§0 Sammanfattning

Spåret är det som återstår när vägen har stängts.

In topological quantum computation the computation is the braiding of anyon world lines, and the answer is read out by closing the braid, which evaluates a Markov trace on a braid-group representation, a Jones-type invariant (Jones 1985; Freedman et al. 2003; Aharonov, Jones & Landau 2009) [E]. From this the post draws one structural claim: a topologically protected record keeps its history only as the isotopy class of a braid, passed through a representation and then through a trace, and each of those steps has a kernel of distinct histories it cannot tell apart [D]. Physical provenance in this sense is always provenance up to the trace kernel [D]. The Braided Substrate (DRK-143) already identified world lines with Wilson lines and Chern–Simons knots; what this post adds is the readout (the trace), what the readout loses (the kernel), and the physical laws that bound any record (§8). The post does not claim that anyonic hardware is mature, that every physical record is topological, or anything outside physics.

Epistemic ledger. Every substantive claim carries one tag:

Tag Meaning
[E] Established: textbook consensus, a proved theorem, direct measurement, or a standard etymology (SAOB, Hellquist, OED, etymonline)
[S] Supported: leading model or majority scholarly reading, strong but incomplete evidence
[H] Hypothesis: open, contested, or without decisive evidence
[D] Draken synthesis: structural claim made by this corpus, submitted for Clinch review
[M] Metaphor or paronomasia: a pointer, not a referent or root (per The Pendragon Source, DRK-165)

§1 Why two dimensions remember

Take $n$ identical particles in $d$ spatial dimensions. Their configuration space is $C_n(\mathbb R^d)$, the set of $n$-point subsets of $\mathbb R^d$, and the possible exchange statistics are the one-dimensional and higher unitary representations of its fundamental group $\pi_1 C_n(\mathbb R^d)$ (Leinaas & Myrheim 1977) [E]. For $d\ge 3$ that group is the symmetric group $S_n$; for $d=2$ it is the braid group $B_n$ (Leinaas & Myrheim 1977; Kassel & Turaev 2008) [E].

The braid group on $n$ strands has generators $\sigma_1,\dots,\sigma_{n-1}$, where $\sigma_i$ exchanges particles $i$ and $i+1$ counter-clockwise, and relations

$$\sigma_i\sigma_j=\sigma_j\sigma_i \ \ (|i-j|\ge 2),\qquad \sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$$

(Kassel & Turaev 2008) [E]. The symmetric group is the quotient of $B_n$ by the extra relations $\sigma_i^2=1$ [E].

That one relation is the whole difference. In three dimensions a double exchange $\sigma_i^2$ is a loop that can be shrunk to nothing, so the history of the exchange is erased up to a permutation and a sign [E]. In two dimensions $\sigma_i^2\neq 1$: one particle winding fully around another is a non-contractible loop, and the winding is counted (Wilczek 1982) [E]. For Abelian anyons an exchange multiplies the state by a phase $e^{i\theta}$ with $\theta$ not restricted to $0$ or $\pi$ (Wilczek 1982), and quasiparticles of the fractional quantum Hall effect were shown to carry such fractional statistics (Arovas, Schrieffer & Wilczek 1984) [E]. For non-Abelian anyons the state space at fixed positions is degenerate, and an exchange acts as a unitary matrix on it rather than a phase (Moore & Read 1991; Kitaev 2006) [E].

A $2+1$-dimensional world therefore keeps a ledger of how its particles moved around each other, and a $3+1$-dimensional one does not [E/D]: the topology is [E], calling it a ledger is the reading [D]. This is where the post starts, because it is the only place in fundamental physics where the order and winding of a history, not only its endpoints, is written into the state by topology alone [D].

§2 Holonomy as memory

Path-dependent phases are older than anyons. An electron taken around a solenoid picks up a phase fixed by the enclosed flux, even though the field vanishes along its path (Aharonov & Bohm 1959) [E]. A quantum system carried adiabatically around a loop in parameter space acquires a geometric phase fixed by the loop's geometry, not its timing (Berry 1984) [E]. When the eigenspace is degenerate, the phase becomes a unitary matrix, a non-Abelian holonomy (Wilczek & Zee 1984), and quantum gates can be built from such holonomies (Zanardi & Rasetti 1999) [E].

The corpus has treated these already, and this post does not repeat them: Berry, Aharonov–Bohm and Wilczek–Zee are worked through in Picking Berries (DRK-172), and world lines as Wilson lines in The Braided Substrate (DRK-143). One distinction from them matters here. A Berry phase depends on the curvature enclosed by the loop and changes under small deformations of it, so it is geometric; the anyon phase depends only on which punctures the loop winds around, because the effective connection is flat away from the particles, so it is topological (Nayak et al. 2008) [E]. Topological protection is that flatness: a perturbation that does not change the winding cannot change the record [E].

§3 World lines as the substrate of the record

A world line is the track of a particle through spacetime. Feynman (1949) treated a positron as an electron world line running backwards in time, so that pair creation and annihilation became a single world line turning in time [E]. In the worldline formalism, amplitudes of quantum field theory are written as path integrals over particle world lines (Schubert 2001) [E].

No Trace, No Section (DRK-153) identified a world line with a global section of a sheaf over spacetime: to leave a trace is to have a section [D]. In $2+1$ dimensions the world lines of $n$ anyons, run from time $0$ to time $T$, form a braid in the slab $\mathbb R^2\times[0,T]$, and what the system keeps of them is the braid's isotopy class (Nayak et al. 2008) [E]. The present post asks a question that post did not: once a section exists, how much of it can be read? The answer in §4–§7 is that it is read through a trace, and the trace forgets [D].

The two senses of trace meet here only as a word. In that post a trace is the presence of a world line; here it is the sum of the diagonal of a matrix, extended to a functional on braid algebras [M]. The argument in §4–§7 uses only the second sense.

§3.1 The Swedish word [E]

In Swedish mathematics the trace of a square matrix, the sum of its diagonal elements, is called spåret (Wikipedia, Swedish edition, entry Spår (matematik)) [E]. Spår in its everyday sense, a track or footprint, comes from Germanic *spura-, from the root sper- also found in spjärna, "to push off with the feet" (Hellquist 1922) [E]. So the matrix term and the footprint are one Swedish word: a trace is literally what the foot leaves [E].

Spå, "to foretell", sounds as if it belonged with spår but does not: Hellquist (1922) derives it from Germanic *spah- and the Indo-European root (s)pek-, "to see", the root of Latin specio and Greek skopéō [E]. A trace read as a prophecy is therefore paronomasia only, and nothing below rests on it [M].

§4 Braid, closure, trace

Join the top end of each strand of a braid $\beta\in B_n$ to its own bottom end, outside the braid, and the result is an oriented link $\hat\beta$, the trace closure of $\beta$ (Kassel & Turaev 2008) [E]. Alexander's theorem says every oriented link is the closure of some braid; Markov's theorem says two braids have isotopic closures if and only if they are related by a finite sequence of conjugations $\beta\mapsto\gamma\beta\gamma^{-1}$ in some $B_n$ and stabilisations $\beta\leftrightarrow\beta\sigma_n^{\pm1}$ between $B_n$ and $B_{n+1}$ (Kassel & Turaev 2008) [E].

So a function on braids is a link invariant exactly when it is constant on conjugacy classes and unchanged by stabilisation [E]. A matrix trace already has the first property, since $\operatorname{tr}(\gamma\beta\gamma^{-1})=\operatorname{tr}(\beta)$ [E]. The second property is what makes a trace Markov.

§4.2 The Markov trace and the Jones polynomial [E]

Jones (1985) built, for each $n$, an algebra $A_n$ generated by projections $e_1,\dots,e_{n-1}$ obeying Temperley–Lieb-type relations, a representation $r_t$ of $B_n$ into $A_n$ sending each $\sigma_i$ to a combination of $1$ and $e_i$, and a trace $\operatorname{tr}$ on $\bigcup_n A_n$ with the Markov property

$$\operatorname{tr}(x\,e_n)=\tau\,\operatorname{tr}(x)\qquad\text{for } x\in A_n ,$$

where $\tau$ is a fixed scalar depending on $t$ [E]. Writing $e(\beta)$ for the exponent sum of $\beta$ (the number of positive crossings minus negative ones), the normalised trace

$$V_{\hat\beta}(t)=\left(-\frac{1+t}{\sqrt t}\right)^{n-1}\left(\sqrt t\right)^{e(\beta)}\operatorname{tr}\big(r_t(\beta)\big)$$

depends only on the link $\hat\beta$, and is its Jones polynomial (Jones 1985) [E]. The construction extends from Temperley–Lieb to Hecke algebra representations, giving the two-variable polynomial of which Jones's is a specialisation (Jones 1987) [E].

§4.3 The same number as a path integral [E]

Witten (1989) showed that the Jones polynomial is the expectation value of Wilson loops in Chern–Simons gauge theory in $2+1$ dimensions: for gauge group $SU(2)$ at level $k$ with the loops in the fundamental representation, the link invariant is evaluated at $q=e^{2\pi i/(k+2)}$ [E]. Here the "loops" are closed world lines in a $2+1$-dimensional spacetime [E].

So the same quantity appears three ways: as a Markov trace on a braid algebra, as a link invariant, and as the amplitude of a topological quantum field theory for closed world lines [E]. The Braided Substrate (DRK-143) used the third face; this post needs the first, because a trace is a readout, and a readout has a kernel [D].

§5 The trace as a computer

Kitaev (2003) proposed that a quantum computer could store information in the global degrees of freedom of a topologically ordered medium, protected from local noise, and compute by moving anyons around one another [E]. Freedman, Larsen & Wang (2002) proved that for the $SU(2)$ theory at the fifth root of unity (the theory of Fibonacci anyons) the braid-group representations are dense in the relevant unitary groups, so braiding alone is universal for quantum computation [E]. Freedman et al. (2003) set out the resulting model: create anyon pairs from the vacuum, braid them, then fuse them back in pairs and record whether the vacuum reappears [E].

That last step is a closure. Capping strands in pairs at the bottom and the top of a braid is a plat closure, and the amplitude that all pairs fuse back to the vacuum is, up to a known normalisation, the Jones polynomial of the plat closure evaluated at the theory's root of unity (Freedman et al. 2003; Aharonov, Jones & Landau 2009) [E]. Aharonov, Jones & Landau (2009) gave an explicit polynomial-time quantum algorithm for an additive approximation of the Jones polynomial at roots of unity $e^{2\pi i/k}$; together with the density results, approximating it is as hard as quantum computation in general (Freedman et al. 2003; Aharonov, Jones & Landau 2009) [E].

The universality result has a mirror. For Ising anyons, the non-Abelian anyons of the leading model of the $\nu=5/2$ quantum Hall state and of Kitaev's honeycomb model in a field, braiding generates only a finite group of gates, the Clifford gates up to phases, and is therefore not universal (Nayak et al. 2008; Kitaev 2006) [E]. This fact returns in §7 as the clearest example of a large kernel.

Read physically: the computation is the history of the world lines, and the output is a trace of that history [D].

§6 The 2020–2026 record

The physics of §1–§5 is decades old. Its experimental record is recent, and almost all of it consists of simulated topological order on non-topological hardware rather than anyons in a natural topological phase [E].

Year Result Platform Status
2020 Braiding phase of Abelian anyons observed in a Fabry–Pérot interferometer (Nakamura et al. 2020) GaAs quantum Hall, $\nu=1/3$ [E]
2020 Fractional statistics from anyon collisions (Bartolomei et al. 2020) GaAs quantum Hall, $\nu=1/3$ [E]
2023 Non-Abelian braiding of graph-vertex defects (Google Quantum AI 2023) superconducting qubits [E]
2024 Non-Abelian $D_4$ topological order and its anyons prepared (Iqbal et al. 2024) trapped ions [E]
2024 Braiding of Fibonacci anyons in a string-net state (Xu et al. 2024) superconducting qubits [E]
2025 Anyonic braiding in a chiral Mach–Zehnder interferometer (Ghosh et al. 2025) GaAs quantum Hall [E]
2026 Universal gate set from braiding and fusing $S_3$ anyons; magic state prepared topologically (Lo et al. 2026) trapped ions, 54 qubits [E]

The first three rows are natural or engineered anyons in semiconductors, and they establish braiding statistics as a measured fact (Nakamura et al. 2020; Bartolomei et al. 2020; Ghosh et al. 2025) [E]. The processor experiments prepare topologically ordered states as wavefunctions of ordinary qubits; their anyons braid as the theory predicts, but the protection is only as good as the qubits that host them (Iqbal et al. 2024; Xu et al. 2024) [S].

The 2026 result is the conceptually sharpest. $S_3$, the smallest non-Abelian group, gives a topological order whose braiding alone is not universal; Lo et al. (2026) made it universal by treating fusion, the measurement of total anyon charge, as a computational primitive alongside braiding, and used the combination to prepare a magic state [E]. Non-Clifford "magic" as the resource that topology alone does not supply was the subject of The Magical Substrate (DRK-156) [D]. In the language of this post, $S_3$ shows that the readout (fusion, a closure) can carry computational weight that the braid (the record) cannot carry by itself [D]. It is a demonstration of a topological gate set on 54 physical qubits, not of a fault-tolerant topological memory at scale (Lo et al. 2026) [S].

§7 The kernel of the trace

§7.1 Three places where histories merge

A protected record of an anyon history goes through three maps, and each one identifies histories that differ [D]. Write $\Sigma$ for the set of world-line histories of $n$ anyons over a time interval, as smooth paths:

$$\Sigma\ \xrightarrow{\ \ [\,\cdot\,]\ \ }\ B_n\ \xrightarrow{\ \ \rho\ \ }\ U(V_n)\ \xrightarrow{\ \ \text{closure + trace}\ \ }\ \mathbb C .$$

Here $[\,\cdot\,]$ takes a history to its braid class, $\rho$ is the unitary representation of $B_n$ on the anyons' fusion space $V_n$ given by the anyon model, and the last map is a closure followed by a Markov trace, as in §4–§5 [E].

Step What it forgets Status
$\Sigma\to B_n$ Timing, speed, shape of the paths within one isotopy class Topological protection itself; by design (§2)
$B_n\to U(V_n)$ Braids in $\ker\rho$, which act identically on the fusion space Large for Ising anyons (finite image); for dense models, faithfulness is a separate question
$U(V_n)\to\mathbb C$ Distinct links with equal invariant, plus everything a single evaluation at one root of unity drops Non-empty for the Jones polynomial

The first step is the point of the hardware, not a flaw: the record is meant to forget everything except topology [E]. The second is a property of the anyon model: Ising braiding generates a finite group (§5), so infinitely many inequivalent braids act as the same gate, and the record cannot distinguish them (Nayak et al. 2008) [E]. Whether a given braid-group representation is faithful is a hard question even in pure mathematics: the Burau representation is not faithful for $n\ge 5$ (Bigelow 1999), while the Lawrence–Krammer representation is, which makes braid groups linear (Bigelow 2001) [E].

§7.2 The Jones polynomial is not complete [E]

The third step forgets the most. Kanenobu (1986) constructed infinitely many families of infinitely many distinct knots that share the same Jones polynomial (and the same two-variable polynomial) while differing in classical invariants [E]. Eliahou, Kauffman & Thistlethwaite (2003) constructed infinite families of non-trivial links whose Jones polynomial equals that of the unlink [E]. Whether some non-trivial knot has the Jones polynomial of the unknot is still open (Kose 2021) [E].

So two anyon histories whose world lines close into different links can return the same amplitude at every root of unity, and therefore the same readout statistics [E]. A physical experiment then adds a fourth loss that mathematics does not: it evaluates at one root of unity fixed by the anyon model, and estimates the amplitude from finitely many shots to additive precision (Aharonov, Jones & Landau 2009) [E].

§7.3 The trace kernel

Call $E:\Sigma\to\mathbb C$ the composite readout in §7.1. The trace kernel of a readout $E$ is the set of pairs of histories that $E$ cannot tell apart:

$$K_E=\{(h,h')\in\Sigma\times\Sigma \;:\; E(h)=E(h')\}.$$

It is an equivalence relation, the kernel pair of $E$, not a subgroup: a trace is not multiplicative, so there is no group kernel to take [E]. The Draken claim is that a protected physical record is provenance up to $K_E$: it certifies the class of a history, never the history [D].

§7.4 Against the warded triple

The Warded Class (DRK-190) proposed the triple rank, distance, soundness, $(b_1,d,\rho)$, as the full invariant of a protected class [D]. (The symbol $\rho$ there is soundness; here it is the braid representation. The two are unrelated.) Soundness measures whether local checks raise an alarm in proportion to damage, that is, to errors that move the state away from the code [E].

The trace kernel measures something the triple does not: pairs of histories that differ and are both legitimate. Nothing was damaged and no check should fire, yet the record cannot say which one happened [D]. A code with perfect soundness still has a trace kernel, because soundness concerns distance from the code, while the kernel concerns distinct paths that land on the same class [D]. The two quantities are complementary: soundness bounds silent corruption, the kernel bounds silent ambiguity [D].

The Invariant (DRK-175) argued that what survives deformation is a knot that must not be untied [D]. §7.2 adds the caution that several knots can wear the same invariant, so keeping the invariant is not the same as keeping the knot [D].

§8 What physics forbids a record from doing

Three results bound any physical record, topological or not.

No-cloning. No physical process can copy an arbitrary unknown quantum state (Wootters & Zurek 1982) [E]. A record held in an anyonic fusion space is such a state, so it cannot be duplicated; it can only be read, and reading is a closure that collapses it to an outcome [E/D]: the theorem is [E], the reading of readout as closure is [D].

No-hiding. If quantum information disappears from a system, it cannot be hidden in the correlations between system and environment; it must be wholly present in the environment (Braunstein & Pati 2007) [E]. A record that seems to have been lost has been moved, not destroyed [E]. The black-hole version of this question was framed as a convergence obstruction in The Dragged Frame (DRK-171) [D].

Landauer. Logically irreversible operations, such as erasing a bit, must dissipate at least $k_B T\ln 2$ of heat per bit into the environment, where $k_B$ is Boltzmann's constant and $T$ the temperature (Landauer 1961) [E]. The bound has been approached experimentally with a single colloidal particle in a double-well potential (Bérut et al. 2012) [E].

Together they fix the physics of provenance [D]. A record cannot be copied, so its authority cannot be multiplied without measuring it [D]. It cannot be erased for free, so wiping a history leaves a thermodynamic receipt in the environment [D]. It cannot be made to vanish into correlations, so lost provenance has gone somewhere [D]. The trace kernel of §7 sits inside these bounds: even a record that is uncopyable, unerasable and conserved is legible only up to $K_E$ [D].

§9 Provenance of the claims themselves

Physics applies the same logic to its own claims, and the recent record of topological and advantage claims shows it.

Retraction. A 2018 report of quantized Majorana conductance, taken as evidence for the zero modes a topological qubit would need, was retracted in 2021 (Zhang et al. 2018) [E].

Open dispute. Microsoft Azure Quantum (2025) reported single-shot interferometric parity measurement in InAs–Al nanowire devices tuned by a transport-based topological gap protocol [E]. Legg (2026) reanalysed the transport data and code behind that tuning and argued that it does not establish a topological gap; Microsoft Quantum (2026) replied that its capacitance measurements do not assume a gap and still constrain non-topological explanations [E]. The question is not settled in print [H].

Overturned advantage. King et al. (2025) reported quantum simulation of spin-glass dynamics on an annealer beyond the reach of classical methods [E]. Tindall et al. (2026) then simulated the same class of dynamics classically with lattice-matched tensor networks and belief propagation, reaching lower error than the annealer on some of the lattices [E]. The scope of that rebuttal is itself still argued, which is how such exchanges usually run [S].

Self-certifying claim. Martiel et al. (2026) proposed sampling circuits that are both hard and encodable in a quantum code, so that the experiment certifies a statistical lower bound on its own fidelity from measured code syndromes [E]. Their own abstract states that the certificate is device-dependent, and the paper is a preprint [E]. Even the version figures move: the posted abstract and the accompanying press release report different qubit counts and bounds, and a reader must say which version was checked (see §13) [E].

The pattern is a structural reading [D]. A claim about a hidden physical state is credible in proportion to how much of its own trace it ships: raw data, analysis code, and a bound computed from the data that an outside party can recompute [D]. Claims that shipped little were the ones overturned slowest, because no one could re-evaluate the trace [H]. §12 states how that could be tested.

§10 Formal sketch (analogy, not derivation)

This section restates §7 in categorical terms. The first two paragraphs are exact; the identification of a "record" with this structure is the analogy.

Let $\mathcal B$ be the braid category: objects are natural numbers $n$, and morphisms $n\to n$ are elements of $B_n$ [E]. An anyon model assigns to each $n$ a finite-dimensional Hilbert space $V_n$, the fusion space of $n$ anyons with fixed total charge, and to each braid a unitary, giving a functor $\rho:\mathcal B\to\mathbf{Hilb}$ that is the braid-group part of a modular functor (Freedman, Larsen & Wang 2002) [E]. Closed world lines are then evaluated by the associated topological quantum field theory (Witten 1989) [E].

A readout is a family of linear functionals $\operatorname{tr}_n:\operatorname{End}(V_n)\to\mathbb C$ with the Markov property, composed with $\rho$ [E]. Its kernel pair $K_E=\Sigma\times_{\mathbb C}\Sigma$ is the pullback of $E$ along itself [E].

The analogy [D]: any physical record is a functor from histories to state spaces, followed by a readout that is a trace-like functional. Its provenance is the quotient $\Sigma/K_E$. Its blind spot is $K_E$ minus the diagonal. Topological records are the case where this structure is exact; for other records the corpus has not shown that the readout is a trace, and §11 does not assume it.

§11 What this does not claim

The post does not claim that topological quantum hardware is mature: every non-Abelian result in §6 runs on qubits that are not themselves topologically protected, and the one natural-topological-qubit programme cited is under open dispute (§9) [E]. It does not claim that every physical record is topological, or that every readout is a Markov trace (§10) [D]. It does not claim that the trace kernel is an obstacle to be removed: forgetting the path within an isotopy class is what topological protection is for [D]. The tempting analogy to provenance of media and documents is left to other posts [D].

§12 Falsification (DRK-131)

F1: Trace sufficiency. §5–§7 claim that a topologically protected readout depends on a history only through its braid class, its image under $\rho$, and the closure. Refuted if an anyonic readout is shown to distinguish two isotopic braids (different timing or path shape, same braid class) by more than the measured non-topological error budget of the device.

F2: Kernel non-triviality. §7 claims that $K_E$ is non-trivial for real anyon models. Refuted for a given experiment if its anyon model's representation is faithful on the braids it actually realises and the readout separates all of them, so that $E$ is injective on that set; the kernel claim is then vacuous there. (For Ising anyons this cannot happen, because their braid image is finite; F2 can only bite for dense models such as Fibonacci.)

F3: Self-certifying claims. §9 predicts that claims shipping their own trace are corrected faster. Sample: quantum-advantage and topological-qubit claims published in Nature, Science, Physical Review Letters and PRX Quantum from 2026-10-01 to 2029-12-31, each classified at publication by whether it releases raw data, analysis code and a recomputable fidelity or error bound. Refuted if, at the end of the window, the classified groups show no difference in the fraction of claims retracted, overturned or substantively revised, or in the median time to such a correction. The sample will be small, so a null result is weak evidence either way.

F4: Scope limit. §1–§5 and §8 are textbook physics and mathematics and survive any failure of F1–F3. §6 is a record of published results. Only the [D] readings in §7.3–§7.4, §8 (last paragraph), §9 (last paragraph) and §10 are at stake.

§13 Provenance and leaks

  1. Authorship. Topic, title, epigraph, section plan, falsifier drafts and source list from Khrug's intake document (prepared 2026-10-03 from the Autumn 2026 Dragon Digest). Post drafted by Claude (Anthropic) on 2026-10-03. Not yet reviewed by the Clinch.
  2. Verification. Every DOI was resolved on 2026-10-03 and its title, authors, year, volume and pages compared against the publisher record; arXiv entries were checked against the arXiv API. Paraphrases of results rest on abstracts and publisher summaries; most papers were not read in full.
  3. Changes from the intake. Author lists marked uncertain in the intake were resolved: Xu et al. (2024), Ghosh et al. (2025), Lo et al. (2026). The Tindall et al. (2026) DOI was found. The 2026 Nature exchange was identified as Legg's Matters Arising and Microsoft Quantum's reply, both now cited. The IBM/UChicago result is cited as the preprint Martiel et al. (2026), not the press release. Kanenobu (1986), Eliahou, Kauffman & Thistlethwaite (2003) and Kose (2021) were added for §7.2; Kassel & Turaev (2008), Jones (1987), Bigelow (1999, 2001) and King et al. (2025) were added for context. The headline claim was sharpened from one kernel to three maps (§7.1).
  4. Not verified. The intake's statement that the $S_3$ experiment "explicitly ignored error correction" was not checked against the full paper; §6 says only what the abstract supports. The Martiel et al. abstract (v1, 64 qubits, 314 $T$ gates, fidelity bound 0.349) and the IBM press release of 2026-07-30 (70 logical qubits, 468 $T$ gates) give different figures; the version difference was not resolved, so §9 quotes no numbers. D-Wave's public response to Tindall et al. was seen only as a press release and is not cited.
  5. Language sources. SAOB's entry spår (sbst. 2) was checked and does not record the mathematical sense, so §3.1 cites Swedish Wikipedia for the matrix term, a weaker source than the standard prefers. Hellquist (1922) was read in the Project Runeberg scan, pages 939–940.
  6. Excluded by scope. Media provenance, disinformation, economics and AI policy, covered in the digest, are left out by instruction.

References

  • Aharonov, D., Jones, V. & Landau, Z. (2009). A polynomial quantum algorithm for approximating the Jones polynomial. Algorithmica 55, 395–421. doi:10.1007/s00453-008-9168-0
  • Aharonov, Y. & Bohm, D. (1959). Significance of electromagnetic potentials in the quantum theory. Physical Review 115, 485–491. doi:10.1103/PhysRev.115.485
  • Arovas, D., Schrieffer, J. R. & Wilczek, F. (1984). Fractional statistics and the quantum Hall effect. Physical Review Letters 53, 722–723. doi:10.1103/PhysRevLett.53.722
  • Bartolomei, H. et al. (2020). Fractional statistics in anyon collisions. Science 368, 173–177. doi:10.1126/science.aaz5601
  • Berry, M. V. (1984). Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society A 392, 45–57. doi:10.1098/rspa.1984.0023
  • Bérut, A., Arakelyan, A., Petrosyan, A., Ciliberto, S., Dillenschneider, R. & Lutz, E. (2012). Experimental verification of Landauer's principle linking information and thermodynamics. Nature 483, 187–189. doi:10.1038/nature10872
  • Bigelow, S. (1999). The Burau representation is not faithful for n = 5. Geometry & Topology 3, 397–404. doi:10.2140/gt.1999.3.397
  • Bigelow, S. (2001). Braid groups are linear. Journal of the American Mathematical Society 14, 471–486. doi:10.1090/S0894-0347-00-00361-1
  • Braunstein, S. L. & Pati, A. K. (2007). Quantum information cannot be completely hidden in correlations: implications for the black-hole information paradox. Physical Review Letters 98, 080502. doi:10.1103/PhysRevLett.98.080502
  • Eliahou, S., Kauffman, L. H. & Thistlethwaite, M. B. (2003). Infinite families of links with trivial Jones polynomial. Topology 42, 155–169. doi:10.1016/S0040-9383(02)00012-5
  • Feynman, R. P. (1949). The theory of positrons. Physical Review 76, 749–759. doi:10.1103/PhysRev.76.749
  • Freedman, M. H., Larsen, M. & Wang, Z. (2002). A modular functor which is universal for quantum computation. Communications in Mathematical Physics 227, 605–622. doi:10.1007/s002200200645
  • Freedman, M. H., Kitaev, A., Larsen, M. J. & Wang, Z. (2003). Topological quantum computation. Bulletin of the American Mathematical Society 40, 31–38. doi:10.1090/S0273-0979-02-00964-3
  • Ghosh, B., Labendik, M., Musina, L., Umansky, V., Heiblum, M. & Mross, D. F. (2025). Anyonic braiding in a chiral Mach–Zehnder interferometer. Nature Physics 21, 1392–1397. doi:10.1038/s41567-025-02960-3
  • Google Quantum AI and Collaborators (Andersen, T. I. et al.) (2023). Non-Abelian braiding of graph vertices in a superconducting processor. Nature 618, 264–269. doi:10.1038/s41586-023-05954-4
  • Hellquist, E. (1922). Svensk etymologisk ordbok. Gleerup. Entries spå and spår. runeberg.org/svetym
  • Iqbal, M. et al. (2024). Non-Abelian topological order and anyons on a trapped-ion processor. Nature 626, 505–511. doi:10.1038/s41586-023-06934-4
  • Jones, V. F. R. (1985). A polynomial invariant for knots via von Neumann algebras. Bulletin of the American Mathematical Society 12, 103–111. doi:10.1090/S0273-0979-1985-15304-2
  • Jones, V. F. R. (1987). Hecke algebra representations of braid groups and link polynomials. Annals of Mathematics 126, 335–388. doi:10.2307/1971403
  • Kanenobu, T. (1986). Infinitely many knots with the same polynomial invariant. Proceedings of the American Mathematical Society 97, 158–162. doi:10.1090/S0002-9939-1986-0831406-7
  • Kassel, C. & Turaev, V. (2008). Braid Groups. Graduate Texts in Mathematics 247. Springer. doi:10.1007/978-0-387-68548-9
  • King, A. D. et al. (2025). Beyond-classical computation in quantum simulation. Science 388, 199–204. doi:10.1126/science.ado6285
  • Kitaev, A. Yu. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics 303, 2–30. doi:10.1016/S0003-4916(02)00018-0
  • Kitaev, A. (2006). Anyons in an exactly solved model and beyond. Annals of Physics 321, 2–111. doi:10.1016/j.aop.2005.10.005
  • Kose, F. C. (2021). On amphichirality of symmetric unions. Preprint. arXiv:2111.08765
  • Landauer, R. (1961). Irreversibility and heat generation in the computing process. IBM Journal of Research and Development 5, 183–191. doi:10.1147/rd.53.0183
  • Legg, H. F. (2026). On the robustness of topological gap detection via transport. Nature 654, E22–E26. doi:10.1038/s41586-026-10567-8
  • Leinaas, J. M. & Myrheim, J. (1977). On the theory of identical particles. Il Nuovo Cimento B 37, 1–23. doi:10.1007/BF02727953
  • Lo, C. F. B., Lyons, A., Gresh, D., Mills, M., Siegfried, P. E., Urmey, M. D., Tantivasadakarn, N., Dreyer, H., Vishwanath, A., Verresen, R. & Iqbal, M. (2026). Universal gates from braiding and fusing anyons on quantum hardware. Nature 655, 591–597. doi:10.1038/s41586-026-10709-y
  • Martiel, S., Chung, J.-U., Seif, A., Ghosh, S., Hincks, I., Deshpande, A., Manabe, H., Gu, H., Pan, F., Fefferman, B., Gambetta, J. M. & Javadi-Abhari, A. (2026). Sampling hard circuits with verifiably high fidelity. Preprint. arXiv:2607.25941
  • Microsoft Azure Quantum (2025). Interferometric single-shot parity measurement in InAs–Al hybrid devices. Nature 638, 651–655. doi:10.1038/s41586-024-08445-2
  • Microsoft Quantum (2026). Reply to: On the robustness of topological gap detection via transport. Nature 654, E27–E28. doi:10.1038/s41586-026-10568-7
  • Moore, G. & Read, N. (1991). Nonabelions in the fractional quantum Hall effect. Nuclear Physics B 360, 362–396. doi:10.1016/0550-3213(91)90407-O
  • Nakamura, J., Liang, S., Gardner, G. C. & Manfra, M. J. (2020). Direct observation of anyonic braiding statistics. Nature Physics 16, 931–936. doi:10.1038/s41567-020-1019-1
  • Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. (2008). Non-Abelian anyons and topological quantum computation. Reviews of Modern Physics 80, 1083–1159. doi:10.1103/RevModPhys.80.1083
  • Schubert, C. (2001). Perturbative quantum field theory in the string-inspired formalism. Physics Reports 355, 73–234. doi:10.1016/S0370-1573(01)00013-8
  • Tindall, J., Mello, A. F., Fishman, M., Stoudenmire, E. M. & Sels, D. (2026). Dynamics of disordered quantum systems with two- and three-dimensional tensor networks. Science 392, 868–872. doi:10.1126/science.adx2728
  • Wikipedia, Swedish edition. Entry Spår (matematik). sv.wikipedia.org, accessed 2026-10-03
  • Wilczek, F. (1982). Quantum mechanics of fractional-spin particles. Physical Review Letters 49, 957–959. doi:10.1103/PhysRevLett.49.957
  • Wilczek, F. & Zee, A. (1984). Appearance of gauge structure in simple dynamical systems. Physical Review Letters 52, 2111–2114. doi:10.1103/PhysRevLett.52.2111
  • Witten, E. (1989). Quantum field theory and the Jones polynomial. Communications in Mathematical Physics 121, 351–399. doi:10.1007/BF01217730
  • Wootters, W. K. & Zurek, W. H. (1982). A single quantum cannot be cloned. Nature 299, 802–803. doi:10.1038/299802a0
  • Xu, S. et al. (2024). Non-Abelian braiding of Fibonacci anyons with a superconducting processor. Nature Physics 20, 1469–1475. doi:10.1038/s41567-024-02529-6
  • Zanardi, P. & Rasetti, M. (1999). Holonomic quantum computation. Physics Letters A 264, 94–99. doi:10.1016/S0375-9601(99)00803-8
  • Zhang, H. et al. (2018). Quantized Majorana conductance. Nature 556, 74–79. doi:10.1038/nature26142. Retracted 2021, retraction note Nature 591, E30, doi:10.1038/s41586-021-03373-x

Operators: $B_n$, $\sigma_i$, $S_n$, $\rho$, $\operatorname{tr}$, $e_i$, $\tau$, $V_{\hat\beta}(t)$, $\hat\beta$, $E$, $K_E$, $k_B T\ln 2$ · Crosslinks: The Braided Substrate (DRK-143) · Picking Berries (DRK-172) · No Trace, No Section (DRK-153) · The Magical Substrate (DRK-156) · The Warded Class (DRK-190) · The Invariant (DRK-175) · The Dragged Frame (DRK-171) · The Pendragon Source (DRK-165)

Spåret minns knuten, inte vägen.

Khrug Engineering · Göteborg · ORCID 0009-0003-8049-7167 · DOI 10.5281/zenodo.23121197 · CC BY-SA 4.0