We thank the referees for their thorough reading of our paper and for their helpful comments and suggestions. Below we address each report separately.

All page numbers mentioned here refer to the page numbers of the revised manuscript, which we are uploading to SciPost in this re-submission. Alternatively, for the convenience of the referees, we have generated a PDF file of our revised manuscript with changes coloured in blue and red, but its page numbering should be ignored (see "List of changes").

Report 1

REFEREE: As the authors state, there are efficient classical algorithms to calculate color factors and this manuscript aims to develop their proposal in this simpler setting (perturbative QCD + only color factor), with the perspective of extending it to the calculation of the kinematic part of Feynman diagrams.

AUTHORS: We would like to highlight that this work's focus on perturbative QCD (rather than lattice QCD) should not be seen as a limitation or undesirable approximation. At present, perturbative QCD is the only viable method for producing theoretical predictions for cross-sections at high-energy colliders like the LHC, where the relevant distance scales are shorter than (hbar/Lambda_{QCD}) by 3-4 orders of magnitude. This is because (in 4 space-time dimensions) the number of lattice points scales like (lattice spacing)^-4, and the cost of (classical) lattice QCD simulations scales with an even larger power of lattice spacing, e.g. (lattice spacing)^-7. A lattice-based simulation of QCD at the length-scales probed by the LHC would hence require at least 10^12 - 10^16 lattice sites and is therefore far beyond the reach of current methods (both classical or quantum).

REFEREE: ... extending it to the calculation of the kinematic part of Feynman diagrams. However the paper lacks concrete discussions about how to make this a viable and practical path, thus limiting the significance of the manuscript. In particular, I see as major hindrances the need for presumably much bigger qubit registers and much deeper circuits when considering kinematic factors.

AUTHORS: To perform quantum computations for high-energy physics processes, the colour parts are a good starting point because the circuits are small enough to be simulated on today's ~30 qubit simulators. Some of the ingredients developed in this work, such as the unitarisation register, could also be applied in future extensions to the kinematic parts. While it is true that the kinematic parts are likely to require much larger circuits than the colour parts, they could certainly be competitive with existing state-of-the-art proposals for the quantum computation of high-energy physics processes. For example, lattice-based proposals like the widely-acclaimed paper [Jordan, Lee, Preskill, (2012) Science 336, 1130-1133] would also require very large circuits to simulate QCD at LHC-like scales, since the lattice would need to be very large for the reasons described above. We therefore believe that the likely larger circuits for the kinematic parts (compared to the circuits for the colour parts) should not be seen as limiting the significance of the manuscript. (We have added a sentence about this to the Conclusion.)

REFEREE: Further, even in this simplified setting, a very high number of repetitions is required to extract the absolute value of the color factors, even in the absence of any realistic noise. The authors could add at least a discussion about noise for the the basic elements of their circuit, considering the high number of entangling gates required.

AUTHORS: Regarding the number of repetitions: our aim was to produce gates to simulate interactions of quarks and gluons, and test it by producing a quantum state encoding the desired colour factor as shown in eq. (12). The repeated-measurements strategy is only intended to be a transparent method for validation, i.e. to confirm that we have produced the state shown in eq. (12). More advanced methods exist for examining, measuring, and exploiting quantum states; Quantum Amplitude Estimation is an example which is quadratically faster than the simple repeated-measurement strategy employed in this paper. We have added a few sentences to the paragraph between eqs (12) and (13) to clarify this, as well as further additions on page 16: we have expanded the first paragraph on page 16, and also added a new paragraph just before the start of section 4 on page 16.

Regarding noise: the present paper is not aimed at near-term noisy quantum computers, but instead aimed at error-corrected quantum computers that are being promised over a medium-term timescale of around 10 years by companies like Google and IBM. We agree that it could be interesting to study noise for specific hardware setups or models. In preparing this revision to the manuscript, we tried to test our implementation using several noise models built into Qiskit, but we did not find a noise model that allowed custom 2- or 3-qubit gates to be defined. Since noise is highly dependent on specific hardware, it is (for now) difficult to make a reliable generic statement about noise. Nonetheless, we hope that following our present work, others can build on this by designing circuits that are customised to the noise characteristics and basis gates available on specific hardware setups (present and future). We have therefore inserted a conservative statement about noise into the Conclusion (page 17).

REFEREE: As a secondary point, the unitarisation register is a core concept of the paper. I think it would be appropriate to have a more explicit and concrete comparison with the existing methods to enforce unitarity mentioned in footnote 1. Why is the proposed unitarization method more appropriate in this case? What are its advanteges?

AUTHORS: To address this, we have added some explanations to: footnote 1, the paragraph after eq. (16), and the paragraph after eq. (19).

Report 2

REFEREE: It is currently not clear what is actually being measured to extract the color factor in a given computation and how that color factor is actually derived then from the measured bit-strings. Please add this information (in Fig. 1 and explain it in the corresponding sections). Further, from the results in Table 1, obtained with 100 million shots, it appears that the precision of the result scales poorly with the complexity of the diagram (c.f. diagram 1 with two color vertices has 0.1% precision versus diagram 7 with 4 color vertices which has ~10% precision). If this scaling would extrapolate to more complex diagrams, then the quantum computer would do worse than a classical computer. It would again help to explain the measurement process (i.e. which register is measured). This will help the reader better understand. The authors should address the measurement cost with growing complexity of the diagram.

AUTHORS: To clarify what quantity is measured, we have added further explanations after Eq. (12) and also added footnote 10 in the paragraph following Eq. (38). This should clarify how the colour factors are extracted and ease the understanding of Eq.(39). Regarding the apparent scaling, please see our response to Report 1 above (the paragraph beginning "Regarding the number of repetitions").

REFEREE: It is my understanding that the computational difficulty of evaluating Feynman diagrams is primarily from the combinatorial complexity, i.e. the number and topologies of Feynman diagrams, increasing with each loop order. Obviously this is then closely related to also computing the kinetic part (i.e. evaluating the loop integrals). This computational complexity comes from the large possibilities / superposition of intermediate many-particle states (`virtual particles’) contributing to a given process. The authors do not address this complexity superposition, instead propose to use compile a quantum circuit for every single Feynman amplitude.

AUTHORS: In this article, to validate our Q and G gates, we use them to build circuits that calculate colour factors for individual diagrams. However, we certainly do not ultimately intend for the Q and G gates to be used to build a separate circuit for each diagram, since such an approach would be unnecessarily costly. Instead, we intend that the Q and G gates will serve as building blocks for future, higher-level algorithms that can calculate multiple diagrams simultaneously. For this reason, we have designed the Q and G gates to maintain quantum coherence, acting linearly on quantum states (see eqs. 14 and 15) so that the state of the quantum computer closely mirrors the quantum state of the simulated quarks and gluons. This means a higher-level algorithm can parallelise over Feynman diagrams, or, even better, recursively calculate intermediate states in a manner similar to today's best tree/loop-level many-particle classical algorithms (see [Berends, Giele, (1988) Nucl. Phys. B306 759-808] for the original idea). While performance details will depend on the specific algorithm design, which we leave to future work, we believe a quantum algorithm using our Q and G gates would be well-suited to handling the complexity from the number of diagrams since, as the referee points out, quantum mechanical superpositions are the origin for the large number of diagrams. We hope all of this is clear in the revised manuscript, particularly in light of some of the additions to pages 6, 7, 8, and 16 (see list of changes for details of changes/additions).

REFEREE: In light of 1., where the measurement cost for every amplitude is not clear, can one really make the point that this is a viable strategy, as opposed to e.g., real-time evolution of a lattice formulation of the model? How many registers are needed for a given loop order? There will be a gluon register for each gluon line, a quark register for each quark and an ancilla (“unitarisation”) register for every vertex?

AUTHORS: Real-time evolution of a lattice formulation is likely to require a very large number of lattice sites and qubits (O(10^12-16)) for the reasons we describe above in our answer to Report 1.

Regarding the unitarisation register: there is only one unitarisation register, regardless of the number of quarks, gluons, and vertices. The size of the unitarisation register does grow with the number of vertices, but only logarithmically. We have added a new sentence in the paragraph between eqs. (19) and (20) to highlight the small size of the unitarisation register.

Regarding the number of registers: There is a gluon register for each gluon, and a pair of quark registers for each quark line. This is explained in sec. 2 [in the paragraph that follows eq. (3)], as well as in the second paragraph of sec. 3. Since the number of quarks and gluons grows linearly with loop order, the number of registers also grows linearly with loop order.

REFEREE: It is unclear what happens when the Q gate is applied a second time. After the first application, the ancilla register is not in its original state (denoted by “term orthogonal to | \Omega_U >” in Eq.9?). From the discussion in 2.2 it is not clear to me what happens if this register is not initially in the state | \Omega_U > when Q is applied a second time. The state \Omega_U > seems to be assumed here. Shouldn’t this register be un-computed, as is usually done for repeated quantum function evaluation? Again, more detailed explanation would help, including what information is contained where and how it is entangled across registered (also important for measurement later).

AUTHORS: To clarify these questions, we have modified and expanded the explanations in sec. 2.2, in the paragraphs between eqs. (19) and (20). Regarding the question of un-computation: we do not un-compute the unitarisation register U, but our unitarisation strategy ensures that U can be very small (logarithmic in the number of times we want to perform non-unitary operations in our circuit). If we fully un-computed the unitarisation register, the overall operation would become unitary, which is contradictory since the quark and gluon interactions that we wish to implement are non-unitary. Despite not performing un-computation, our strategy preserves the quantum coherence of states produced by the non-unitary operations, and thus allows these operations to be combined, as well as used as building blocks in higher-level algorithms; indeed this is why the results in Table 1 match the analytic predictions.

REFEREE: Most controlled operations are presented only in a very high level way. Can the authors show the decomposition into fundamental gates at least of some of them, e.g., the operator M in Eq. 31, or Gamma, which involve quite complicated control operations. I know one can set these by hand in qiskit, but in an actual circuit this will factor into the complexity of the circuit.

AUTHORS: We have added Fig. 12 for an explicit representation of the gate Lambda in the appendix, along with a new paragraph above it in which we discuss Lambda and also M. Regarding the complexity of the circuit: it is true that each Q or G gate requires dozens of fundamental gates, but these circuits are still practical to implement on suitable simulators or future hardware, since the circuit depth is linear in the number of Q and G gates (i.e. in the number of interactions of gluons or quarks), whereas the difficulty of calculating a Feynman diagram (at least classically) grows extremely quickly (faster than exponentially) with the number of interactions.

REFEREE: Please expand on Eq. 16 especially in connection to my point 3. Is this to mean that the state of the ancilla register is unchanged? Or are the additional terms. If so, left and right hand side could differ by a constant. Please provide more details.

AUTHORS: The quantum state contains additional terms, but in eq. (16) we have contracted with |Omega> so that we emphasise only the part which is of interest. We have added a few sentences to the explanations in the paragraphs between eq.(19) and eq. (20) to provide more details about this.

REFEREE: The increment operation is clearly explained in Fig. 3, the corresponding discussing around Eq. 24 however is unclear. Make clear that in each recursion step there is actually a control operation involved.

AUTHORS: There are two ways in which the increment operation A_n can be defined recursively: either splitting off the single-qubit X gate on the top right of fig 3, or instead splitting off the left-most gate (X_n, which is by definition an X gate with n-1 control qubits) on the RHS of fig 3. As the referee points out, the first choice of definition would mean defining A_n to be a controlled version of A_{n-1} followed by a (non-controlled) X gate. However, in this article we instead made the second choice (to simplify the notation) and this means defining A_n as the X_n gate followed by the A_{n-1} gate. Unlike the first way, this second way does not require any additional controls on the A_{n-1} gate. The two possible definitions are equivalent and both lead to the same circuit (the one in fig. 3). We have added a footnote to the discussion surrounding eq. (24) to clarify this.

All page numbers in the list below refer to the page numbers of the revised manuscript, which we are uploading to SciPost in this re-submission.

In addition, for the convenience of the referees, we have generated a PDF file of our revised manuscript with changes coloured in blue and red, which can be found at the following URL (but its page numbers necessarily diverge from those in the list below): https://ora.ox.ac.uk/objects/uuid:0f1ce9a6-1c04-403a-8ca3-e52bc4775a43/files/rtt44pn652

On page 3, expanded footnote 1.

On page 6, edited and expanded the two paragraphs that follow eq. (12).

On page 7, expanded the sentence "In this work, we have sought a way ..." in the paragraph after eq. (16).

On page 8, edited and expanded the first paragraph.

On page 9, added a footnote to the sentence before eq. (24).

On page 14, added a footnote to the last line on the page.

On page 16, added new sentence ("Since the output (37) is ...") to the first paragraph.
On page 16, added new paragraph ("It may be observed ...") to end of Section 3.

On page 17, expanded the text by adding two new sentences ("This is likely to require ..." and "Finally, although the present work ..."). Also changed a word: "Finally, it would also" -> "Furthermore, it would also".
On page 17, added sentence ("We also provide a circuit diagram ...") to first sentence of the Appendix.

On page 18, added new paragraph ("Finally, the explicit form of the ...") and new figure (Fig. 12).

On page 21, added new reference (Ref. [46]), to which we refer in the first paragraph on page 16.