Topological Quantum Computation

Topological quantum computation represents a finite computation by states in a fixed anyon fusion sector and by operations assembled from braids, fusion measurements, classical adaptation, and explicitly declared non-braid resources. The central object is an anyon program record: it fixes the anyon theory and conventions, identifies the logical subspace, orders every operation, derives the resulting logical channel, separates projective gate error from leakage and rejected branches, and counts the resources that the claim actually uses.

This page owns that device-independent computation model. It does not derive a fusion category, classify a topological phase, certify a material platform, or turn passive suppression into a fault-tolerance theorem. Its worked records use ideal Ising, Fibonacci, and semion data to make the program semantics finite and reproducible while keeping physical protection and hardware evidence outside the licensed conclusions.

Required background. Anyons and Braiding supplies fusion spaces, $F$- and $R$-moves, braid topology, and adiabatic worldline operations. Universal Gate Sets supplies exact and approximate universality, projective target metrics, synthesis hypotheses, and the distinction between native, compiled, and logical gates.

Topological Computation as an Anyon Program

Fix ordered anyon charges $a_1,\ldots,a_n$ and a total charge $c$. A finite program acts on a declared subspace of the fusion space

$$\mathcal H_{a_1\cdots a_n}^{c}.$$

The input is not specified merely by drawing worldlines. A computational contract must also declare a fusion tree and basis, any multiplicity indices, the logical projector $P$, the prepared sector, the allowed braid generators, the measurement instruments, the classical record, and the output decoder. Only after those choices are fixed does a braid word define a logical operation that can be compared with a target.

The abstract workflow is

$$\text{import anyon data} \longrightarrow \text{encode and prepare} \longrightarrow \text{braid or measure} \longrightarrow \text{consume the record} \longrightarrow \text{decode and verify}.$$

This is a specialization of the carrier-independent process semantics owned by the Circuit Model. An anyon worldline is not automatically a circuit wire, however: its endpoints can be permuted, fusion channels can be superselected, a fusion-space dimension need not factor into qubits, and an $F$ matrix used in a calculation can be only a coordinate change rather than a physical operation.

A complete claim therefore distinguishes four layers:

• algebra: the imported fusion multiplicities and $F$ and $R$ data;
• program: the encoding, braid or measurement word, branch rule, and decoder;
• verification: logical action, projective error, leakage, success probability, and reproducible numerical certificate;
• implementation: gaps, paths, temperature, readout, calibration, error correction, and hardware resources.

This page owns the middle two layers. The first and fourth have separate canonical homes.

The Ten-Field Topological Computation Record

Every worked program on this page is reported vertically with the same ten fields. A field may be N/A only when the record explains why the corresponding operation or claim is outside its licensed model.

1. Computational task, input family, and licensed claim — State the input family, desired logical transformation or classical output, exact or approximate claim, and every capability used by the conclusion.
2. Anyon theory, topological data, conventions, and promises — Name the charges, vacuum, duals, fusion multiplicities, imported $F$ and $R$ data, braid orientation, gauge, and assumptions under which those data apply.
3. Encoding, fusion tree, total charge, basis, and logical projector — Fix the ordered anyons, total sector, fusion tree, basis labels, logical subspace, and projector.
4. Preparation, ancillas, initialization, and superselection sector — State pair creation, initial fusion outcomes, ancilla charges, accepted sectors, and whether preparation is assumed or costed.
5. Braid word, fusion measurements, and non-topological operations — Give generator signs, written and execution order, fusion projectors, and every injected or otherwise non-braid operation.
6. Classical record, adaptivity, byproducts, and frame convention — State outcomes, branch probabilities, corrections, retries, accepted records, and whether a byproduct is physically corrected or tracked.
7. Output, decoder, accepted event, and error metric — Define the output, endpoint ordering, decoder, success event, leakage, logical channel, projective metric, and any conditional normalization.
8. Protection assumptions, leakage model, resources, and comparator boundary — Separate ideal algebra from physical assumptions; count anyons, exchanges, depth, measurements, ancillas, feedforward, non-braid resources, and repeats; and identify the comparator.
9. Verification data, uncertainty, and reproducibility — Supply exact identities, numerical matrices and probabilities when used, precision, tolerance, evaluator convention, and experimental uncertainty or its N/A justification.
10. Conclusion, stopping point, and canonical handoff — State only what the record establishes, identify what remains unproved, and route the next phase, protection, hardware, code, compiler, or evidence question.

The schema prevents a common substitution: reporting an appealing braid matrix while leaving its encoding, preparation, readout, leakage, and resource boundary implicit. It also makes two records comparable without pretending that algebraic word length is a hardware runtime.

Fusion-Space Encodings and Total-Charge Sectors

For simple charges $a,b,c$, vacuum $1$, and fusion multiplicities $N_{ab}^{\;c}$,

$$a\times b = \sum_c N_{ab}^{\;c}\,c, \qquad \dim \mathcal V_{ab}^{c} = N_{ab}^{\;c}.$$

Multiplicity-free examples have $N_{ab}^{\;c}\in\{0,1\}$, but that is an extra property, not part of the definition. If multiplicities exceed one, the corresponding basis indices belong in the program record and must be transformed with the fusion-channel labels.

For a left-associated tree with ordered leaves $a_1,\ldots,a_n$ and total charge $c$,

$$\begin{aligned} \mathcal H_{a_1\cdots a_n}^{c} &= \bigoplus_{x_1,\ldots,x_{n-2}} \mathcal V_{a_1a_2}^{x_1} \otimes \mathcal V_{x_1a_3}^{x_2} \otimes\cdots \\ &\qquad\otimes \mathcal V_{x_{n-2}a_n}^{c}. \end{aligned}$$

The intermediate charges $x_j$ label a basis only after the tree and any multiplicity bases are fixed. An $F$ move changes that basis. It does not create an anyon, change the total sector, or automatically describe a physical recoupling pulse.

An encoding selects a logical subspace with projector $P$ inside the declared total-charge sector. Write

$$Q = I-P$$

for states counted as leakage relative to that encoding. Three distinctions matter:

• a fixed total charge does not by itself identify a logical tensor product;
• a fusion space whose dimension exceeds $2^k$ is not automatically a $k$-qubit register with harmless spare states;
• a process can preserve total charge while leaving the chosen code subspace.

Total charge is also a superselection label. A record must not assume coherent superpositions of different total sectors unless it declares the physical reference structure that makes those coherences operational. Preparing two vacuum-created pairs, for example, supplies a known total charge but not every desired intermediate fusion outcome.

A fusion-tree gauge change acts simultaneously on coordinates, states, operators, projectors, and readout effects. If $G$ is the unitary coordinate change, then a consistent convention uses

$$|\psi'\rangle = G|\psi\rangle, \qquad U' = GUG^\dagger, \qquad P' = GPG^\dagger.$$

Changing only a displayed braid matrix and retaining an untransformed target or fusion projector is not a gauge comparison.

Braid Words, Fusion-Tree Bases, and Projective Gates

The braid group $B_n$ is generated by adjacent exchanges $\sigma_i$ satisfying

$$\sigma_i\sigma_j = \sigma_j\sigma_i \quad (|i-j|\ge 2),$$

and

$$\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}.$$

Fix a unitary representation on the declared fusion sector,

$$\rho:B_n\longrightarrow U(\mathcal H), \qquad B_i = \rho(\sigma_i).$$

This page takes a positive generator to be a counterclockwise exchange and uses

$$B_i^{-1} = B_i^\dagger.$$

It also freezes rightmost-first multiplication. For

$$w = \sigma_{i_L}^{\epsilon_L}\cdots \sigma_{i_2}^{\epsilon_2} \sigma_{i_1}^{\epsilon_1},$$

the rightmost generator executes first and

$$U(w) = B_{i_L}^{\epsilon_L}\cdots B_{i_2}^{\epsilon_2} B_{i_1}^{\epsilon_1}.$$

A source using clockwise-positive braids or leftmost-first execution must translate the entire word and every derived matrix. Reversing the written symbols without inverting each generator does not produce the inverse braid.

If the exchanged pair is diagonal in the chosen fusion tree, its generator is built from the imported $R$ data. A neighboring pair that is not diagonal requires a basis change, schematically

$$B_i = F^{-1}RF.$$

The two $F$ factors here are coordinate transformations used to evaluate the same physical exchange. They are not additional exchanges in the resource ledger. A protocol that physically fuses, measures, or reassociates charges must instead record and count those operations.

For an isolated logical unitary $V$, a common phase is operationally irrelevant. The projective operator distance is

$$d_{\mathrm{proj}}(M,V) = \min_{\alpha\in\mathbb R} \left\| M-e^{i\alpha}V \right\|_{\mathrm{op}},$$

where $M=P U(w)P$ is the restricted map. This minimization does not erase relative phases between logical basis states, fusion channels, or coherent branches. Nor does it excuse lost norm: leakage and rejected records remain separate quantities.

Initialization, Fusion Measurement, and Adaptive Frames

Local operations normally create charges in combinations with allowed total charge, such as $a$ and $\bar a$. A program must therefore state whether its initial fusion channel is prepared, measured and accepted, corrected, or merely assumed. A cold topological phase does not by itself initialize the desired logical basis state.

An outcome-resolved fusion measurement is described by branch operators $K_y$. For input $\rho$,

$$p_y = \operatorname{Tr} \left( K_y\rho K_y^\dagger \right).$$

If outcome $y$ produces a known byproduct, a correction $C_y$ can be applied physically or accumulated in a classical logical frame. For accepted records $y\in\mathcal A$, the unnormalized accepted map is

$$\mathcal E_{\mathcal A}(\rho) = \sum_{y\in\mathcal A} C_y P K_y\rho K_y^\dagger P C_y^\dagger.$$

Its trace is the accepted probability. Dividing by that trace produces a conditional state for the declared input; if the probability depends on the input, the normalized rule is not itself a linear trace-preserving channel. The general distinction between an outcome probability and its conditional state update belongs to Quantum Instruments.

Every adaptive program must state:

• which fusion or parity result is stored in each classical bit;
• the branch probability and whether any branch is rejected;
• the next measurement or braid selected by that bit;
• the byproduct convention and whether it is corrected or tracked;
• the retry rule, stopping condition, and total accepted cost.

A tracked frame is computationally meaningful only if the decoder and all subsequent operations consume it. Calling a correction “virtual” does not make its classical latency, memory, or eventual readout reinterpretation disappear. Conversely, a basis-label relabeling is not automatically a physical gate.

Measurement-only topological computation replaces spatial braids with charge measurements and ancillas under a proved branch equivalence. It still needs a quantum instrument, initialization, feedforward, reset or retry, and a complete accepted-channel calculation. “No motion” is not “no control.”

Universality, Compilation, and Non-Braid Resources

An anyon model can be non-Abelian without being braid-universal. A braid-only claim must name the encoded family and state the relevant projective closure, for example

$$\overline{\pi\!\left(\rho(B_n)\right)} = PU(d_{\mathrm L}),$$

where $\pi$ removes common phase and $d_{\mathrm L}$ is the declared logical dimension. A noncommuting image is weaker than a dense image, and density on one fixed sector is weaker than a scalable encoded architecture.

Ideal Ising-anyon braids generate protected Clifford operations on standard encodings, so braiding alone is not universal. A universal Ising architecture must name and cost an additional non-Clifford resource, such as an injected state, a calibrated non-topological phase operation, a richer measurement primitive, or topology change. Magic State Distillation owns the general injection and factory workflow; citing it does not make its cost topologically protected.

Fibonacci braid representations are dense on suitable encodings, but that algebraic fact does not supply a short word for a chosen target. A finite compiler record must state:

• target unitary and projective metric;
• tolerance and verification precision;
• generator alphabet and inverse availability;
• encoding, ancillas, measurements, and non-braid operations;
• braid word, word length, and parallel depth;
• search or synthesis method and its certificate.

The generic transformation from a target unitary to a gate word belongs to Gate Decomposition. This page owns the braid-specific representation, word, and verification ledger once the imported anyon data are fixed.

Solovay–Kitaev scaling applies only under its stated hypotheses: a finite, inverse-closed, dense generator set acting on a compact finite-dimensional group, together with a suitable base approximation procedure. It must not be invoked for a leaking map, an infinite unbounded alphabet, a generator set not shown dense, or a hardware process whose inverses are unavailable. Even when the theorem applies, its asymptotic word-length statement is not a claim about routing, braid time, thermal errors, or end-to-end advantage.

Logical Channels, Leakage, and Verification

Let $P$ project onto the selected logical subspace and $Q=I-P$. For a unitary program $U(w)$ and logical input $\rho=P\rho P$, define

$$p_{\mathrm{leak}}(\rho,w) = \operatorname{Tr} \left[ Q U(w)\rho U(w)^\dagger \right].$$

A worst-case claim requires

$$p_{\mathrm{leak}}^{\max}(w) = \sup_{\rho=P\rho P} p_{\mathrm{leak}}(\rho,w),$$

not one favorable input. Projecting the output back into the code and renormalizing can give unit conditional fidelity while hiding a large loss probability.

For two $d$-dimensional unitaries $U$ and $V$, one may also report

$$F_{\mathrm{pro}} = \frac{ \left| \operatorname{Tr}(V^\dagger U) \right|^2 }{d^2},$$

and

$$F_{\mathrm{avg}} = \frac{dF_{\mathrm{pro}}+1}{d+1}.$$

These unitary fidelities are not substitutes for the declared projective operator metric. They also cannot be applied without qualification to a trace-decreasing, leaking, postselected, or outcome-dependent map. A complete adaptive verification instead compares the corrected accepted channel, its trace, the rejected branches, and leakage with the declared target.

A finite program should pass at least these checks:

1. every imported $F$, $R$, and braid generator used numerically is unitary;
2. the braid relations hold in the stated gauge and sector;
3. word order and inverse conventions reproduce an independent matrix product;
4. probabilities are nonnegative and sum to the declared accepted plus rejected plus leakage probability;
5. target, states, and readout effects transform consistently under a gauge change;
6. exact identities are distinguished from rounded numerical witnesses;
7. the conclusion uses the same metric and resource boundary as the record.

One input-output state overlap does not certify a logical gate. One word does not certify a dense image. One fixed-size representation does not certify a scalable family.

Protection Assumptions and Fault-Tolerance Boundaries

In an ideal anyon theory, smoothly deforming a braid without collisions or a change of topological class preserves its projective action. A physical use of that statement additionally assumes an appropriate gapped sector, separated excitations, controlled worldlines, no unintended enclosed charge, and an operating time compatible with nonadiabatic and environmental errors. The Adiabatic Theorem owns the general spectral and runtime conditions; a topological label alone supplies no runtime bound.

Passive protection can suppress selected local matrix elements or residual splittings. It does not guarantee correct initialization, fusion readout, absence of thermally created anyons, immunity to quasiparticle poisoning, leakage-free transport, or a universal native gate set. The Topological Quantum Computation Bridge owns this protection hierarchy and the distinction between intrinsic, engineered, and programmed anyons. The Topological Qubits page owns device modules, control and readout, calibration, noise metrics, scaling, and current hardware evidence.

Topological matter and an actively stabilized topological code are different architectures. The Surface Code owns stabilizer patches, repeated syndrome histories, decoding, thresholds, and lattice surgery. Fault-Tolerant Gates owns the general fault-containment contract for encoded operations. A decreasing logical error rate under a stated noise model requires those active architecture ingredients; it cannot be inferred from an exact ideal braid matrix.

Similarly, Topological Order owns phase classification, modular and response data, and phase-level evidence. This page imports only the finite anyon data needed to specify a computation. It does not infer a phase from a processor simulation or infer a protected computation from a phase signature.

Resource Accounting and Evidence Boundaries

Braid word length is one resource currency, not a complete cost. Every record should report the subset of the following quantities that it actually uses:

• number and types of computational anyons;
• fixed total sector and unused fusion-space dimensions;
• preparation pairs, ancilla charges, and accepted initialization probability;
• elementary exchanges, signed word length, parallel braid depth, and routing assumptions;
• fusion or parity measurements, classical record bits, feedforward depth, retries, and reset;
• physical or tracked byproduct corrections;
• non-braid gates, injected states, distillation, or topology changes;
• leakage checks, rejected branches, repetitions, and accepted throughput;
• compiler time, verification precision, and target tolerance when claimed.

An $F$ matrix used to change coordinates is not an exchange. A simultaneous layer of disjoint exchanges has word length greater than one but ideal parallel depth one. A repeat-until-success branch must be converted into expected or tail-bounded accepted cost rather than reported as one attempt. If physical time, qubit equivalents, factory cost, or control bandwidth is not licensed by the abstract model, the record marks it N/A instead of silently setting it to zero.

Model-level resource counts can be handed to Resource Estimation only after the logical operations are mapped to a declared fault-tolerant architecture. Claims about experimental realization, protection scaling, or computational advantage require the evidence vocabulary developed in Claims, Hype, and Evidence Standards.

Keep four conclusions separate:

1. a matrix is the image of the stated braid word;
2. that image approximates a target under a stated metric;
3. an operation is protected under stated physical assumptions;
4. a complete architecture is fault tolerant or advantageous under a stated comparator.

The first two are licensed by the finite records below. The latter two require their own owners and evidence.

Worked Audit: An Exact Ising-Anyon Hadamard Braid

This audit uses one fixed Ising gauge to turn a three-exchange word into an exact logical statement. The phase, basis, decoder, and resource boundary are part of the result.

1. Computational task, input family, and licensed claim — On a logical qubit encoded by four Ising $\sigma$ anyons of total charge $1$, implement a Hadamard on input $|0_{\mathrm L}\rangle$ and verify the final fusion distribution. The licensed claim is exact equality up to global phase in the stated ideal gauge.

2. Anyon theory, topological data, conventions, and promises — Use charges $1,\psi,\sigma$ with

   $$\sigma\times\sigma = 1+\psi,$$

   and the imported data

   $$F = \frac{1}{\sqrt2} \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix},$$ $$R = \begin{pmatrix} e^{-i\pi/8}&0\\ 0&e^{3i\pi/8} \end{pmatrix} = e^{-i\pi/8} \begin{pmatrix} 1&0\\ 0&i \end{pmatrix}.$$

   A positive generator is counterclockwise and products act rightmost first. These are exact ideal anyon data, not a material certificate.

3. Encoding, fusion tree, total charge, basis, and logical projector — Order four $\sigma$ anyons, fix total charge $1$, and label the first-pair fusion channel $x\in\{1,\psi\}$. Use

   $$|0_{\mathrm L}\rangle = |x=1\rangle, \qquad |1_{\mathrm L}\rangle = |x=\psi\rangle.$$

   The selected ideal sector is exactly two-dimensional, so its logical projector is $P=I_2$.

4. Preparation, ancillas, initialization, and superselection sector — Prepare exactly four $\sigma$ anyons in total sector $1$ and initialize $x=1$. No ancilla is used. Ideal sector preparation is assumed; its physical mechanism, error, and cost are N/A in this algebra audit.

5. Braid word, fusion measurements, and non-topological operations — Set

   $$B_1 = R, \qquad B_2 = F^{-1}RF.$$

   Use

   $$w = \sigma_1\sigma_2\sigma_1,$$

   so that

   $$U(w) = B_1B_2B_1 = e^{-i\pi/8}H, \qquad H=F.$$

   Fuse the first pair at the end. There is no non-topological gate. The exact common phase is $e^{-i\pi/8}$, not $e^{-3i\pi/8}$.

6. Classical record, adaptivity, byproducts, and frame convention — Record the final $x\in\{1,\psi\}$. There is no mid-program measurement, adaptive branch, byproduct, retry, or frame update, so those costs are zero rather than hidden.

7. Output, decoder, accepted event, and error metric — Decode $x=1$ as bit $0$ and $x=\psi$ as bit $1$. Both outcomes are accepted. Because

   $$U(w)|0_{\mathrm L}\rangle = e^{-i\pi/8} \frac{ |0_{\mathrm L}\rangle + |1_{\mathrm L}\rangle }{\sqrt2},$$

   the output statistics and logical metrics are

   $$p(x=1) = p(x=\psi) = \frac12,$$ $$p_{\mathrm{leak}} = 0, \qquad d_{\mathrm{proj}}(U,H) = 0,$$

   with $F_{\mathrm{pro}}=F_{\mathrm{avg}}=1$.

8. Protection assumptions, leakage model, resources, and comparator boundary — Count four anyons, three elementary exchanges, sequential braid depth three, and one final fusion measurement. Ancillas, feedforward, non-braid gates, and repeats are zero. Leakage is zero inside the exact two-dimensional representation; physical out-of-sector leakage is N/A. The record licenses no gap, path, temperature, lifetime, readout fidelity, or competing implementation comparator.

9. Verification data, uncertainty, and reproducibility — Verify

   $$F^\dagger F = R^\dagger R = I,$$

   and

   $$B_1B_2B_1 = B_2B_1B_2$$

   exactly. The rounded matrix witness is

   $$U \approx \begin{pmatrix} 0.653281482438188-0.270598050073098i& 0.653281482438188-0.270598050073098i\\ 0.653281482438188-0.270598050073098i& -0.653281482438188+0.270598050073098i \end{pmatrix}.$$

   Exact symbolic evaluation is authoritative. At-least-30-digit and independent binary64 evaluations must agree componentwise and in the probabilities to $10^{-12}$. Experimental uncertainty is N/A because this is an ideal algebra audit.

10. Conclusion, stopping point, and canonical handoff — The word implements an exact projective Hadamard in the declared Ising encoding. Ising braids remain Clifford-only and do not form a universal braid gate set. Protection, hardware, non-Clifford completion, and fault-tolerant overhead stop at their canonical owners.

The output is exact projectively because the common phase multiplies both logical amplitudes. Dropping that phase is legitimate here; dropping the relative sign in the second column of $H$ would change the gate.

Worked Audit: A Fibonacci Short-Word Compilation

The second audit applies the same three-exchange word to a Fibonacci encoding. It produces a close but nonzero approximation to $H$, so the record must state the metric rather than calling the word “a Hadamard.”

1. Computational task, input family, and licensed claim — Approximate a logical Hadamard on $|0_{\mathrm L}\rangle$ using a three-exchange Fibonacci braid, then report the projective operator error, two unitary fidelities, and the final first-pair fusion distribution.

2. Anyon theory, topological data, conventions, and promises — Use

   $$\tau\times\tau = 1+\tau, \qquad \varphi = \frac{1+\sqrt5}{2},$$

   with

   $$F = \begin{pmatrix} \varphi^{-1}&\varphi^{-1/2}\\ \varphi^{-1/2}&-\varphi^{-1} \end{pmatrix}, \qquad F^2=I,$$

   and

   $$R = \begin{pmatrix} e^{-4\pi i/5}&0\\ 0&e^{3\pi i/5} \end{pmatrix}.$$

   Retain the counterclockwise, rightmost-first convention. These exact matrices define the audit gauge and are not rederived here.

3. Encoding, fusion tree, total charge, basis, and logical projector — Use three $\tau$ anyons with total charge $\tau$ and first-pair basis $x\in\{1,\tau\}$, where

   $$|0_{\mathrm L}\rangle = |x=1\rangle, \qquad |1_{\mathrm L}\rangle = |x=\tau\rangle.$$

   This ideal sector is two-dimensional, so $P=I_2$.

4. Preparation, ancillas, initialization, and superselection sector — Prepare the fixed total-$\tau$ sector in $|0_{\mathrm L}\rangle$. No ancilla or postselected preparation branch is included. Preparation cost and physical sector errors are N/A.

5. Braid word, fusion measurements, and non-topological operations — Set

   $$B_1 = R, \qquad B_2 = FRF,$$

   and use

   $$w = \sigma_1\sigma_2\sigma_1, \qquad U(w) = B_1B_2B_1.$$

   Measure the first-pair fusion channel at the end. No non-braid operation is inserted.

6. Classical record, adaptivity, byproducts, and frame convention — Record $x\in\{1,\tau\}$. Both outcomes are accepted; there is no correction, byproduct, retry, or adaptive branch.

7. Output, decoder, accepted event, and error metric — Decode $x=1$ and $x=\tau$ as the two logical bits. For target $H$, the minimizing common phase is

   $$\alpha = -\frac{4\pi}{5},$$

   and

   $$d_{\mathrm{proj}}(U,H) \approx 0.119088247109089,$$ $$F_{\mathrm{pro}} \approx 0.985868271756646, \qquad F_{\mathrm{avg}} \approx 0.990578847837764.$$

   On $|0_{\mathrm L}\rangle$,

   $$p(x=1) = \varphi^{-2} \approx 0.381966011250105,$$ $$p(x=\tau) = \varphi^{-1} \approx 0.618033988749895, \qquad p_{\mathrm{leak}} = 0.$$

8. Protection assumptions, leakage model, resources, and comparator boundary — Count three anyons, three elementary exchanges, sequential depth three, and one final fusion measurement. Ancillas, feedforward, non-braid gates, and repeats are zero. Leakage is zero only inside this exact sector. The comparator is the exact logical Hadamard in the same basis and metric, not a physical gate stack.

9. Verification data, uncertainty, and reproducibility — Verify unitarity and the braid relation exactly. The frozen rounded matrix is

   $$U \approx \begin{pmatrix} -0.500000000000000-0.363271264002680i& -0.636009824757034-0.462088185915222i\\ -0.636009824757034-0.462088185915222i& 0.500000000000000+0.363271264002680i \end{pmatrix}.$$

   Recompute from exact $\varphi$, $F$, and $R$ to at least 30 digits and in an independent binary64 evaluator. Matrix entries, probabilities, distance, and fidelities must agree to $10^{-12}$. Experimental uncertainty is N/A.

10. Conclusion, stopping point, and canonical handoff — This three-exchange word is a quantitatively certified approximation to $H$ in one Fibonacci sector. One word does not prove density, efficient compilation, scalable multi-qubit action, protection, hardware performance, or advantage. Those claims stop at their theorem, compiler, protection, hardware, and evidence owners.

For two-dimensional unitary targets, the reported distance can be reproduced from

$$d_{\mathrm{proj}}(U,H) = \sqrt{ 2- \left| \operatorname{Tr}(H^\dagger U) \right| }.$$

The high gate fidelities and unequal output probabilities answer different questions. Neither number licenses a physical protection or compiler-efficiency claim.

Common Failure Modes and Canonical Handoffs

Treating non-Abelian statistics as universality. Noncommuting braid matrices establish a non-Abelian image, not a dense image, an efficient compiler, or a scalable encoded family. State each stronger claim separately.

Omitting the fusion-tree convention. Matrix entries are meaningless for a cross-source comparison unless total charge, fusion tree, multiplicity basis, gauge, braid orientation, and word order are aligned. Transform the target, state, projector, and readout with the generators.

Reading a braid word in the wrong direction. In this page’s convention the rightmost factor acts first. The inverse reverses the order and inverts every generator. Merely reversing symbols or merely changing every sign gives a different word in general.

Counting coordinate changes as exchanges. An $F^{-1}RF$ matrix product can evaluate one exchange. The $F$ factors are not additional physical braids unless the protocol actually performs and licenses a recoupling operation.

Discarding an observable relative phase. A common phase on an isolated logical unitary is projectively irrelevant. A phase that differs by logical basis state, fusion channel, or coherent branch is not common and must remain in the record.

Renormalizing away failure. A conditional state can have fidelity one even when leakage or rejection is large. Report the accepted probability, unconditioned channel, rejected records, and leakage before quoting conditional fidelity.

Invoking a compiler theorem without its hypotheses. A finite dense inverse-closed generator set and a compact finite-dimensional target group are substantive assumptions. Word-length scaling does not set physical braid time or fault-tolerant overhead.

Calling Ising braids universal. Standard Ising braids generate Clifford operations. Any non-Clifford injection, measurement, topology change, or calibrated phase gate must be named, costed, and assigned its own error and protection status.

Calling measurement-only computation hardware-free. Measurement-only protocols replace motion with ancillas, quantum instruments, feedforward, reset, and sometimes retry. Those operations remain in the resource and error ledger.

Promoting ideal algebra into physical evidence. An exact braid matrix does not prove a gap, correct adiabatic transport, suppressed thermal errors, protected readout, a threshold, or scalable hardware. Topological Superconductors and the Fractional Quantum Hall Effect own phase- and platform-specific evidence.

Inferring advantage from universality. A universal model can simulate a broad class of computations. It says nothing by itself about algorithms, input access, error-corrected cost, classical comparators, or observed advantage.

Use the site’s navigation owners without duplicating their content:

• Gates, Circuits, and Computation Models locates this model among circuit, measurement-based, adiabatic, annealing, and continuous-variable alternatives.
• Quantum Information and Technology supplies the volume-level path from information foundations through computation, error correction, hardware, and evidence.
• What Is Quantum Information? distinguishes information-processing claims from carrier and implementation claims.
• The Quantum Information Roadmap places the anyon program after its mathematical prerequisites and before algorithm and architecture work.
• Math Needed for Quantum Information routes braid representations, direct sums, operator norms, projective metrics, channels, and probability checks to their mathematical foundations.

Exercises

Exercise 1: Decode a Braid Word and Its Inverse

In the Ising gauge of the first audit, take

$$w = \sigma_2^{-1}\sigma_1.$$

Identify the execution order, derive $\rho(w)$ and $\rho(w^{-1})$, verify the inverse, and find the first-pair fusion probabilities on $|0_{\mathrm L}\rangle$. Count the word length and sequential depth.

Solution

The rightmost factor executes first: perform the counterclockwise $\sigma_1$, followed by the clockwise $\sigma_2^{-1}$. Therefore

$$\rho(w) = B_2^\dagger B_1 = \frac12 \begin{pmatrix} 1-i&-1+i\\ 1+i&1+i \end{pmatrix}.$$

The group inverse reverses the factor order and inverts both generators:

$$w^{-1} = \sigma_1^{-1}\sigma_2.$$

Hence

$$\rho(w^{-1}) = B_1^\dagger B_2 = \rho(w)^\dagger,$$

and direct multiplication gives

$$\rho(w^{-1})\rho(w) = \rho(w)\rho(w^{-1}) = I.$$

The first column of $\rho(w)$ gives

$$\rho(w)|0_{\mathrm L}\rangle = \frac{1-i}{2}|0_{\mathrm L}\rangle + \frac{1+i}{2}|1_{\mathrm L}\rangle.$$

Both squared amplitudes are $1/2$, so the two fusion outcomes are equally likely. The word and its inverse each have length two and sequential depth two because the two exchanges share an anyon. Reversing the written symbols without inverting them, or applying the matrices leftmost first, produces a different operation.

Exercise 2: Rephase a Fusion-Tree Basis

Let

$$G = \operatorname{diag} \left( 1,e^{i\pi/3} \right).$$

Transform the Ising generators, states, target, logical projector, and fusion projectors into the rephased basis. Prove that a complete protocol has the same fusion probabilities, leakage, and projective gate error.

Solution

Use one coordinate transformation everywhere:

$$B_i' = GB_iG^\dagger, \qquad |\psi'\rangle = G|\psi\rangle,$$ $$V' = GVG^\dagger, \qquad P' = GPG^\dagger, \qquad M_y' = GM_yG^\dagger.$$

For any word,

$$U' = GUG^\dagger,$$

because adjacent $G^\dagger G$ factors cancel. The first Ising generator is diagonal, so $B_1'=B_1$. If

$$B_2 = e^{-i\pi/8} \begin{pmatrix} A&C\\ C&A \end{pmatrix}, \qquad A=\frac{1+i}{2}, \quad C=\frac{1-i}{2},$$

then

$$B_2' = e^{-i\pi/8} \begin{pmatrix} A&C e^{-i\pi/3}\\ C e^{i\pi/3}&A \end{pmatrix}.$$

The matrix entries changed, but a complete measurement amplitude did not:

$$\langle m_y'|U'|\psi'\rangle = \langle m_y|U|\psi\rangle.$$

Similarly, $Q'=GQG^\dagger$ makes the leakage trace invariant. Unitary invariance of the operator norm gives

$$d_{\mathrm{proj}}(U',V') = d_{\mathrm{proj}}(U,V).$$

Rephasing only $B_i$ while leaving the target or readout unchanged compares different protocols, not different gauges for one protocol.

Exercise 3: Count a Fibonacci Encoding Sector

Starting from one $\tau$ anyon and $\tau\times\tau=1+\tau$, count the fusion-space dimensions for six $\tau$ anyons in total sectors $1$ and $\tau$. Also give the dimension when the total charge is not fixed, and explain what the count does not establish about a qubit encoding.

Solution

Let

$$a_n = \dim\mathcal H_{\tau^n}^{1}, \qquad b_n = \dim\mathcal H_{\tau^n}^{\tau}.$$

For one anyon, $(a_1,b_1)=(0,1)$. Adding one $\tau$ and using the fusion rule gives

$$a_{n+1} = b_n, \qquad b_{n+1} = a_n+b_n.$$

Thus

$$a_n = F_{n-1}, \qquad b_n = F_n,$$

with $F_1=F_2=1$. For six anyons,

$$a_6 = F_5 = 5, \qquad b_6 = F_6 = 8.$$

Without a fixed total sector the direct-sum dimension is

$$a_6+b_6 = 13.$$

A five-dimensional sector does not itself select a two-qubit tensor factor, choose the unused state, or prove that a proposed braid preserves a chosen four-dimensional code. Those are encoding and leakage questions beyond the dimension count.

Exercise 4: Consume an Adaptive Fusion Record

Consider the abstract two-outcome instrument

$$K_0 = \sqrt{\frac35}I, \qquad K_1 = \sqrt{\frac25}X,$$

with corrections $C_0=I$ and $C_1=X$. Verify completeness, compute the corrected channel, and find the channel and target fidelity on $|0\rangle$ if the classical record is forgotten before correction.

Solution

Completeness follows from

$$K_0^\dagger K_0 + K_1^\dagger K_1 = \frac35 I + \frac25 I = I.$$

The outcome probabilities are state independent:

$$p_0 = \frac35, \qquad p_1 = \frac25.$$

After consuming the record,

$$\begin{aligned} \mathcal E_{\mathrm{corr}}(\rho) &= C_0K_0\rho K_0^\dagger C_0^\dagger + C_1K_1\rho K_1^\dagger C_1^\dagger \\ &= \frac35\rho + \frac25 X(X\rho X)X \\ &= \rho. \end{aligned}$$

If the record is discarded before correction, the channel is instead

$$\mathcal E_{\mathrm{forget}}(\rho) = \frac35\rho + \frac25X\rho X.$$

For input $|0\rangle$ this yields

$$\frac35|0\rangle\langle0| + \frac25|1\rangle\langle1|,$$

whose fidelity with $|0\rangle$ is $3/5$. This fixture is an abstract licensed instrument with a known byproduct. It is not a claim that a particular device implements a fusion measurement with these Kraus operators.

Exercise 5: Compare Three Logical-Gate Metrics

Suppose $V^\dagger U$ is a two-dimensional unitary whose eigenvalues are $e^{i(\theta\pm\delta/2)}$, with principal eigenphase spread $\delta=\pi/5$. Compute the projective operator distance, process fidelity, and average gate fidelity. Explain why these values are not interchangeable.

Solution

The optimal common phase removes $\theta$. The two residual eigenphases are $\pm\delta/2$, so

$$d_{\mathrm{proj}} = 2\sin\frac{\delta}{4} = 2\sin\frac{\pi}{20} \approx 0.312868930080462.$$

The trace magnitude is $2\cos(\delta/2)$, giving

$$F_{\mathrm{pro}} = \cos^2\frac{\delta}{2} = \cos^2\frac{\pi}{10} \approx 0.904508497187474.$$

For $d=2$,

$$F_{\mathrm{avg}} = \frac{2F_{\mathrm{pro}}+1}{3} \approx 0.936338998124982.$$

The operator distance is a worst-direction norm after optimizing global phase; the process fidelity is a normalized trace overlap; and the average gate fidelity averages state fidelity. Their numerical values and operational uses differ. This fixture assumes two unitaries on one code space, so none of the three values licenses any statement about leakage or rejected branches.

Exercise 6: Separate Leakage from Conditional Fidelity

In a Hilbert space containing a code state $|+\rangle$ and an orthogonal leakage state $|\ell\rangle$, analyze

$$|\Psi\rangle = \frac{\sqrt3}{2}|+\rangle + \frac12|\ell\rangle.$$

Find the leakage, code acceptance probability, conditional code-state fidelity, and unconditioned overlap with the target $|+\rangle$.

Solution

Let $P$ project onto the code and $Q$ onto the leakage sector. Orthogonality gives

$$p_{\mathrm{leak}} = \langle\Psi|Q|\Psi\rangle = \frac14.$$

The code acceptance probability is

$$p_{\mathrm{code}} = \langle\Psi|P|\Psi\rangle = \frac34.$$

Conditioned on code acceptance, the normalized state is exactly $|+\rangle$, so its conditional fidelity is $1$. Without conditioning, the squared target overlap is

$$|\langle+|\Psi\rangle|^2 = \frac34.$$

Quoting only the unit conditional fidelity would hide the $1/4$ leakage. Even the full calculation concerns one input; it does not certify a deterministic or worst-case logical gate.

Exercise 7: Schedule an Eight-Anyon Braid Word

On eight ordered anyons, execute the commuting layer

$$\sigma_1\sigma_3\sigma_5\sigma_7$$

first and then the commuting layer

$$\sigma_2^{-1}\sigma_4^{-1}\sigma_6^{-1}.$$

Count anyons, elementary exchanges, word length, and ideal parallel depth. State what logical claim the schedule licenses without further data.

Solution

The first layer exchanges disjoint pairs $(1,2)$, $(3,4)$, $(5,6)$, and $(7,8)$. Because the generator indices differ by at least two, all four commute and can run in one ideal layer. The second layer exchanges $(2,3)$, $(4,5)$, and $(6,7)$ clockwise; those three generators also commute.

The ledger is therefore:

• eight anyons;
• seven elementary exchanges;
• signed word length seven;
• ideal parallel braid depth two.

Fusion-basis $F$ changes used to evaluate the word are coordinate operations, not extra exchanges. Physical routing or unequal braid durations could make the implementation depth larger and are N/A here. Without an anyon model, total sector, encoding, input, target, and final measurement, this schedule licenses no logical gate or fidelity claim.

Exercise 8: Complete a Ten-Field Abelian-Anyon Program Record

Use semion charges $\{1,s\}$ with

$$s\times s = 1, \qquad R_{ss}^{1} = i.$$

Prepare four semions of total charge $1$, execute $\sigma_1$ and $\sigma_3$ in parallel followed by $\sigma_2^{-1}$, and complete the full program record.

Solution

1. Computational task, input family, and licensed claim — Evaluate a finite ideal semion braid-and-fuse program on its unique four-anyon state. The only licensed claim is its scalar braid action, deterministic fusion output, and exact resource count.

2. Anyon theory, topological data, conventions, and promises — Use charges $\{1,s\}$ with $s\times s=1$ and the declared gauge $R_{ss}^{1}=i$. Positive generators are counterclockwise and products act rightmost first.

3. Encoding, fusion tree, total charge, basis, and logical projector — Four $s$ anyons of total charge $1$ have a one-dimensional fusion space. Use the pairwise tree and $P=I_1$; there is no logical qubit or unused code state.

4. Preparation, ancillas, initialization, and superselection sector — Prepare two vacuum-created $s$ pairs in total sector $1$. No ancilla, alternative sector, or preparation acceptance branch is licensed.

5. Braid word, fusion measurements, and non-topological operations — First execute $\sigma_1$ and $\sigma_3$ in parallel, then $\sigma_2^{-1}$. Thus

   $$w = \sigma_2^{-1}\sigma_1\sigma_3, \qquad U(w) = (-i)(i)(i) = i.$$

   Track the endpoint permutation and fuse in the declared output tree. There is no non-topological operation.

6. Classical record, adaptivity, byproducts, and frame convention — The sole fusion record occurs with probability $1$. There is no adaptive branch, retry, byproduct, or frame update.

7. Output, decoder, accepted event, and error metric — Accept the unique fusion outcome and return the sole label. Leakage is zero and the projective error relative to the identity on the one-dimensional space is zero; the scalar $i$ is a global phase.

8. Protection assumptions, leakage model, resources, and comparator boundary — Count four anyons, three exchanges, word length three, parallel depth two, and one final fusion measurement. Ancillas, feedforward, non-braid gates, and repeats are zero. The comparator is only the identity on the same ideal one-dimensional space. Physical gap, path, timing, and error quantities are N/A.

9. Verification data, uncertainty, and reproducibility — Verify the scalar product exactly and independently as complex binary64 to $10^{-12}$. Experimental uncertainty is N/A because no physical implementation is claimed.

10. Conclusion, stopping point, and canonical handoff — The record verifies a nontrivial scalar anyon braid but no encoded qubit or logical transformation beyond projective identity. It establishes neither non-Abelian statistics, universality, protection, hardware, fault tolerance, nor speedup; those questions stop at their canonical owners.

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