Percolation thresholds for photonic quantum computing

Despite linear-optical fusion (Bell measurement) being probabilistic, photonic cluster states for universal quantum computation can be prepared without feed-forward by fusing small n-photon entangled clusters, if the success probability of each fusion attempt is above a threshold, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{\lambda }}_{\mathrm{c}}^{(n)}$$\end{document}λc(n). We prove a general bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{\lambda }}_{\mathrm{c}}^{(n)} \ge 1/(n - 1)$$\end{document}λc(n)≥1∕(n-1), and develop a conceptual method to construct long-range-connected clusters where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{\lambda }}_{\mathrm{c}}^{(n)}$$\end{document}λc(n) becomes the bond percolation threshold of a logical graph. This mapping lets us find constructions that require lower fusion success probabilities than currently known, and settle a heretofore open question by showing that a universal cluster state can be created by fusing 3-photon clusters over a 2D lattice with a fusion success probability that is achievable with linear optics and single photons, making this attractive for integrated-photonic realizations.

10. At the very end of page 5 of the Supplementary Material, the authors refer to "yellow dashed lines". The dashed lines in Fig. 5 don't look yellow to me. This may just be a problem with my printer, but if they're not actually yellow, the authors should correct this.
11. In the first bit of text on page 6 of the Supplementary Material, the phrase "Z measurement followed by a Hadamard gate" should probably be reversed to become "Hadamard gate followed by a Z measurement" since it makes no sense to do a gate on a qubit that has been measured -and because the effect of an X measurement (H followed by Z-meas.) is what is being described.
12. The authors should ensure that the abbreviations 'GCC' and 'LCC' have been properly defined before use in all sections of the main text and also separately in the Supplementary Material. 13. Typos * (main) page 5, right column, near the top: "black bonds" should probably be "black nodes" (?) * (main) page 6, left column, near middle: "it's lifetime" should be "its lifetime" * (main) page 7, left column, near top: "high-Fidelity" should be "high-fidelity" * (supplement) page 1, left column, near top: "if there are a path" should be "if there is a path" * (supplement) page 2, left column, about 1/3 of the way down: within "L_{0k} = \{ v_{al}, v_{bl} \}", the subscript {bl} should probably be {bm} (I think). * (supplement) page 2, left column, final paragraph: "if \{ v_a, v_b \} if" should be "if \{ v_a, v_b \} is" * (supplement) page 2, right column, near top: The sentence "From Lemma 3..." is grammatically incorrect. * (supplement) page 2, right column, about 2/3 of the way down: "every vertex is" should probably be "every vertex" (?) 14. Lemmas and Theorems in the Supplementary Material: The authors should use the 'amsthm' LaTeX package and its included {theorem}, {proof}, etc. environments. This ensures that lemmas and theorems are numbered uniquely throughout the document and that they and their proofs are formatted correctly (including ending proofs with a \qed symbol).
Once these changes are properly addressed, I strongly recommend publication.
Reviewer #3 (Remarks to the Author): In the manuscript, titled "Percolation thresholds for photonic quantum computing", authors studied linear optical quantum computing using the ballistic strategy and presented some interesting results.
Using linear optics to entangle photons is one of the most promising way of realising quantum computing. Preparing scalable cluster states is the critical step in linear optical quantum computing. Ballistic protocols are attractive protocols for preparing scalable cluster states. Ballistic protocols are not well studied as the alternative family of protocols, KLM and its variants. It is still an open question whether the performance of ballistic protocols can be superior to KLM family protocols. However, we know that KLM family protocols require a huge number of linear optical instruments [e.g. see PRX 5, 041007 (2015)]. It may turn out to be the case that the optimal protocol combines two strategies. General conclusions about the ballistic strategy are important.
The manuscript presented two general conclusions, which are the first two of four results as summarised by authors. However, these two conclusions are not sufficiently justified. Therefore, I do not recommend the publication of the manuscript.
About the first general conclusion: The first general conclusion (main result 1) has two parts. The first part is a lower bound of the success probability threshold. If the success probability is higher than the threshold, scalable cluster states can be prepared using a ballistic protocol. The second part is a conjecture that the bound is tight. If the conjecture is true, a success probability higher than the lower bound is enough for preparing scalable cluster states.
All evidences supporting the second part are given in the appendix. I do not find that these evidences can support a reasonable conjecture accounted as a main result of a paper. I recommend authors to either move the second part from the first main result or provide more evidences. As a main result, authors should at least explain in the main text why they conjecture that the second part is true.
About the second general conclusion: The second general conclusion may be more useful. Authors find that the success probability threshold of a ballistic protocol is the standard bound percolation threshold on a lattice. This result can help us in looking for the optimal protocol of linear optical quantum computing.
The second general conclusion is sound. However, why the bound percolation threshold of the fusion-operation lattice is the success probability threshold is not proved or explicitly explained. Actually, it may not be difficult to prove their conclusion. I recomend authors to explicitly give the proof rather than hand-waving.
Other comments and questions: 1. It is costly to generate n-photon states starting from single photons. If this stage is considered, is it still that case that a larger n has advantage, because the lower bound decreases with n? Generating n-photon states using emitters is also discussed in the manuscript. How good the emitters should be in terms of the loss rate and fidelity?

Reviewer #1 (Remarks to the Author):
This paper applies ideas from percolation theory to develop bounds on the fusion statistics for qubit fusion. The authors use state-of-the art methods in percolation theory to analyze lattices not previously analyzed, in two, three, four and infinite (Bethe lattice) dimensions. The results should be useful for future design of qubit systems. I highly recommend publication.
We thank reviewer 1 for taking the time to read our paper and the highly positive review.

SUMMARY As the title indicates, the authors study percolation thresholds for photonic quantum computing. This work provides important general bounds for successful generation of photonic universal resource states for quantum computing, answers important open questions previously posed as conjectures, and provides an elegant graph-theoretic framework for studying the creation of general graph states from microclusters and probabilistic fusion operations (2mode and beyond).
We thank review 2 for taking the time to read our manuscript, and the constructive feedback.

REVIEW
This is an important, significant, well-written contribution to the literature regarding percolation-based approaches to photonic quantum computing. For all of the reasons given in my summary above, I recommend publication after the following are properly addressed: We thank reviewer 2 for recommending publication of our manuscript upon successfully addressing the comments below. Please see our response in line with the feedback below, and description of changes we have made to the manuscript, as necessary to address those.
1. I question the wording immediately below Eq. (6) --i.e., the claim that the authors have "almost clos [ed] the gap" to the general lower bound for n=3. After all, 0.5898 is closer to 0.611 than it is to 0.5, so there is still a large (proportional) distance to go to close this gap. Perhaps the authors could soften this language.
The 0.611 achievability threshold in Eq. (5) is one of our (new) results in this paper. This threshold corresponds to our new graphical method applied to the 4-dimensional extension of the generalization of the (10,3)-b lattice family we propose in this paper. Before this paper, the best-known threshold was ~0.625 (diamond lattice from Rudolph and colleagues). We changed the language to: "thereby bringing the best-known achievable threshold with $3$-photon clusters as the initial resource closer to our general lower bound applied to $n=3$, $\lambda_c^{(3)} \ge 0.5$."

At the beginning of Section II.E, the authors should double check the direction of the inequality on \lambda_c^{(3)} \geq 0.746. Should it perhaps be \leq instead? (That is, isn't it possible that there could be a lower value of \lambda_c^{(3)} obtained from an even-more-clever choice of graph?)
We apologize for the confusion wording in that paragraph. Please see below the modified wording of that paragraph: We found a 2D logical lattice, a modified brickwork (see Fig.~\ref{fig:2Dmodifiedbrickwork}), for which $\lambda_c = 0.746$. This gives a bound $\lambda_c^{(3)} \leq 0.746$, which is looser compared to the ones we obtained from the constructions in the previous section. However, since $0.746 < 25/32$, this result shows that single-photon-boosted linear optical fusion and $3$-photon GHZ state clusters are sufficient to generate a universal lattice ballistically by fusing the clusters on a {\em two-dimensional} lattice. Fusion on a 2D lattice is significantly simpler compared to 3D (or higher dimensional) lattices from an experimental standpoint, since 2D programmable linear optics is a fastmaturing technology. Whether or not the above is possible was posed as a challenge problem by Rudolph and colleagues in~\cite{2015.PRL.Gimeno-Segovia-Rudolph.3GHZtoBallisticQC,2015.PRA.Zaidi-Rudolph.BallisticLOQC}.
3. In Section II.F, it isn't quite accurate to say that " [p]hoton loss is *equivalent* to measurement by the environment in the Z basis, with an unknown measurement result..." (emphasis mine). The operation described is merely dephasing in Z. Photon loss would be equivalent to complete depolarisation, with a heralding cue or not depending on whether the loss is heralded. The reason this analogy works for the purposes to which the authors are using are putting it is that (a) the state being measured is unbiased in the Z basis, which means that dephasing results in a maximally mixed state anyway, and (b) the noise model can be weakened as much as one likes because it is being used to prove the impossibility of creating a giant connected component under certain conditions. Nevertheless, the authors should correct this language --lest an unfortunate graduate student think this is how loss can be modelled in all cases.
We have clarified this in the revised draft. Please see below the modified wording: Furthermore, for our bound, we treat loss as the removal of the corresponding node from the cluster state or a measurement in the $Z$ basis. In general, loss of a qubit is not equivalent to a $Z$ basis measurement. Loss results in tracing out the qubit, but treating it as a $Z$ basis measurement leads to a lower value of $\lambda_c$ ($Z$ basis measurement gives us more information than tracing out over the $Z$ basis) and is sufficient for a lower bound on $\lambda_c^{(n)}$.

Section I of the Supplementary Material would really benefit from a few diagrams. I was able to follow the proof, but an illustration or two would help. So would a brief outline of the method before diving into the mathematics. As such... (next item)
5. Also in Section I of the supplementary Material, I would recommend adding a sentence or two at the beginning giving some intuition for how the F_n, L, and S are going to be used. This becomes clear as the proof progresses, but some foreshadowing of their roles as they are introduced would really help the reader.
We have added a proof outline to the beginning of Section I which now foreshadows how the roles of the different pieces of the proof and also provide intuition to the reader. Furthermore, we have added Fig. 1 to the supplementary material which depicts the mapping between the different operations used in the proof to guide the reader.

The final step in proving (the first-named) Lemma 3 involves a claim about bond percolation on a graph with maximum degree n. The authors should add a rigorous proof --or a citation to such a proof --justifying the particular claim being used (regarding o(N) scaling).
We have added citations which contain the proof (Ref. 1, 2 in the supplementary information).

The end of Section I of the Supplementary Material states that "[t]he three lemmas can be proved with the same reasoning...", but the "proof" of (the second-named) Lemma 3 (which should receive a new, unique number, as requested below) should be more specific since there is a new parameter m and a new claim being made about what happens when \lambda < 1/(n-1)(m-1). It's plausible that (this second instance of) Lemma 3 might be true, but the authors should add an explicit proof or a citation to such a proof.
We have changed the numbering of the Lemmas used in the generalization to the m node fusions to 4,5 and 6. Furthermore, these Lemmas now have their own separate proofs in the revised draft that can be read independently. We have also added Fig. 2 to the supplementary material to aid the explanation of Lemma 6.

On page 5 of the Supplementary Material, it is argued that "it is simple to argue that there are O(N) red nodes in the GCC." The authors should state this "simple" argument explicitly.
We have modified the following paragraphs leading up to the argument, to make this explicit.

~~
Let us say there are $N$ unmeasured photons in the lattice in Fig.~\ref{fig:Keilingtonow}(b). It is clear from the figure that there are $8N$ photons that get measured, making it a total of $9N$ photons. The logical graph in Fig.~\ref{fig:Keilingtonow}(e) has $3N$ nodes, where each node represents a $3$-photon (GHZ) cluster.
Each bond of the logical graph $G$ represents a fusion attempt. All the photons within the black nodes disappear after the fusion attempts have been made but help provide long-range connections, if more than $\lambda_c$ fraction of the bonds (fusion attempts) were successful. Only the red nodes, which in the example of These $O(N)$ nodes have both red nodes and black nodes. However, since every black node has a red node at a constant distance (one bond away) from it, the probability of a red node being connected to the nearest black node is a constant. Therefore, the presence of $O(N)$ black nodes in the GCC implies the presence of $O(N)$ red nodes in the GCC, and hence, there must be $O(N)$ red nodes in the GCC whenever there are $O(N)$ nodes in the GCC. ~~ 9. Section III.A of the Supplementary Material refers only once to "the Bethe lattice above". Using this terminology in isolation is confusing. If the authors wish to retain it, then they should state exactly where "above" they are referring, and at that "above" location, they should label the lattice as a "Bethe lattice" (I'm assuming it's the one in Fig. 3).
Referring to the graph as the Bethe lattice is not necessary and the reference has just been changed to "the lattice above" 10. At the very end of page 5 of the Supplementary Material, the authors refer to "yellow dashed lines". The dashed lines in Fig. 5 don't look yellow to me. This may just be a problem with my printer, but if they're not actually yellow, the authors should correct this.
The lines are yellow, but to avoid any confusion, we've changed it to simply "dashed lines", since they are the only dashed lines in the figure. 11. In the first bit of text on page 6 of the Supplementary Material, the phrase "Z measurement followed by a Hadamard gate" should probably be reversed to become "Hadamard gate followed by a Z measurement" since it makes no sense to do a gate on a qubit that has been measured --and because the effect of an X measurement (H followed by Z-meas.) is what is being described.
The referee is correct in stating that an X measurement is the same as a Hadamard followed by a Z measurement. However, in the particular case of X measurement of a qubit in a clique cluster state, an X measurement has the same effect as a Z measurement on a qubit and a Hadamard on one of its neighbors. In the previous version, it was unclear that the Z measurement and Hadamard were on different qubits. To prevent any confusion, we have rephrased the statement, and have also added a line pointing the reader to a reference which shows why this is true: "The $X$ measurement of a qubit in a clique removes the measured qubit and all its edges, and applies a Hadamard gate to one of the qubits that neighbored the measured qubit (before it was removed). This can be seen by looking at the set of stabilizers obtained when an a qubit in a clique is measured in the $X$ basis as detailed in \cite{2015.PRA.Zaidi-Rudolph.BallisticLOQC}"

The authors should ensure that the abbreviations 'GCC' and 'LCC' have been properly defined before use in all sections of the main text and also separately in the Supplementary Material.
Please see the definitions in the response to question 8 above.

Lemmas and Theorems in the Supplementary
Once these changes are properly addressed, I strongly recommend publication.

Reviewer #3 (Remarks to the Author):
In the manuscript, titled "Percolation thresholds for photonic quantum computing", authors studied linear optical quantum computing using the ballistic strategy and presented some interesting results.
Using linear optics to entangle photons is one of the most promising way of realising quantum computing. Preparing scalable cluster states is the critical step in linear optical quantum computing. Ballistic protocols are attractive protocols for preparing scalable cluster states. Ballistic protocols are not well studied as the alternative family of protocols, KLM and its variants. It is still an open question whether the performance of ballistic protocols can be superior to KLM family protocols. However, we know that KLM family protocols require a huge number of linear optical instruments [e.g. see PRX 5, 041007 (2015)]. It may turn out to be the case that the optimal protocol combines two strategies. General conclusions about the ballistic strategy are important.
The manuscript presented two general conclusions, which are the first two of four results as summarised by authors. However, these two conclusions are not sufficiently justified. Therefore, I do not recommend the publication of the manuscript.
About the first general conclusion: The first general conclusion (main result 1) has two parts. The first part is a lower bound of the success probability threshold. If the success probability is higher than the threshold, scalable cluster states can be prepared using a ballistic protocol. The second part is a conjecture that the bound is tight. If the conjecture is true, a success probability higher than the lower bound is enough for preparing scalable cluster states.
All evidences supporting the second part are given in the appendix. I do not find that these evidences can support a reasonable conjecture accounted as a main result of a paper. I recommend authors to either move the second part from the first main result or provide more evidences. As a main result, authors should at least explain in the main text why they conjecture that the second part is true.
About the second general conclusion: The second general conclusion may be more useful. Authors find that the success probability threshold of a ballistic protocol is the standard bound percolation threshold on a lattice. This result can help us in looking for the optimal protocol of linear optical quantum computing.
The second general conclusion is sound. However, why the bound percolation threshold of the fusion-operation lattice is the success probability threshold is not proved or explicitly explained. Actually, it may not be difficult to prove their conclusion. I recommend authors to explicitly give the proof rather than hand-waving.