Introduction to Topological Quantum Computation

1- Good introductory survey of the field of topological quantum computing for a non-expert

2- Covers appropriate definitions and examples (Fibonacci and Ising anyons) without going into unnecessary detail.

3- Gives complete non-technical description of how to use anyons to perform quantum computation (at least to the point of Ising anyons), up to the level of detailing Majorana wires in the Kitaev model.

Nil.

This is a well-written review article that was a pleasure to read, and a nice addition to the existing literature for non-experts wishing to learn more about topological quantum computing. Though other similar reviews do exist (Freedman et. al., Nayak et. al., Trebts et. al., as well as lecture notes by Roy and DiVincenzo and by Preskill, and books by Wang and by Pachos), this distinguishes itself by being short, non-technical, and having a specific focus on anyon computation rather than specific physical realizations.

As it stands, none of the requested changes below are critical, but instead comments to further improve the manuscript. As it is aimed at an audience not well-versed in the field, I feel that readibility and accessibility is key, and hope that these suggestions can help with this.

1- In general, as this review is for non-experts in the field, it would benefit from more interconnection between the various sections. For example, Sec.5.1-5.2 gives a physical model which realizes the Ising anyons of Sec.2.4; a forward reference to the latter section in Sec.2.4 would help a more physically-minded reader who is finding it difficult to grasp the notion of Fusion channels.

Similarly, in Sec.2.1 the fusion channels are described as the only options for the order in which anyons are to be fused. A naive reader might ask why in Fig.3 particles a and c cannot first be fused. Tying this section together with Sec.2.2 to answer why this case is not considered (because such a fusion would first require braiding of a and b or c and b) would probably be of assistance to the reader.

Also, the 'sigma anyons' in Sec.4.1 should probably carry a reference to Sec.2.4 for a reader who has skipped this example.

Also, a reference to Sec.5 in the last paragraph of Sec.2 where experimental realizations of Ising anyons exist might be useful.

2- In the definition of fusion channels (Sec.2.1), the authors assert that the Fusion rules give rise to a basis for the Hilbert space in which a topological quantum computation would be performed. Although this is unequivocally true, it is very non-trivial and somewhat counter-intuitive to a new reader (after all, fusion is a process occurring over a span of time, whilst a Hilbert space is usually associated with a system at a fixed point in time). The nature of the basis vectors is described at (in appropriate detail) in later sections (in particular 5.2.1), but it might be preferable to either link to this in the earlier section, or give a longer description of what a fusion space describes physically (this difficult given the fairly detailed maths underpinning this section, but if a few sentences with a reference were added here it would add much insight to a key part of the text).

3- Section 4.2 describes the link between quantum dimension and entanglement entropy, which gives means by which the anyon model for a given physical system can be partially understood from entropy scaling. However, the purpose of this section is not made clear in the first few paragraphs, which instead discuss long-range correlations and the physical system topology. At this point in the review, this discussion is a slight diversion, and the entire section is somewhat difficult to follow. I would suggest dropping the second paragraph (which could potentially be moved to a separate section and expanded on, but this is entirely optional), and combining the first and third paragraphs to immediately draw the reader's attention to equation (26) and the relationship $\gamma=\log \mathcal{D}_M$, as these are the key points of this section.

4- In Sec.1.1, the fact that one particle encircling the other is equivalent to two exchanges is somewhat non-trivial. I do not know a good way of explicitly demonstrating this without using diagrams (i.e. showing a deformation of the two exchanges to a loop). But perhaps the sentence below equation 1 can be rewritten to describe this in more detail. The word 'encircle' here should probably be replaced by 'pass' as well. On a related note, it is somewhat difficult to see that the line in the 3D paths of Fig.1 should go behind the right-hand particle; perhaps this should be noted in the caption?

5- Recent work on Majorana modes in 2D heterostructures (theory: https://arxiv.org/abs/1608.08769 and https://arxiv.org/abs/1609.09482 , expt: https://arxiv.org/abs/1705.05049) seems somewhat promising and should probably be mentioned along with the wire experiments.

6- I would switch Sec.2.4 and Sec.2.3, as Ising anyons are a bit easier to understand as a first example than Fibonacci anyons.

7- In Sec.2.4, below equation 14, the Hilbert space dimension is only given for systems with 2N Ising anyons, whilst the example above uses an odd number thereof. It should probably be explained that Ising anyons only come in pairs, and that the example given in eq (13) considers a subsystem where an additional anyon would also be present.

8- Sec.5.2.2 could include some more details on proposals for universal quantum computation in Majoranas, as well as perhaps some more details of the evolution of the quantum state whilst Fig.10 evolves, so as to make better connection between the braiding and the underlying physical model.

9- In equation 3, it could be noted that if $2q\phi=2\pi n$ the same evolution occurs, but it is bosonic/fermionic.

10- When describing the classification scheme for systems that support anyons in Sec.1.2, a sentence on how general this scheme is would be useful.

11- There are a very large number of grammar errors throughout the paper. Below I have written down a list of those I noticed whilst reading, however it is quite possible that I missed many myself, and so it would be advisable to further run the text through a grammar editor.

Abstract,
s.4: 'general steps how to use' -> 'general steps for using', 'as well as discuss' -> 'as well as discussing the', 'various ways' -> 'various ways that'.

Sec.1:
par.1, s.3: 'statistical behaviour' -> 'statistical behaviour,'

Sec.1.1:
par.2, s.1: 'However, in the real world small definitions matter.'
par.2, s.6: 'error in implementing quantum gates' -> 'erroneous manipulation of the physical system'.
par.4, s.2: 'it dictates' -> 'this dictates'.
par.7, s.1: 'is not equivalent to actually existing' -> 'is not equivalent to it actually existing'.
par.8, s.1: 'The study of the strongly correlated system' -> 'The study of strongly correlated systems'.
par.8, s.3: 'This enables to' -> 'This enables us to'

Sec.1.2:
par.1, s.2: 'It is possible to characterise them' -> 'It is possible to characterise this distinction', 'enable to classify' -> 'enable a classification of'.
par.1, s.3: 'topological classification is only defined given that' -> 'topological classification requires only that'.
par.1, s.7: 'but due to the particle-hole symmetry' -> 'but due to particle-hole symmetry'.
par.1, s.9: 'the most experimentally accessible ones' -> 'the most experimentally accessible anyons'.
par.5 s.1: 'FQH states can also emerge' -> 'FQH states can emerge'
par.6, s.4: 'study physically most common' -> 'study (the physically most common)'
par.6, s.5: 'could exist [34]' -> 'could exist [34],'
par.9, s.1: '(iii)' -> '(iv)'.
par.9, s.3: 'cold atoms [68]' -> 'cold atoms [68],'

Sec.2:
par.1, s.1: 'what are their defining properties' -> 'what their defining properties are'

Sec.2.1:
par.2, s.1: 'fusion channel degree of freedom enables to define' -> 'fusion channel degree of freedom enables the definition of'
par.3, s.6: 'set of the fusion matrices' -> 'set of fusion matrices', 'the pentagon equations, the $F$-matrices as they are often called,' -> 'the pentagon equations (often called $F$-matrices)'

Sec.2.2:
par.1, s.2: 'In particular, anyons are exchanged, or braided as the process is often called due to their worldlines forming braids.' -> 'In particular, anyons are exchanged or braided (as the process is often called due to their worldlines forming braids).'
par.1 s.9: 'Thus when there is fusion space degeneracy' -> 'Thus, when there is a fusion space degeneracy', 'of the exchanges but' -> 'of the exchanges, but'.

Sec.2.3:
par.2, s.5: 'enable to test and develop' -> 'enable testing and development of'

Sec.2.4:
par.1, s.2: 'implies that when brought together two' -> 'implies that, when brought together, two'
par.1, s.2: 'the last fusion rule which' -> 'the last fusion rule, which'.
last par, last sentence: 'be operated and then Section 5 illustrate' -> 'be operated, and then in Section 5 we illustrate'

Sec.3:
par.2, last sentence: 'present in laboratory' -> 'present in a laboratory'.

Sec.3.1:
par.2, last sentence: 'choose a computational basis as the pairwise fusion basis' -> 'choose the pairwise fusion basis as a computational basis'.

Sec.4.2:
par.2, last sentence: 'can not unambiguously' -> 'can not be unambiguously'.

Sec. 5.2.1:
par.1, s.3: 'braiding statistics is only defined projectively, i.e. up to the overall phase' -> 'braiding statistics are only defined projectively, i.e. up to an overall phase'

Sec. 6:
par.3, s.1: 'More realistic route' -> 'A more realistic route'.

- Readability and style
- Length

- Unclear intended audience and prerequisites (for an introductory text)
- Unclear placement with respect to existing literature and online material
- In several points, lack of (short) explanations / definitions of several advanced concepts

This work is an introduction on topological quantum computation. There are already several reviews/monographs/notes/lectures available on this topic, with different degrees of length, depth, prerequisites. The table of contents and lenght of this work may make it an appropriate early entry point to the subject, and in many parts the text is of good quality. However, it is not clear which type of reader the Authors had in mind when writing the text. This work often seems to assume advanced knowledge of condensed matter physics, and the topological aspects of it in particular, but (of course) no knowledge in topological quantum computation. This is a rather peculiar combination of background knowledge and may make this work not as useful as it could be to the truly uninitiated reader (the initiated readers already have plenty of material and references to increase their knowledge, and this work would likely not be used for that purpose).

Many concepts are introduced without proper explanation, and many references or remarks are made which cannot be useful to someone without an advanced level of research experience in the topic. Sometimes distinct concepts are conflated or thrown around in the same paragraphs, in a way which is easily interpretable by the experts but which may be confusing to beginners. I will point out several occurrences of these problematic passages in the list which follows below.

I believe that re-focusing several passages this work can find its place as an accessible, high-quality entry point to the field. Without such revision, it would remain a text without a clear purpose and without a clear audience, so I hope the Authors will consider this advice. Altogether, I believe SciPost is the right venue for this work.

The list of "requested changes" that follows contains some more open-ended remarks, and some more detailed ones.

This is a list of points where I believe the exposition is problematic. I do not always request an explicit change, but I believe that all these points should somehow be addressed in a revisited exposition.

1 - In the introduction, Sec. 1, the Authors manage to avoid mentioning two fundamental concepts behind the idea of topological quantum computation, adiabaticity and degeneracy. While these are discussed later, I think the two concepts are so important (while also being understandable to anyone with a good basic knowledge of quantum mechanics) that it would be beneficial to mention their crucial role topological quantum computation.

2 - Page 2, "(at least what comes to our every day energy scales)". This is a rather obscure remark, not sure what is meant by that.

3 - The paragraph below Eq. 3 is also quite obscure for the uninitiated reader. First, the concept of fractionalization is introduced here first without any explanation. It is linked immediately to strong interactions, but the connection is not clear. This is an example of a paragraph which would be unintelligible for someone without the proper condensed matter background, for instance a knowledge of fractional quantum Hall effect. Also, note Majoranas do not require strong interactions, yet they are the most treated example in the rest of the work; they are also not so easily linked to fractionalization or the Aharonov-Bohm effect. So it is also not clear that the concept quickly introduced here are beneficial to the reader to understand the material that follows.

4 - In the paragraph below Eq. 3, the sentence "This happens if the charge or the flux is fractionalised, i.e. $q \neq= n$ (in units of the electron charge) or $\Phi\neq 2\pi n$, respectively" is unclear. First, Eq. 3 is only a joint condition on the product $\Phi q$. Also, physical units for flux are off, or a definition of the flux quantum is missing.

5 - The first two paragraphs of Sec. 1.3 are too dense and can not be followed without knowing several advanced concepts of condensed matter physics. At this point in the manuscript a definition or description of topological order has not been attempted yet (not even a heuristic one), yet these paragraphs dwell on the distinction between SPTs and intrinsic topological order. Again, it's not clear what the benefit is for the reader. The role of topology in condensed matter is very broad and diverse and it is not always directly relevant for TQC, especially in a supposedly gentle introduction such as this one.

6 - Last sentence of page 5, "topological insulator heterostructures". This term seem to exclude semiconductor heterostructures, which are actually an even most prominent candidate to engineer topological superconductivity. I would suggest to reformulate the wording to make the term more encompassing of all the different proposals.

7 - Page 6, first paragraph: "off-diagonal conductance" is a confusing and unusual term for the familiar Hall conductance.

8 - Page 6, "Employing the intricate connection to conformal field theory,...". Which connection? No mention of such connection was made so far in the text. This sentence comes out of the blue and, again, is very obscure for someone not knowledgeable about the topic.

9 - Page 7, "... it has been known that if the pairing in a 2D superconductor is so-called p-wave type, i.e. time reversal is broken by the pairing term, then..." Breaking TRS is a necessary but not sufficient condition to obtain p-wave pairing, hence this sentence is slightly misleading. This passage should be expanded not to make things sound more simple than they are.

10 - Page 7, last paragraph before Sec. II. It is not clear from this paragraph if these quantum simulations are only equivalent to topologically ordered states via a non-local mapping, or if they realize the ordered state directly. Judging from the list of references, it probably conflates the two things, which conceptually is not helpful.

11 - In Sec. 2, it would be nice to give some more information about pentagon and hexagon equations. These are mentioned several times but never really discussed, not even at a heuristic level, so a sense of mystr is left in the reader. As a consequence, some passage are obscure. For instance, why do the hexagon equation have more than one solution, and to the pentagon equation also share this property?

12 - Sec. 2.4: "Unlike Fibonacci anyons, the fusion space of Ising anyons has a natural tensor product structure.". This sentence is crucial but also mysterious, because the strucutre of the Hilbert space of Fibonacci anyons was not discussed in the section regarding Fibonacci anyons. It should be explained how the Hilbert space of Fibonacci anyons has no tensor product structure, and why this is special. The same problem features in a similar sentence in the first paragraph of Sec. 3.1. The structure of the fusion space of Fibonacci anyons is invoked twice without prior discussion of the notion.

13 - Page 13, second paragraph, "means that it is also protected against any uncorrelated local perturbations." What does uncorrelated mean here? Is it a redundant adjective or is it essential? Do "correlated" local operations do something in the fusion space?

14 - After Eq. (21), "While this is likely to be challenging in the laboratory..." I would say it is more likely to be very noisy, than challenging. Such a coupling can be easily achieved at least in the case of Coulomb-coupled Majorana modes.

15 - Page 18, "In the presence of thermally excited anyons the statistics of anyons as is expected to become ill-defined and..." First, this sentence is grammatically broken. Second, it is not clear why the statistics of anyons becomes ill-defined.

16 - Sec. IV, first paragraphs. The use of the adjective "massive" is counterproductive here. First, this word has many heuristic meanings depending on the context. Second, I think there is no easy sense in which Majorana zero modes are "massive". What is missing here is a thorough discussion of the distinction between anyons as true quasiparticles (as in the FQHE) or as quasiaparticles associated with topological defects. So again, without further elaborations (which may go beyond the scope of the paper) the use of the term is confusing.

17 - The first pargraph of Sec.4.2 needs serious improvement: first, "the degeneracy of the ground
state in the presence of anyonic quasiparticle excitations" is an oxymoron (ground state in presence of excitations); second, anyons may not arise as true excitations of the system (topological superconductors have no anyonic excitations in the strict sense of the term); third, ground state property and properties of the excitations are mixed throughout the paragraph.

18 - "The degeneracy equals the number of distinct anyons (including the vacuum)." This statement is given without a heuristic explanation, and as such it is not very helpful and becomes quite forgettable.

19 - "The degeneracy is most conveniently obtained from the Verlinde formula [124]." This is another concept/formula which is name-dropped without explanation or clear benefit to the reader.

20 - Entanglement entropy is also never defined, not even heuristically. It is quite an advanced concept to assume a beginner reader to be familiar with. Same problem with "quantum dimension" and "topological twists" later in this section.

21 - Page 21, "Since the work of Ivanov, ..." While of course Ivanov's paper is a seminal one, I think the braiding properties of vortices in p+ip superconductor where also discussed in Read & Green.

22 - Page 22, "...implies that Majorana fermions could...". I feel that by now Majorana fermion is a discouraged term in this context. Also, terminology should be consistent and it seems in the rest of the work authors prefer the more precise "Majorana zero modes".

23 - Page 22, "It took 10 years to discover that Kitaev’s simple model could be realized by depositing a spin-orbit coupled semiconducting nanowire on top of a normal s-wave superconductor [56, 57]." I would mention the necessary presence of a magnetic field.

24 - Page 25, "These include switching domains of the wires between Coulomb energy and charging energy dominated regimes enabling Majorana modes to jump between different domain walls [137, 138]". This sentence is wrong, since charging and Coulomb energy dominated regimes are the same thing. The Authors probably meant charging vs Josephson regimes. Also, the References are wrongly placed here; the Coulomb coupling of Majorana modes was primarily proposed by the Leiden group.

25 - Page 26, "the cleanness of the wires is paramount to implement robust topological quantum computation". This may not be true depending on the protocol implemented. For instance, use of Coulomb coupling-based protocols may be resilient to spurious Majorana modes, see for instance https://journals.aps.org/prb/abstract/10.1103/PhysRevB.88.155435. Similar arguments may hold for other proposals as well.