Fault-Tolerant Quantum Computation by Anyons

Alexei Yu. Kitaev · Annals of Physics / arXiv · 2003 · physics · 10.1016/S0003-4916(02)00018-0 · record kitaev2003-anyons
tag layer · index · concepts · authors · עברית

Where this record's text came from. faceted 2026-08 by reading the locally held text (pdftext); every evidence string verified as an exact substring of that text before merging. The method vocabulary has no term for a theoretical physics model-construction paper, so I used 'theoretical-construction'. Claim type is 'method' because the deliverable is a computation scheme, but the paper also carries an existence claim (that anyons arise from a local Hamiltonian with no symmetry) that the single-valued facet cannot hold. The paper additionally criticises the existing quasi-triangular quasi-Hopf-algebra theory of anyons [10] for merely postulating anyon properties; I left that out of claim relations because it is a critique of a body of theory rather than a builds-on/extends/contradicts/replicates relation to a specific result. Discipline could equally be condensed-matter-physics. 2026-08-14, 2026-08-17 and 2026-08-25 schema-0.3 passes: every quote verified as an exact substring of the staged text and anchored at merge time; concept terms verified present; every edge endpoint an existing node.

How to read this page. The article itself is not reproduced here — this paper is not cleared for redistribution, so what follows is the tag layer alone. The evidence sentences are withheld with the article. The table at the foot lists every tag the record carries; each row names the facet it belongs to, and the whose words column says whether the tag records the paper's own wording or the tagger's sentence about the paper. Every tag links to each other paper in the corpus carrying it. The machine-readable edition of the same thing is the TEI file beside this one, where the tags live in standOff, and the article is at its publisher via the link below. When we have changed our mind. Some entries in the table begin with a word in capitals and the reason for it. EVIDENCE CORRECTED — the finding still stands, but we had quoted the wrong sentence from the paper, and the quotation has been replaced with the right one. RETRACTED — the entry itself was wrong; it stays on the page, marked, with the reason, rather than being deleted. WITHDRAWN AND REPLACED — the entry was wrong, and a corrected version of it appears in the entries below. Each mark carries its own reason, in brackets, because the reasons differ from one entry to the next. An entry we have WITHDRAWN is removed from this page and kept in the record, with its reason. A line reading Audit and a date is a later re-reading of the entry against its whole sentence: DOUBTFUL means it was kept, and the reason says what is loose about it. Quotation marks inside the tagger's sentence. Where a tag is the tagger's own sentence, a phrase in quotation marks inside it is borrowed from the paper — the tagger's wording around it is not. The paper's own sentence is the evidence quote for that row; on this page it is withheld with the article, and the article is at its publisher. Nothing inside quotation marks anywhere on this page is ours: if it is quoted, it is the paper's. Whose words a concept carries. A concept entry records a term the paper itself uses, so the term, the sense given for it, and its evidence are the paper's own words, quoted where the paper fixes them. This is the opposite of a sense entry in the tag table, which is the tagger's sentence about how the paper uses a term. Where a concept entry could not be grounded in the paper's words, the entry says so on its own row.

Tags on this paper

Every tag the record carries, each linking to every other paper in the corpus that carries it. A tag appears here whether or not its words occur in the article — the spans below are only the ones a sentence in this paper licenses. Of the 10 concept tags on this paper, 5 are grounded in a quoted sentence.

topological quantum computation anyon fault-tolerance quantum error correction braiding protected space ground state stabilizer operators anyonic excitations fault-tolerant A PHYSICAL SYSTEM IS RE-DESCRIBED AS A COMPUTER, AND THE RE-DESCRIPTION IS MARKED AS ONE: Quotation withheld, with unitary transformations performed BY MOVING THE EXCITATIONS AROUND EACH OTHER and measurements BY JOINING EXCITATIONS IN PAIRS. `Can be considered as` is a way of regarding, not an identity, and the whole paper rests on it A PROPERTY IS LOCATED IN THE PHYSICS RATHER THAN IN THE PROTOCOL: such computation is said to be FAULT-TOLERANT BY ITS PHYSICAL NATURE - so what is elsewhere achieved by a procedure is here claimed as a feature of the material. The same word, fault-tolerance, is being attached to a different kind of thing A PHYSICAL PROCESS IS RE-DESCRIBED AS AN ALGORITHM AND THE HEDGE IS LEFT IN: the spins' mutual interaction forcing a flipped spin back is said to be QUITE SIMILAR TO THE STANDARD ERROR CORRECTION PROCEDURE FOR THE REPETITION CODE, after which - WE MAY SAY THAT ERRORS ARE BEING CORRECTED AT THE PHYSICAL LEVEL. The identification is offered as a manner of speaking THE ARGUMENT PROCEEDS BY ASKING WHY AN EXISTING PRACTICE IS UNNECESSARY: classical computation CAN ALSO be performed fault-tolerantly, HOWEVER IT IS RARELY DONE IN PRACTICE BECAUSE CLASSICAL GATES ARE RELIABLE ENOUGH - and the next sentence is WHY IS IT POSSIBLE. The paper's question is generated by noticing that a technique is available and unused THE EXPLANATION IS SOUGHT IN THE EASIEST CASE FIRST AND THE CHOICE IS ANNOUNCED: LET US TRY TO UNDERSTAND THE EASIEST THING - why classical information can be stored reliably on magnetic media - before the quantum case is attempted. The order of explanation is stated rather than left implicit, and the chosen starting point is a fact about ordinary technology A THRESHOLD IS REPORTED AS THE OBSTACLE, AND ITS SIZE IS WHAT MATTERS RATHER THAN ITS EXISTENCE: an arbitrary circuit CAN be simulated with imperfect gates PROVIDED they are close to ideal up to a constant precision - and UNFORTUNATELY THE THRESHOLD VALUE IS RATHER SMALL, IT IS VERY DIFFICULT TO ACHIEVE THIS PRECISION. The theoretical solution is granted and then declared impractical by a number A PROBLEM IS DECLARED SOLVED FROM ONE STANDPOINT AND OPEN FROM ANOTHER, WITH BOTH NAMED: physical realization is A BIG CHALLENGE and decoherence is one important problem - but FROM THE PURELY THEORETICAL POINT OF VIEW, THIS PROBLEM HAS BEEN SOLVED. Solved and unsolved at once, distinguished by which standpoint the word is being used from AN EXPLICIT CAUTION AGAINST CONFLATING TWO THINGS THE SAME MATHEMATICS DESCRIBES: the local operators SHOULD NOT BE INTERPRETED AS symmetry transformations - the true symmetry transformations are a different set, and MATHEMATICALLY THEY ARE DESCRIBED BY THE SAME ALGEBRA, BUT THEIR ACTION ON PHYSICAL STATES IS DIFFERENT. One formalism, two referents, and the paper stops to say so A TERM OF ART IS CONCEDED AND THEN SET ASIDE AS NOT THE RELEVANT PROPERTY: ONE MAY SAY that these codes have QUITE POOR PARAMETERS, and WELL, THEY ARE NOT GOOD CODES IN THE SENSE OF a cited definition - HOWEVER the code corrects almost any multiple error of a stated size, SO THESE CODES WORK. The established criterion is granted, and then shown not to be the criterion that matters here quantum excitations topological unitary stabilizer degenerate anyons error nonabelian anyons degenerate state mean average of standard-error standard error theory

Method, as described: theoretical-construction

Legend

second reading	`alternatives`	an independent re-reading of a facet, recorded whether or not it agrees
relation to other work	`claim_relations`	how this paper stands to another: agrees, contradicts, extends, qualifies
claim type	`claim_type`	the shape of the claim: existence, mechanism, method, critique, prediction, null
concept edge	`concept_edges`	an asserted relation between two concepts, with the sentence asserting it
concept	`concepts`	an idea the paper works with, quoted where the paper uses it
content tag	`content_tags`	a finding, definition, quantity, statistic or variable named in the paper
limitation	`limitations`	what the paper says its evidence cannot support
mechanism	`mechanisms`	how the reported effect is said to come about
method	`method`	how the paper produced its evidence
phenomenon	`phenomenon`	the effect or regularity under study
sense	`senses`	a distinct use of a term, quoted where the paper fixes it
transfer	`transfers`	a method or concept carried in from another field

The paper

The full text is not reproduced here. This article is not cleared for redistribution, so the page carries the tag layer only. Its evidence sentences are withheld; the grounding status remains with each tag, and the article itself is at https://doi.org/10.1016/S0003-4916(02)00018-0. The reading view exists for the papers whose licence permits it.

Every tag on this page

alternatives 3 · claim_relations 4 · claim_type 1 · concept_edges 2 · concepts 5 · content_tags 10 · limitations 9 · mechanisms 9 · method 1 · phenomenon 1 · senses 3 · transfers 1

Relations on this page carry a quotation for the claim. Where a relation was separately evidenced — and so tested against the vocabulary for its type — the row says so; none of 2 relations here carry that mark. The rest are the tagger’s reading of the sentence quoted beside them.

This record holds no controls. A layer is empty when the tagger recorded nothing in it — which is not the same as the paper having nothing to record.

facet	tag	field	conf.	whose words
claim_type	method Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_type`		the paper's words
phenomenon	anyonic excitations in two-dimensional lattice spin models Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`phenomenon`		the paper's words
method	theory Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: theory	`method`		the paper's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: A PHYSICAL SYSTEM IS RE-DESCRIBED AS A COMPUTER, AND THE RE-DESCRIPTION IS MARKED AS ONE: Quotation withheld, with unitary transformations performed BY MOVING THE EXCITATIONS AROUND EACH OTHER and measurements BY JOINING EXCITATIONS IN PAIRS. `Can be considered as` is a way of regarding, not an identity, and the whole paper rests on it	`mechanisms[0]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: A PROPERTY IS LOCATED IN THE PHYSICS RATHER THAN IN THE PROTOCOL: such computation is said to be FAULT-TOLERANT BY ITS PHYSICAL NATURE - so what is elsewhere achieved by a procedure is here claimed as a feature of the material. The same word, fault-tolerance, is being attached to a different kind of thing	`mechanisms[1]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: A PHYSICAL PROCESS IS RE-DESCRIBED AS AN ALGORITHM AND THE HEDGE IS LEFT IN: the spins' mutual interaction forcing a flipped spin back is said to be QUITE SIMILAR TO THE STANDARD ERROR CORRECTION PROCEDURE FOR THE REPETITION CODE, after which - WE MAY SAY THAT ERRORS ARE BEING CORRECTED AT THE PHYSICAL LEVEL. The identification is offered as a manner of speaking	`mechanisms[2]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: THE ARGUMENT PROCEEDS BY ASKING WHY AN EXISTING PRACTICE IS UNNECESSARY: classical computation CAN ALSO be performed fault-tolerantly, HOWEVER IT IS RARELY DONE IN PRACTICE BECAUSE CLASSICAL GATES ARE RELIABLE ENOUGH - and the next sentence is WHY IS IT POSSIBLE. The paper's question is generated by noticing that a technique is available and unused	`mechanisms[3]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: THE EXPLANATION IS SOUGHT IN THE EASIEST CASE FIRST AND THE CHOICE IS ANNOUNCED: LET US TRY TO UNDERSTAND THE EASIEST THING - why classical information can be stored reliably on magnetic media - before the quantum case is attempted. The order of explanation is stated rather than left implicit, and the chosen starting point is a fact about ordinary technology	`mechanisms[4]`	0.85	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. about this quote: The span contains 'small1;' - a footnote marker glued to the preceding word by the extraction. The quote is an exact substring and anchors correctly. also in: A THRESHOLD IS REPORTED AS THE OBSTACLE, AND ITS SIZE IS WHAT MATTERS RATHER THAN ITS EXISTENCE: an arbitrary circuit CAN be simulated with imperfect gates PROVIDED they are close to ideal up to a constant precision - and UNFORTUNATELY THE THRESHOLD VALUE IS RATHER SMALL, IT IS VERY DIFFICULT TO ACHIEVE THIS PRECISION. The theoretical solution is granted and then declared impractical by a number	`mechanisms[5]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: A PROBLEM IS DECLARED SOLVED FROM ONE STANDPOINT AND OPEN FROM ANOTHER, WITH BOTH NAMED: physical realization is A BIG CHALLENGE and decoherence is one important problem - but FROM THE PURELY THEORETICAL POINT OF VIEW, THIS PROBLEM HAS BEEN SOLVED. Solved and unsolved at once, distinguished by which standpoint the word is being used from	`mechanisms[6]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. about this quote: DAMAGE SHAPE 5 INSIDE THIS SPAN, and it removes a name. The sentence reads "described by the same algebra , but their action on physical states is different" - with a space before the comma where the algebra's symbol should be. The symbol was displaced past the end of the sentence by the extraction: "...is different. D 15 a b Figure 7:". So the quote says "the same algebra [ ]" and the algebra it names is unrecoverable from the span. The tag does not name it. Found by reading; no guard sees this. also in: AN EXPLICIT CAUTION AGAINST CONFLATING TWO THINGS THE SAME MATHEMATICS DESCRIBES: the local operators SHOULD NOT BE INTERPRETED AS symmetry transformations - the true symmetry transformations are a different set, and MATHEMATICALLY THEY ARE DESCRIBED BY THE SAME ALGEBRA, BUT THEIR ACTION ON PHYSICAL STATES IS DIFFERENT. One formalism, two referents, and the paper stops to say so	`mechanisms[7]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: A TERM OF ART IS CONCEDED AND THEN SET ASIDE AS NOT THE RELEVANT PROPERTY: ONE MAY SAY that these codes have QUITE POOR PARAMETERS, and WELL, THEY ARE NOT GOOD CODES IN THE SENSE OF a cited definition - HOWEVER the code corrects almost any multiple error of a stated size, SO THESE CODES WORK. The established criterion is granted, and then shown not to be the criterion that matters here	`mechanisms[8]`	0.95	the tagger's words
limitations	real quantum systems suffer decoherence and systematic errors Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: every fault-tolerance claim: DECOHERENCE AND SYSTEMATIC ERRORS IN UNITARY TRANSFORMATIONS OCCUR IN REAL QUANTUM SYSTEMS	`limitations[0]`		the tagger's words
limitations	For the abelian toric-code memory the author has no way to read information into or out of the protected space. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: the abelian toric-code memory as a usable memory: UNFORTUNATELY, I DO NOT KNOW ANY WAY THIS QUANTUM INFORMATION CAN GET IN OR OUT	`limitations[1]`		the tagger's words
limitations	The gate set obtainable by braiding abelian anyons is not universal. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: any universality claim for braiding abelian anyons: TOO FEW THINGS CAN BE DONE BY MOVING ABELIAN ANYONS	`limitations[2]`		the tagger's words
limitations	The group used for the universal-computation construction is physically implausible. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: the universal-computation construction: (THE GROUP S5 IS QUITE UNREALISTIC FOR PHYSICAL IMPLEMENTATION)	`limitations[3]`		the tagger's words
limitations	No general theory underlies the constructions given. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: the generality of the constructions: A GENERAL THEORY OF ANYONS AND TOPOLOGICAL QUANTUM ORDER IS LACKING	`limitations[4]`		the tagger's words
limitations	The universal-computation results are stated without full derivation, to appear elsewhere. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: the universal-computation results as presented here: (THIS SECTION SHOULD BE CONSIDERED AS AN ABSTRACT OF RESULTS TO BE PRESENTED ELSEWHERE)	`limitations[5]`		the tagger's words
limitations	The number the whole fault-tolerance argument depends on is declared unknown in a footnote, with estimates three orders of magnitude apart. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: any use of the fault-tolerance threshold cited here: a footnote states the threshold is not known and that published estimates vary across several orders of magnitude	`limitations[6]`	1.0	the tagger's words
limitations	The case that errors can be corrected by physics rather than by procedure is made by analogy with magnetic storage, and hedged. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: the claim that this scheme corrects errors at the physical level: the supporting argument is an analogy with spins on magnetic media, which the author introduces with 'we may say'	`limitations[7]`	1.0	the tagger's words
limitations	The protection is degeneracy of a ground state on a surface, so the quantity protected depends on the topology chosen. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: the amount of protected quantum information claimed here: the protected space is the ground state of the Hamiltonian and is 4g-fold degenerate, where g is the genus of the surface the lattice is placed on	`limitations[8]`	1.0	the tagger's words
transfers	1-homologies and 1-cohomologies over Z2 used to describe the protected subspace of the lattice model, so that the stored information is a topological invariant rather than a local one Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`transfers[0]`	0.8	the tagger's words
concepts	protected space Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: protected space	`concepts[5]`		the paper's words
concepts	ground state Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: ground state	`concepts[6]`		the paper's words
concepts	stabilizer operators Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: stabilizer operators	`concepts[7]`		the paper's words
concepts	anyonic excitations Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: anyonic excitations	`concepts[8]`		the paper's words
concepts	fault-tolerant Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: fault-tolerant	`concepts[9]`		the paper's words
senses	fault tolerance - BY PHYSICS RATHER THAN BY PROCEDURE: Shor's construction simulates a circuit with imperfect gates given a precision threshold, whereas here the computation is said to be fault-tolerant by its physical nature, so the two uses of the phrase name a software guarantee and a property of a material Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[0]`	1.0	the tagger's words
senses	the threshold - A NUMBER THE ARGUMENT NEEDS AND NOBODY HAS: the text calls it rather small and very difficult to achieve, and the footnote says outright that it is not known, giving estimates that differ by orders of magnitude Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[1]`	1.0	the tagger's words
senses	the protected space - AS A DEGENERATE GROUND STATE: it coincides with the ground state of a Hamiltonian built from local stabilizer operators, and its size is set by the genus of the surface rather than by anything about the computation Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[2]`	1.0	the tagger's words
alternatives	claim_type: method Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. a second reading reaching the same value. Recorded: method — note: A BLIND second reading by the project on 2026-08-25: the record's own claim type/method/phenomenon were not looked at until every value here was written and substring-verified. Not applied - a person decides. Sampling frame: the 504 staged records carrying no alternatives block, so this never re-measures the 202. All 11 records with 130+ tags were taken as a census and 19 more spread one per discipline across 22 disciplines - the dense non-neuroscience cell is the one the 202 do not contain (inside them, 94-100% of the densest bands are neuroscience; among these 11, one is).,	`alternatives.claim_type`	0.7	the tagger's words
alternatives	method: formal-model Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. a second reading DIFFERING from the recorded value. Recorded: theory — note: A BLIND second reading by the project on 2026-08-25: the record's own claim type/method/phenomenon were not looked at until every value here was written and substring-verified. Not applied - a person decides. Sampling frame: the 504 staged records carrying no alternatives block, so this never re-measures the 202. All 11 records with 130+ tags were taken as a census and 19 more spread one per discipline across 22 disciplines - the dense non-neuroscience cell is the one the 202 do not contain (inside them, 94-100% of the densest bands are neuroscience; among these 11, one is).,	`alternatives.method`	0.8	the tagger's words
alternatives	phenomenon: decoherence and systematic gate error as the obstacle to physically realising a quantum computer Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. a second reading DIFFERING from the recorded value. Recorded: anyonic excitations in two-dimensional lattice spin models — note: A BLIND second reading by the project on 2026-08-25: the record's own claim type/method/phenomenon were not looked at until every value here was written and substring-verified. Not applied - a person decides. Sampling frame: the 504 staged records carrying no alternatives block, so this never re-measures the 202. All 11 records with 130+ tags were taken as a census and 19 more spread one per discipline across 22 disciplines - the dense non-neuroscience cell is the one the 202 do not contain (inside them, 94-100% of the densest bands are neuroscience; among these 11, one is).,	`alternatives.phenomenon`	0.85	the tagger's words
claim_relations	builds-on → quantum error-correcting codes (stabilizer codes) Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[0]`		the tagger's words
claim_relations	builds-on → stabilizer quantum codes on lattices on the torus and other 2-D surfaces [6, 8] Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[1]`		the tagger's words
claim_relations	extends → G. Castagnoli and M. Rasetti's use of one-dimensional anyons for quantum computation [14] Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[2]`		the tagger's words
claim_relations	builds-on → Einarsson's proof that ground state degeneracy on the torus follows from the existence of anyons [19] Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[3]`		the tagger's words
concept_edges	protected space —same-as→ ground state Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. strength of the relation: 1 also in: protected space · ground state	`concept_edges[0]`		the tagger's words
concept_edges	anyonic excitations —causes→ fault-tolerant Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. strength of the relation: 1 also in: anyonic excitations · fault-tolerant	`concept_edges[1]`		the tagger's words
content_tags	4g-fold — ground-state degeneracy of the toric-code Hamiltonian, g = genus of the surface Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.quantities[0]`		the tagger's words
content_tags	ΔE = 2 — energy gap separating excited states (eigenvalue difference of the stabilizer operators) Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.quantities[1]`		the tagger's words
content_tags	A two-dimensional quantum system with anyonic excitations can serve as a quantum computer, with unitary transformations performed by moving the excitations around each other. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.findings[0]`		the tagger's words
content_tags	The ground-state degeneracy persists under local perturbation. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.findings[1]`		the tagger's words
content_tags	braiding Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: braiding	`content_tags.variables[0]`		the tagger's words
content_tags	degenerate state Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: degenerate state	`content_tags.variables[1]`		the tagger's words
content_tags	nonabelian anyons Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: nonabelian anyons	`content_tags.definitions[0]`		the tagger's words
content_tags	braiding Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: braiding	`content_tags.definitions[1]`		the tagger's words
content_tags	mean Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: mean	`content_tags.statistics[0]`		the tagger's words
content_tags	standard-error Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: standard-error	`content_tags.statistics[2]`		the tagger's words

Interpretive choices

Tags that could not be placed in the text:

none — every tag was located in the text

What was deliberately not tagged, and why, is recorded per paper by the tagger. Where a paper has no such record, this page says so rather than implying that nothing was declined.