioned, they should be emphasized more clearly and discussed in greater depth.


- Generality of the conceptual advance:

While the reformulation from matrix inversion to a binary spectral filtering procedure is technically interesting, it remains unclear whether this represents a sufficiently deep or broadly applicable conceptual advance in quantum algorithms. The approach appears strongly tailored to the specific structure of the constructed Hamiltonian and to idealized spectral separation assumptions. The authors should clarify to what extent this methodology generalizes beyond the presented toy model and whether it provides insights that could be transferred to other classes of problems. A more explicit positioning of the work within the broader landscape of quantum algorithm development would help justify its significance for a general physics audience.


Minor Comments:

- The algorithm has only been tested using emulators, and no real quantum devices have been used, even though the reported circuit depths and qubit numbers might allow execution on current quantum hardware. Including at least a small proof-of-principle implementation on actual hardware would strengthen the manuscript and support claims of near-term applicability.

- In Figure 2(c), using logarithmic scales on both axes would make the scaling differences clearer. Additionally, the labeling in the caption and the figure panels appears inconsistent (the references to (a) and (b) do not match).

- Above Equation (6), the state $|b\rangle$ should be written as a superposition with all qubits prepared in the $|+\rangle$ state, as one would expect from applying Hadamard gates to qubits initialized in the state $|0\rangle$. As written, this does not appear to be the case and should be corrected or clarified.

- In Step 4 (Equation (10)), adding one intermediate equation showing explicitly how the time register is restored would improve clarity.

- A brief comment on whether or not error mitigation techniques were applied in the simulations would be helpful.



Reviewer #3 (Remarks to the Author):

1. Summary

The authors present a tailored implementation of the HHL algorithm in the context of track reconstruction in high-luminosity scenarios, with a focus on the LHCb Vertex Locator. With their implementation they significantly reduce the circuit depth and achieve a considerable improvement on the asymptotic gate complexity. The resulting polynomial gate complexity is also better than the one corresponding to previous implementations. Furthermore, authors benchmark their circuit in two different platforms, discussing the performance on both of them in terms of two-qubit gate count, Signal Separation Index and the Hellinger Fidelity.

2. Originality and significance

The results presented in this work constitute a significant step forward towards a realistic implementation of one of the most challenging computational tasks for HEP experiments, track reconstruction, algorithm using Quantum Computing. The results, while quite tailored to a particular use-case (the Vertex Locator at LHCb) are undoubtedly very useful for the field, as they target a common problem that is the overhead introduced in circuits when using QPE via Direct Structural Synthesis. This is especially important considering that this is partially what makes the HHL algorithm notably challenging for near-term devices. The analysis and interpretation of their results is robust and well-documented by their metrics assessment. The work is presented in a clear and precise manner.

3. General comments

Authors study the asymptotic gate complexity and part of their claim is the improvement of such complexity with respect to previous studies, which is of course correct from a computational standpoint. At what value of N does this asymptotic advantage become significant compared to state-of-the-art methods? And how is this N relate with the HL-LHC expected luminosity?
In a similar way, it would be useful if authors explicitly state the timeline for HL-LHC in the introduction. This will help give context when considering whether ‘near-term’ or ‘fault-tolerant’ align better with such timeline.
I understand the focus of the hardware discussion is on actual and near-term devices. However, I think it would be useful for authors to include in the Results section a short discussion about fault-tolerant scenarios as well, and whether their conclusions e.g. regarding connectivity would hold for those, also considering the expected ‘scaling’ in size quantum computers for both of the technologies that are considered and the effect of ‘noise floor’.


4. Comments on the draft

Methods:
Definition of theta is missing before Eq. 2.
Authors refer to the ‘resulting matrix’ in L. 68 before defining the matrix in Eq. 9. I suggest moving the remark ‘ensuring the resulting matrix is positive semi-definite’ until after the definition of the matrix and how it maps to HHL.
I find it difficult to follow the reasoning in L. 129-130 and Eq. 4: do the diagonal coefficients determine the combinatorial noise because of their position in the matrix?
Is beta in Eq. 5 the same as the beta term in Eq. 1? I understand it isn’t, since beta was set to 1 beforehand. I suggest rephrasing this to explain it and/or changing this beta if it doesn’t refer to the same thing.
L. 179: ‘controlled global phase rotation’ is repetition from L. 177
L. 183: remove ‘and’
L. 221: define n_s (also in Fig. 3)
L. 240: at least one reference is needed for ‘classical exhaustive search methods’
L. 263: it would be helpful for the reader to give an example of ‘dense physics observable’
Caption Figure 4: please make sure to indicate what the labels of the points ‘#p’ mean.
Figure 4: The uncertainties from c_filter seem too small - what are the uncertainties from a?
L. 282: using Qiskit’s Aer simulator —> with Qiskit’s Aer simulator
Figure 5: modify linestyles to make it BW friendly.
L. 316-317: ‘These false positives satisfy the Hamiltonian constraints and contribute to the final solution state, increasing the P_succ even though they do not correspond to any real tracks’. —> This sentence should be further developed, discussing potential ways to mitigate this and/or what percentage does this rate represent, also compared to SOTA. Moreover, I think here or the Discussion section are good places where to comment on how well does the algorithm perform with realistic simulation, e.g. missing hits in layers.
L. 277 and 340 have inconsistent way of writing ‘r1, r2, r3’

Discussion:
This section would benefit for more explanation on why are Primary Vertices or z-vertices useful information on their own. It may seem straightforward for an LHC experimentalist but it’s not necessarily the case for a wider QC4HEP audience.

I would also suggest authors comment on how generalizable is this method for tracking, and whether the sparsity of the matrix holds for HL-LHC. I am aware this was indicated in previous works, but it would add completeness to this paper.

References:
The way of formatting DOIs needs to be made uniform across references.
DOI is missing from refs. 23, 24, 30
arXiv references also need to be made uniform, see e.g. refs. 20 and 22
Add journal to ref. 29
Some references are missing capital letters when compared to the publish papers. Please check. I have found it to be the case for refs. 1, 5, 6, 10, 11, 12, 15, 16, 17, 19, 20, 23, 31, 33