Fault-Tolerant Quantum Computation by Anyons

Fault-Tolerant Quantum Computation by Anyons Alexei Yu. Kitaev

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Fault-Tolerant Quantum Computation by Anyons 2003 Annals of Physics / arXiv 10.1016/S0003-4916(02)00018-0 quant-ph/9707021 kitaev2003-anyons kitaev2003-anyons.txt, the project's staged copy efc6ee02529c20d67012c32af5ec777972199d94 arxiv-open physics

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0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them method Quotation withheld because this record is not cleared for redistribution. anyonic excitations in two-dimensional lattice spin models Quotation withheld because this record is not cleared for redistribution. theory theory Quotation withheld because this record is not cleared for redistribution. mechanisms 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 Quotation withheld because this record is not cleared for redistribution. mechanisms 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 Quotation withheld because this record is not cleared for redistribution. mechanisms 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 Quotation withheld because this record is not cleared for redistribution. mechanisms 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 Quotation withheld because this record is not cleared for redistribution. mechanisms 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 Quotation withheld because this record is not cleared for redistribution. mechanisms 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 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. Quotation withheld because this record is not cleared for redistribution. mechanisms 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 Quotation withheld because this record is not cleared for redistribution. mechanisms 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 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. Quotation withheld because this record is not cleared for redistribution. mechanisms 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 Quotation withheld because this record is not cleared for redistribution. real quantum systems suffer decoherence and systematic errors what this limits: every fault-tolerance claim: DECOHERENCE AND SYSTEMATIC ERRORS IN UNITARY TRANSFORMATIONS OCCUR IN REAL QUANTUM SYSTEMS Quotation withheld because this record is not cleared for redistribution. For the abelian toric-code memory the author has no way to read information into or out of the protected space. 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 Quotation withheld because this record is not cleared for redistribution. The gate set obtainable by braiding abelian anyons is not universal. what this limits: any universality claim for braiding abelian anyons: TOO FEW THINGS CAN BE DONE BY MOVING ABELIAN ANYONS Quotation withheld because this record is not cleared for redistribution. The group used for the universal-computation construction is physically implausible. what this limits: the universal-computation construction: (THE GROUP S5 IS QUITE UNREALISTIC FOR PHYSICAL IMPLEMENTATION) Quotation withheld because this record is not cleared for redistribution. No general theory underlies the constructions given. what this limits: the generality of the constructions: A GENERAL THEORY OF ANYONS AND TOPOLOGICAL QUANTUM ORDER IS LACKING Quotation withheld because this record is not cleared for redistribution. The universal-computation results are stated without full derivation, to appear elsewhere. what this limits: the universal-computation results as presented here: (THIS SECTION SHOULD BE CONSIDERED AS AN ABSTRACT OF RESULTS TO BE PRESENTED ELSEWHERE) Quotation withheld because this record is not cleared for redistribution. The number the whole fault-tolerance argument depends on is declared unknown in a footnote, with estimates three orders of magnitude apart. 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 Quotation withheld because this record is not cleared for redistribution. The case that errors can be corrected by physics rather than by procedure is made by analogy with magnetic storage, and hedged. 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' Quotation withheld because this record is not cleared for redistribution. The protection is degeneracy of a ground state on a surface, so the quantity protected depends on the topology chosen. 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 Quotation withheld because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. protected space protected space Quotation withheld because this record is not cleared for redistribution. ground state ground state Quotation withheld because this record is not cleared for redistribution. stabilizer operators stabilizer operators Quotation withheld because this record is not cleared for redistribution. anyonic excitations anyonic excitations Quotation withheld because this record is not cleared for redistribution. fault-tolerant fault-tolerant Quotation withheld because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. claim_type: method 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)., Quotation withheld because this record is not cleared for redistribution. method: formal-model 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)., Quotation withheld because this record is not cleared for redistribution. phenomenon: decoherence and systematic gate error as the obstacle to physically realising a quantum computer 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)., Quotation withheld because this record is not cleared for redistribution. builds-on → quantum error-correcting codes (stabilizer codes) Quotation withheld because this record is not cleared for redistribution. builds-on → stabilizer quantum codes on lattices on the torus and other 2-D surfaces [6, 8] Quotation withheld because this record is not cleared for redistribution. extends → G. Castagnoli and M. Rasetti's use of one-dimensional anyons for quantum computation [14] Quotation withheld because this record is not cleared for redistribution. builds-on → Einarsson's proof that ground state degeneracy on the torus follows from the existence of anyons [19] Quotation withheld because this record is not cleared for redistribution. protected space —same-as→ ground state protected space ground state strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. anyonic excitations —causes→ fault-tolerant anyonic excitations fault-tolerant strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. 4g-fold — ground-state degeneracy of the toric-code Hamiltonian, g = genus of the surface Quotation withheld because this record is not cleared for redistribution. ΔE = 2 — energy gap separating excited states (eigenvalue difference of the stabilizer operators) Quotation withheld because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. The ground-state degeneracy persists under local perturbation. Quotation withheld because this record is not cleared for redistribution. braiding braiding Quotation withheld because this record is not cleared for redistribution. degenerate state degenerate state Quotation withheld because this record is not cleared for redistribution. nonabelian anyons nonabelian anyons Quotation withheld because this record is not cleared for redistribution. braiding braiding Quotation withheld because this record is not cleared for redistribution. mean mean Quotation withheld because this record is not cleared for redistribution. standard-error standard-error Quotation withheld because this record is not cleared for redistribution.

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