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Quantum computation with Turaev-Viro codes

Robert Koenig, Greg Kuperberg, Ben W. Reichardt

arXiv:1002.2816v1quant-ph

TL;DR

The paper tackles how to implement TQFT computations in the Turaev–Viro spin-lattice code space, including computations involving non-abelian anyons. It identifies fusion bases and gives procedures for braids, state preparation, and charge measurement; the resulting schemes support universal quantum computation with geometrically local operations on locally stabilized codewords.

  • Problem

    The paper addresses how to implement TQFT computations for modular tensor categories in the Turaev–Viro spin-lattice code space, beyond the toric code’s abelian anyon model, which does not allow universal quantum computation.

    Source: §1 · p. 3

  • Method

    It identifies fusion-diagram bases for the doubled theory and implements encoded braids through protected local gates corresponding to triangulation Pachner moves.

    Source: §1 · p. 3

  • Results

    The paper provides schemes for universal quantum computation using geometrically local operations on locally stabilized codewords.

    Source: §1 · p. 3

  • Takeaways & Limitations

    These schemes may serve as a starting point for fault-tolerance schemes using continuous stabilizer measurements and active error correction.

    Source: §1 · p. 3

    With noise and periodic error correction, both measurement and preparation expose encoded information.

    Source: §5.1 · p. 23

Abstract

from arXiv · show

The Turaev-Viro invariant for a closed 3-manifold is defined as the contraction of a certain tensor network. The tensors correspond to tetrahedra in a triangulation of the manifold, with values determined by a fixed spherical category. For a manifold with boundary, the tensor network has free indices that can be associated to qudits, and its contraction gives the coefficients of a quantum error-correcting code. The code has local stabilizers determined by Levin and Wen. For example, applied to the genus-one handlebody using the Z_2 category, this construction yields the well-known toric code. For other categories, such as the Fibonacci category, the construction realizes a non-abelian anyon model over a discrete lattice. By studying braid group representations acting on equivalence classes of colored ribbon graphs embedded in a punctured sphere, we identify the anyons, and give a simple recipe for mapping fusion basis states of the doubled category to ribbon graphs. We explain how suitable initial states can be prepared efficiently, how to implement braids, by successively changing the triangulation using a fixed five-qudit local unitary gate, and how to measure the topological charge. Combined with known universality results for anyonic systems, this provides a large family of schemes for quantum computation based on local deformations of stabilizer codes. These schemes may serve as a starting point for developing fault-tolerance schemes using continuous stabilizer measurements and active error-correction.

1 Introduction

The paper addresses how to implement TQFT computations in a locally stabilized lattice code, extending beyond the abelian anyon model of the toric code. It identifies the doubled-theory basis, implements braids, and describes state preparation and topological-charge measurement.

1 Introduction

  • The scheme translates more general anyon models into error-correcting codes, addressing the toric code’s limitation to abelian anyons and its inability to support universal quantum computation.The Turaev–Viro construction realizes TQFT Hilbert spaces as code spaces of commuting local stabilizers [LW05b].Source: §1 · p. 3
  • The paper addresses implementing TQFT computations for modular tensor categories in the Turaev–Viro spin-lattice code space.The introduction organizes this problem around three steps.Source: §1 · p. 3
  • It identifies bases of HΣ corresponding to fusion diagrams for the doubled theory by decomposing the braid-group representation on colored ribbon graphs.The ribbon graphs are considered modulo local equivalence relations.Source: §1 · p. 3
  • It realizes each encoded braid unitary as a sequence of protected local gates, using Pachner moves that correspond to spin F-moves.Each move transforms the code space between local stabilizer-Hamiltonian ground spaces by adiabatically changing three Hamiltonian terms.Source: §1 · p. 3
  • The paper also gives procedures for preparing initial states in HΣ and measuring topological charge within a region.The paper’s outline places these procedures alongside encoding, logical operations, and readout of encoded information.Source: §1 · p. 3, Outline · p. 4
  • The resulting schemes provide universal quantum computation using geometrically local operations on locally stabilized codewords, and may inform fault-tolerance schemes based on continuous stabilizer measurements and active error correction.The presentation focuses on Fibonacci through Sections 2–5, then extends results to arbitrary ribbon categories in Section 6.Source: §1 · p. 3, Outline · p. 4

2 The Hilbert space HΣ of Fibonacci ribbon graphs on Σ

The section defines the ribbon-graph Hilbert space on a surface, constructs computational bases for encoding it, and describes how gluing and surface deformations act on that space.

2.1 Definition of Fibonacci ribbon graphs and HΣ

  • A Fibonacci ribbon graph is an embedded graph with degree-two or degree-three vertices, allowing a degree-one vertex at each marked boundary point; HΣ consists of their linear combinations modulo local relations.These graphs are also called skeins, spin networks, or string nets.Source: §2.1 · p. 4

2.2 Computational bases for and inner product on HΣ

  • For punctured spheres, computational bases arise from dual graphs of triangulations, with edge bits encoding ribbon-graph edges and assignments constrained to have no dead ends.The basis states are orthonormal under an inner product that does not depend on the triangulation.Source: §2.2 · p. 5
  • For n ≥ 2, these bases embed HΣn into (C2)^(5n−6), giving a computational encoding with 5n − 6 qubits.The pants-decomposition construction associates qubits to leaves, the root, and internal pant segments.Source: §2.2 · p. 6

2.3 Actions on HΣ

  • Gluing ribbon-graph states with matching boundary labels defines a linear operator between the corresponding surface Hilbert spaces.The operator acts by gluing constituent graphs and reducing them under the local equivalences.Source: §2.3 · p. 7
  • The mapping class group acts unitarily on HΣ; on punctured spheres it is generated by Dehn twists and braid moves that deform ribbon graphs.The section describes these operations through cutting along curves and inserting cylindrical segments.Source: §2.3 · pp. 7–8

3 Doubled Fibonacci anyons in HΣ

The standard computational basis is inconvenient for Dehn-twists and braid-moves; the paper introduces an orthonormal anyonic fusion basis that makes these actions and the structure needed for universal computation clearer.

3 Doubled Fibonacci anyons in HΣ

  • The orthonormal anyonic fusion basis diagonalizes Dehn-twists, addressing the standard basis’s inconvenient transformations and asymmetry.The authors say this basis also clarifies the steps needed for universal quantum computation.Source: §3 · p. 8

3.1 Anyonic fusion basis for HΣn

  • Basis states are indexed by ℓ-consistent doubled fusion diagrams, constructed as three-dimensional ribbon graphs and reduced to two dimensions.The construction adds vacuum lines and closes the graphs at boundaries according to their labels.Source: §3.1 · p. 9

Example: Anyonic fusion basis states for HΣ2

3.2 Action of Dehn-twists and braid-moves

  • Fusion-basis states are simultaneous eigenstates of Dehn-twists, while a braid move swaps diagram portions and adds a phase corresponding to the doubled model’s R-matrix.For the stated labelings, the Dehn-twist eigenvalue is e−4πi/5 for (1, 1), τ1 and e+4πi/5 for 1τ.Source: §3.2 · pp. 11–12

3.3 Interpretation as a doubled Fibonacci anyon model

  • The doubled Fibonacci category consists of two Fibonacci copies with opposite chiralities; the two ribbon graphs transform independently, and swapping them negates twist or braid phases.Source: §3.3 · p. 13

3.4 Changing the pants decomposition: The F-matrix

  • Changing pants decompositions requires a move that reconnects a subtree; its basis-change unitary is the F-matrix, a tensor product of the Fibonacci and conjugate-category F-matrices.The Fibonacci F-matrix is real-valued, so it equals the F-matrix for Fib∗.Source: §3.4 · p. 14

3.5 Recursive construction of the anyonic fusion basis states by gluing

  • Gluing surfaces combines their fusion-basis states into a superposition indexed by the label on the connecting boundary.The coefficient depends on the doubled anyon labels assigned to the connecting edge.Source: §3.5 · p. 15

4 Realizing the Fibonacci ribbon-graph Hilbert space HΣ

The Fibonacci code realizes the ribbon-graph Hilbert space as a code space on qubits placed along a surface lattice. Its commuting local projections enforce the ribbon-graph constraints, and local triangulation changes relate the resulting ground spaces.

4.1 The Fibonacci code

  • Vertex projections enforce legal ribbon-graph vertices, while plaquette projections add vacuum loops; the resulting operators commute and their joint +1 eigenspace is isomorphic to HℓΣ.Adding a vacuum loop removes the puncture from its plaquette, allowing ribbons to pass across it.Source: §4.1 · pp. 16–17

4 Realizing the Fibonacci ribbon-graph Hilbert space HΣ

  • The Fibonacci code is the joint +1 eigenspace of commuting projections and encodes the ribbon-graph Hilbert space HℓΣ.The construction places qubits on triangulation edges and incorporates fixed boundary conditions.Source: §4 · p. 15, §4.1 · p. 17

4.2 Anyonic fusion basis states and gluing

  • For a gluing with one connecting edge, its qubit directly reveals some information about the doubled anyon labels on that edge.Source: §4.2 · p. 18

4.3 Relating ground spaces for different triangulations

  • Flipping an interior triangulation edge changes up to five dual-lattice edges, and the associated F-move maps one ground space to the other.The move is defined as a unitary between the qubit Hilbert spaces for the two triangulations.Source: §4.3 · p. 19

5 Computation with doubled Fibonacci anyons in HΣ

The section describes how to prepare and measure doubled Fibonacci anyon states and implement diffeomorphisms on the code through triangulation changes. These procedures use F-moves, with braids and Dehn-twists realized as local unitaries.

5 Computation with doubled Fibonacci anyons in HΣ

  • The computation scheme prepares paired τ anyons, executes braids, and measures an anyon label in a fixed fusion-tree basis.The section uses doubled versions 11 and τ1 for computation, leaving 1τ and ττ unused.Source: §5 · p. 20

5.1 State preparation and measurement

  • The cut(γ) procedure isolates a region by sequential F-moves along a closed dual-graph path, yielding a tadpole-like structure that can be reversed by glue(γ).The selected curve should enclose no holes, ensuring the associated fusion diagrams have trivial total charge.Source: §5.1 · pp. 21–22, Figure 4 · p. 21
  • The cut(γ) F-moves do not commute and can produce a degenerate triangulation, but extra moves can keep the bounding plaquette degree small.Source: §5.1 · p. 21
  • With periodic error correction, preparation and measurement expose encoded information; below threshold, majority voting reduces measurement error, while distillation addresses preparation errors.The cited preparation-error method is composite Fibonacci anyon distillation [Kön09].Source: §5.1 · p. 23

5.2 Implementing diffeomorphisms on Σ by changes of triangulation

  • For a Dehn-twist, each round applies F-moves along edges linking boundary vertices to vertices off the curve, and the resulting operator implements the ribbon-graph twist.The construction repeats for 2(n −1) rounds when the curve consists of n edges.Source: §5.2 · p. 23
  • Arbitrary braids and Dehn-twists on ribbon-graph states can be implemented by local unitary F-moves on the lattice qudits.The Dehn-twist procedure applies F-moves along edges bordering a curve over successive rounds; the moves within each round commute.Source: §5.2 · pp. 23–24

6 Quantum computation with Turaev-Viro codes from general anyon models

The section generalizes Turaev-Viro codes to spherical categories and identifies the doubled anyon theory associated with modular categories. It constructs ribbon-graph representations of the doubled fusion basis and verifies that their braiding and fusion data match the doubled theory.

6.2 Quantum codes from categories

  • For any finite unitary spherical category and triangulated surface, the construction gives an edge-qudit quantum error-correcting code; triangles also need qudits when fusion multiplicities are present.The edge-qudit dimension equals the number of simple objects in the category.Source: §6.2 · p. 26
  • For modular C, the code anyons belong to C ⊗ C*, and the Turaev-Viro theory agrees with the Reshetikhin-Turaev theory of the quantum double.The paper notes a broader theorem for finite semisimple spherical categories.Source: §6.2 · p. 26

6.3 Parameters of an anyon model

  • A modular category's double is C+ ⊗ C−, where C− has opposite chirality and conjugated topological phases and braiding data.The paper assumes modularity for this product form; for a general category, the double need not be isomorphic to C+ ⊗ C−.Source: §6.3 · p. 28

6.4 The ribbon graph Hilbert space HΣ

  • The ribbon-graph Hilbert space consists of formal combinations of fusion-consistent colored graphs modulo local relations, and it can be represented using dual graphs of triangulations.Different triangulations with the same number of edges yield equivalent inner products because edge flips correspond to unitary F-moves.Source: §6.4 · pp. 28–29

6.5 Ribbon graphs on Σ × [−1, 1] and fusion basis states of the doubled model

  • Doubled fusion-basis states are represented by placing the two category components on opposite boundaries of Σ × [−1, 1] and adding vacuum lines around punctures.The thickened-surface ribbon graphs are then reduced to ribbon graphs on Σ by connecting boundary ends and resolving crossings.Source: §6.5 · pp. 29–30
  • The doubled ribbon-graph representation reproduces the doubled theory's R- and F-matrices, with the two ribbon graphs transforming independently.The single-anyon construction is also checked against the expected idempotency and topological phase.Source: §6.5 · pp. 29–30

Appendix A.2 proves that the anyonic fusion basis states form a complete, orthonormal basis.

This appendix section extends the Levin-Wen code-space construction to general anyon models without fusion multiplicities. It identifies the code space as the simultaneous +1 eigenspace of local vertex and plaquette projections, isomorphic to the ribbon-graph space.

6.6 The Turaev-Viro code as the ground space of the Levin-Wen Hamiltonian

  • The general construction uses one qudit per dual-graph edge, with basis labels given by the anyon types.The section restricts this construction to anyon models without fusion multiplicities.Source: §6.6 · p. 31
  • Vertex projections act on incident edges and enforce the fusion rules at each vertex.Source: §6.6 · p. 31
  • The vertex and plaquette operators are commuting projections, and their simultaneous +1 eigenspace is the code space.Source: §6.6 · p. 31
  • Plaquette projections commute with one another and with the vertex projections.The plaquette operator adds a vacuum loop divided by the total quantum dimension D.Source: §6.6 · p. 31
  • The code space is isomorphic to the ribbon-graph space HℓΣ.Source: §6.6 · p. 32
  • Computations in the code space use F-moves implemented as five-qudit gates; generalized gluing supports state preparation and topological-charge measurement.Source: §6.6 · p. 32

7 The Turaev-Viro invariant and the Turaev-Viro code

The section defines the Turaev–Viro invariant and uses it to construct a code on a triangulated surface. It relates that code to the ribbon-graph Hilbert space and to Levin–Wen plaquette operators.

7 The Turaev-Viro invariant and the Turaev-Viro code

  • The section motivates the code construction as an extension of the Turaev–Viro invariant to a TQFT, and outlines its derivation from the invariant of a thickened surface.It also notes that an alternative invariant definition is useful background for a proof that approximating the invariant is BQP-complete.Source: §7 · p. 32

7.1 Definition of the Turaev-Viro invariant

  • The invariant is defined for triangulated 3-manifolds using tetrahedron tensors, with boundary values depending on the edge labeling.The section assumes tetrahedral symmetry of the category’s F tensor for simplicity; the invariant is independent of interior triangulation.Source: §7.1 · pp. 32–33

7.2 The Turaev-Viro code

  • The Turaev–Viro code is the range of a projection and coincides with the Hilbert space of colored ribbon graphs embedded in the surface.The projection is obtained from the invariant of the thickened surface.Source: §7.2 · p. 33

7.3 Local stabilizers for the Turaev-Viro code: The Levin-Wen Hamiltonian

  • The local B_p projections commute, and their simultaneous +1 eigenspace is the Turaev–Viro code.Each B_p is identified with a Levin–Wen plaquette operator by evaluating the invariant for a blister attached at a vertex.Source: §7.3 · pp. 34–35

A Ribbon graphs for general categories

A.1 Ribbon graph identities

The appendix states identities for ribbon graphs and vacuum lines, then derives them using local diagrammatic manipulations.

A.1 Ribbon graph identities

  • The appendix summarizes properties of vacuum lines and gives a consequence of the identity in Lemma A.1.Source: §A.1 · pp. 35–36
  • The stated ribbon-graph identities hold regardless of the contents of the shaded region.Source: §A.1 · p. 36
  • The proofs use F-moves and symmetry to establish the displayed identities.Source: §A.1 · p. 36

A.2 The anyonic fusion basis states form an orthonormal basis

This appendix establishes that the anyonic fusion basis is complete and orthonormal. It introduces a trace inner product that can be evaluated directly on ribbon graphs and agrees with the original inner product.

A.2 The anyonic fusion basis states form an orthonormal basis

  • For a fixed pants decomposition, the anyonic fusion basis states form a complete, orthonormal basis of HΣ.Source: §A.2 · p. 37
  • The trace inner product is computed by stacking ribbon diagrams, connecting their boundary ribbons, and extracting the vacuum coefficient with a normalization factor.The paper states that this inner product agrees with the original definition and avoids first reducing diagrams to two dimensions.Source: §A.2 · p. 37
  • The trace inner product agrees with the original inner product on computational basis states, yielding Kronecker deltas for matching labels.Source: §A.2 · p. 38
  • Fusion basis states with satisfied fusion constraints are mutually orthogonal and unit-normalized for n-punctured spheres.Source: §A.2 · p. 39
  • The appendix proves the orthogonality result by reducing inner products with F-moves and separating the calculation into component inner products.Source: §A.2 · p. 39

Completeness of anyonic fusion basis for HΣn

  • Completeness is established by showing that computational-basis and fusion-basis state counts satisfy the same recursion.The argument compares the counts for a fixed root label and uses associativity of fusion.Source: Completeness of anyonic fusion basis… · pp. 40–41

A.3 Application to the Fibonacci model

The appendix illustrates the fusion-basis construction in the Fibonacci model. Its examples include annular idempotents and fusion states on the three-punctured sphere.

A.3 Application to the Fibonacci model

  • The appendix verifies that the annular expressions agree with the idempotents defined earlier in the paper.The calculation uses the Fibonacci F and R tensors to resolve crossings.Source: §A.3 · pp. 41–42
  • In the Fibonacci model, annulus fusion basis states with identical boundary labels are equivalent to idempotents up to normalization.Source: §A.3 · p. 41
  • For the three-punctured sphere, the appendix gives fusion-basis examples and checks that one reduced state has unit norm.The norm calculation is stated as an instance of Lemma A.3.Source: §A.3 · p. 42

A.4 Generalization to higher-genus surfaces

For higher-genus surfaces, the paper extends the anyonic fusion basis by attaching a handle diagram to each handle of a punctured sphere. The two handle choices give bases diagonal with respect to different pairs of Dehn twists.

A.4 Generalization to higher-genus surfaces

  • A higher-genus surface is decomposed into a punctured sphere with handles attached to some holes, with a handle diagram added for each handle.The basis combines these diagrams with a doubled anyon fusion tree in the punctured sphere.Source: §A.4 · pp. 42–43, Figure 12 · p. 43
  • Each handle admits two basis choices, diagonalizing Dehn twists along either {α, β} or {α, γ}.The choices are determined by fusion-consistent triples from C+ and C−.Source: §A.4 · p. 43
  • The two choices of handle basis provide distinct descriptions of the Dehn-twist action on the higher-genus surface.The section says a similar change-of-basis matrix specifies the torus case.Source: §A.4 · p. 43

B Discretizing and Gluing: Proofs

The proofs connect the lattice Levin–Wen code space with ribbon-graph spaces and show how gluing and puncturing act on the corresponding states. Vacuum-loop insertions make these operations independent of local deformations and preserve norms under the stated map.

B Discretizing and Gluing: Proofs

  • The simultaneous +1 eigenspace of the Levin–Wen operators is isomorphic to the corresponding ribbon-graph space.The source identifies the eigenspace as the code subspace represented by ribbon graphs.Source: B Discretizing and Gluing: Proofs · p. 43
  • Adding a vacuum loop around a puncture makes the projected state independent of the initial local deformation.The map that adds the loop, divided by D, is a norm-preserving isomorphism.Source: B Discretizing and Gluing: Proofs · p. 44
  • Applying the plaquette operators inserts vacuum loops around punctures, yielding the stated puncture-related code-state construction.The proof obtains the result by applying the puncture lemma inductively.Source: B Discretizing and Gluing: Proofs · p. 44
  • Gluing along a curve is described by the corresponding anyonic fusion-basis state, with vacuum loops introduced along the gluing curve.The proof uses doubled vacuum loops and an F-move to obtain the glued state.Source: B Discretizing and Gluing: Proofs · pp. 44–45

C The Witten-Reshetikhin-Turaev invariant

The section relates the Turaev–Viro invariant to the doubled-category mapping-class representation by expressing a Heegaard splitting through handlebody states and a mapping cylinder. It identifies the handlebody state with a normalized anyonic fusion-basis state, establishing the link used in the invariant relation.

C The Witten-Reshetikhin-Turaev invariant

  • The Turaev–Viro and Witten–Reshetikhin–Turaev invariants are related through the categorical double.For modular C, the double takes the form C ⊗ C*; the cited result also holds more generally for spherical categories.Source: C The Witten-Reshetikhin-Turaev… · p. 46
  • A Heegaard splitting expresses the manifold as two handlebodies joined by a mapping cylinder, whose map acts on their boundary Hilbert spaces.The map is the representation ρC⊗C*(x), and it can be interpreted through local F-moves.Source: C The Witten-Reshetikhin-Turaev… · p. 47
  • Up to a phase, the genus-g handlebody state is D^(g−1) times the normalized doubled-category anyonic fusion-basis state.Its ribbon graph contains a vacuum loop around each handle.Source: C The Witten-Reshetikhin-Turaev… · p. 47
  • The handlebody-state identification follows by induction on genus, using gluing of boundary disks to add a handle.The resulting handlebody state has a vacuum loop around each handle.Source: C The Witten-Reshetikhin-Turaev… · p. 47, Figure 15 · p. 48
  • The handlebody state is proportional to the fusion-basis state after vacuum lines are added along the constrictions of a pants decomposition.The normalization fixes the proportionality constant.Source: C The Witten-Reshetikhin-Turaev… · pp. 47, 50
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