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A Short Introduction to Topological Quantum Computation

by Ville Lahtinen, Jiannis K. Pachos

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Submission summary

Authors (as registered SciPost users): Ville Lahtinen · Jiannis Pachos
Submission information
Preprint Link:  (pdf)
Date submitted: 2017-07-26 02:00
Submitted by: Lahtinen, Ville
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical


This review presents an entry-level introduction to topological quantum computation -- quantum computing with anyons. We introduce anyons at the system-independent level of anyon models and discuss the key concepts of protected fusion spaces and statistical quantum evolutions for encoding and processing quantum information. Both the encoding and the processing are inherently resilient against errors due to their topological nature, thus promising to overcome one of the main obstacles for the realisation of quantum computers. We outline the general steps of topological quantum computation, as well as discuss various challenges faced it. We also review the literature on condensed matter systems where anyons can emerge. Finally, the appearance of anyons and employing them for quantum computation is demonstrated in the context of a simple microscopic model -- the topological superconducting nanowire -- that describes the low-energy physics of several experimentally relevant settings. This model supports localised Majorana zero modes that are the simplest and the experimentally most tractable types of anyons that are needed to perform topological quantum computation.

Author comments upon resubmission

Thank you very much for the high quality refereeing process.

According to the editorial recommendation, we have carried out extensive revisions to clarify the intended audience and to make our more work accessible for them. We also decided to slightly change the title in order to avoid confusion, but also to make connection with the book by Pachos, where many of the condensed matter topics are discussed in more detail. The list of major changes can be found attached.

Sincerely yours,
Ville Lahtinen and Jiannis K. Pachos

List of changes

List of changes

We changed the title to “A Short Introduction to Topological Quantum Computation” in order to have a unique title that is not confused with Pachos's book.

We have revised the abstract to more clearly communicate the contents and the intended audience.

After deliberating on the optimal structure of the review, we decided move Section 5 (“Manifestations of anyons...”) to be a subsection of Section 2. In this way it does not interrupt the flow of the discussion that progresses from anyons models (Sec 3) to quantum computating with them (Sec 4) to the example (Sec 5).

We also revised the titles of the Sections 5.1-5.5 to better reflect their content.

Section 1:
As suggested by referee 1, we have added a sentence to the first paragraph to immediately introduce the key concepts of a degenerate protected subspace and adiabatic transport to manipulate the states within it.

We have also added two new paragraphs. The first explains the intended audience of the present review and puts our work in the context of other reviews/books on topological quantum computation. The second new paragraph explains the structure of the review.

Section 2:
We have completely revised the three paragraphs before Sec 2.1. Without going deeper into the mathematics, our intention here is to make clear the difference between SPT order and intrinsic topological order and clarify that the first requires defects of some type to support anyons, while the latter does not. In the third paragraph we also now anticipate the concepts of topological entanglement entropy and topological degeneracy that arise in intrinsically ordered systems and which are explained in Section 2.2.

Section 2.2:
This section is the old Section 5. The first paragraph explains the scope and purpose of the section.

Section 2.2.1:
Throughout this section we make it clear that when talking about excitations we refer to intrinsically ordered systems where anyons are massive excitations, but that similar manifestations of protected degeneracies apply also to SPT states with defects. We also try to anticipate the discussion ahead and connect the protected degenerate subspaces and Berry phases to encoding and processing of quantum information that is described in the following sections.

Section 2.2.2:
This section has now been shortened and we explicitly mention that the concepts of topological entanglement entropy and topological degeneracy only apply to intrinsic topological states. Regarding the nature of topological degeneracy, we no longer discuss how it depends on the anyon model, but only give references. After all, our intention is just to introduce the reader to these two concepts that appear often in the literature, but that do not directly feature in topological quantum computation.

In the same spirit, we now define how entanglement entropy of a system is calculated, but omit the details how it depends on the anyon model via the total quantum dimension. We also no longer talk of “topological twists” when extracting the full data.

We also made the decision to drop the paragraph related to edge states and their relation entanglement spectrum in order to cut unnecessary concepts that are not directly relevant to quantum computation.

Section 3.1:
Regarding pentagon and hexagon equations, we now mention explicitly that their role is to classify possible consistent anyon models, but refrain from talking about them any deeper since their solutions can be looked up from literature for all anyon models of interest. We don’t feel that discussing them in depth provides any significant understanding of anyons from the point of view of quantum computation.

Section 3.3:
We now give as eq. (16) the fusion rules for few different numbers of Fibonacci anyons to show how the dimensionality of the fusion space grows and why it implies a lack of tensor product structure.

To clarify this key difference to Ising anyons that do have a tensor product structure, we also added the fusion rules for many Ising anyons as eq. (18).

Section 4.3:
We have revised the paragraphs discussing the challenge posed by finite temperature to topological quantum computation. We omit any discussion about “ill-defined statistics” and merely state the results from the given references that discuss Abelian anyon based quantum memories. Any discussion about temperature in nanowires is deferred to Section 5.4.

Section 5.2:
We have clarified how the microscopic properties of the wire, such as fermion parity conservation, relate to the Ising anyon model and how they are used in the protected encoding of the Majorana qubit.

Section 5.3:
We thought how to expand and provide more details about the calculation of braid evolutions in the wire network, but in the end decided that providing any of the mathematics behind the braiding calculations does not serve the purpose here. Our aim is merely to justify that topological quantum computation is indeed possible in nanowire arrays along the principles outlined in Section 4. We have revised the section though the make all our statements clearer and we now also explicitly point to references where details how braiding and measurement is carried in a particular realization of nanowire arrays could be carried out.

Furthermore, we added a paragraph to describe how the general measurement protocol of Section 4.3 is to be carried out in nanowire arrays.

Further revisions:
Proof-reading to catch typos

We have added all the suggested references as well as few other that we came across.

Improved inter-connectedness by referring back and forth to different sections.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 1 on 2017-8-14 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1705.04103v2, delivered 2017-08-14, doi: 10.21468/SciPost.Report.209


1 - Readability.
2 - Length.
3 - Good entry point to the field and to the rest of the literature.


I find that the weaknesses I indicated in my first report have been successfully addressed by the Authors.


I read the new version and, in particular, went through the list of changes submitted by the Authors. I thank the Authors for considering the remarks that popped up in the first round of reviews. I find that the manuscript has considerably improved in the new version. The text has increased by a few pages, but this has allowed more breath and clarity in the exposition. The concepts that were not properly introduced before are now explained or at least briefly defined with pointers to the literature.

Overall, I think this review work is ready for publication and that it will be a valuable addition to the literature in general and to the journal in particular. I have one minor revision to be considered before the article is published, see below.

Requested changes

10 - In the last paragraph before Sec. II, I would emphasize more that the proposals in Refs. 81-83 are qualitatively different than those in the preceding references. While interesting, I don't think these works can lead to a platform to topological quantum computation, since as also mentioned by the Authors they do not really realize topological systems. Hence, they are sort of misplaced here. If the Authors want to refrain from explaining Jordan-Wigner transformations or other non-local mappings - a sentiment which I agree with - they can still state that these proposals are only unitarily equivalent to a system with anyons, in a way which is not protected by local errors and is thus non-topological in nature.

  • validity: good
  • significance: good
  • originality: ok
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Author:  Ville Lahtinen  on 2017-08-15  [id 161]

(in reply to Report 1 on 2017-08-14)

Thank you very much again to carefully go through our manuscript and helping us to improve. We are happy to hear that our revisions were satisfactory and we have now also addressed the final remark according to the suggestion. Details can be found in the list of changes in the resubmission.

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