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Sipser-Spielman meets Dijkgraaf-Witten: non-Abelian qLDPC codes via twisted sheaf gauge theory and almost-constant-overhead magic state fountain

Guanyu Zhu, Shi Jie Samuel Tan, Ryohei Kobayashi, Po-Shen Hsin

arXiv:2609.31541v1quant-phcond-mat.str-elcs.IThep-thmath-phmath.ITmath.MP

TL;DR

The paper asks whether qLDPC codes can combine non-Abelian topological order with the low space overhead and large distance of expander-based constructions. It develops sheaf-based gauging constructions and demonstrates magic-state fountains, including an almost-constant magic rate.

  • Problem

    The work asks whether qLDPC codes can pair non-Abelian topological order with the low space overhead and large distance of expander-based constructions.

    Source: §I · p. 3

  • Method

    The paper uses sheaf cohomology and 0-form subcomplex symmetry gauging to construct twisted qLDPC codes and measure addressable logical CZ gates.

    Source: §I · pp. 4–5, A. Summary of results · p. 6

  • Results

    s ≥ n^(1−ϵ) logical CZ magic states are prepared in one run in the almost-constant-rate family; the constant-rate construction prepares Θ(√n).

    Source: A. Summary of results · pp. 6–7

  • Takeaways & Limitations

    For p = 2, the field-extension construction remains qubit-based, with CZ dressings realizable as depth-one qubit circuits.

    Source: B. The Fq path integral · p. 38

    As ϵ approaches zero, the alphabet size grows; constant magic rate requires a different mechanism for satisfying the local cycle condition.

    Source: Code parameters · p. 60

Abstract

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Sipser-Spielman codes and Dijkgraaf-Witten twisted gauge theories are among the most profound ideas in the last few decades in the areas of computer science and mathematical physics respectively. This work unifies them in the same framework using the language of sheaf cohomology, and produces a new framework of twisted sheaf guage theories describing a class of quantum low-density parity-check (qLDPC) codes with non-Abelian $D_4$ topological order. The corresponding twisted qLDPC code can be obtained from gauging a new type of 0-form sub-complex $\mathbb{Z}_2^3$ symmetry from a qLDPC code defined on a sheaf complex, and can be interpreted as a topological defect network of non-Abelain $D_4$ patches glued together with proper gapped interfaces. As an application, one can use this to realize a \textit{magic state fountain} via the gauging measurement of the addressable logical CZ gates as 0-form subcomplex symmetries in a 2D hypergraph-product on a sheaf complex. This includes a scheme of subdividing an arbitrary constant-rate 2D hypergraph-product code with parameter $[[n,\Theta(n), \Omega(n^{1/2})]]$ into a quantum sheaf code which allows preparation of $\Theta(n^{1/2})$ in parallel, equivalent to the recent geometric construction using the code-to-manifold mapping in (arXiv:2601.06736). Moreover, using the recent sheaf complex and algebraic code constructions by Golowich-Tamo-Zhu (arXiv:2609.27801) with parameter $[[n,\Theta(n^{1-\epsilon}), \Omega(n^{(1-\epsilon)/2})]]$ for arbitrary small $\epsilon$, one can prepare $\Theta(n^{1-\epsilon})$ CZ magic states in parallel and hence achieve an almost-constant magic rate.

I. INTRODUCTION

The introduction relates Sipser–Spielman graph codes and their vertex-local checks to generalized sheaf codes and first sheaf homology, developing a sheaf-cohomological framework for twisted qLDPC codes with non-Abelian D4 topological order. It then applies the framework to addressable logical CZ measurements for a magic-state fountain, including an almost-constant magic rate.

A. Summary of results

  • Gauging a 0-form subcomplex symmetry produces a Clifford-stabilizer qLDPC code whose dressed X-generators commute on the zero-flux subspace and carry non-Abelian D4 topological order.The twist is derived either by gauging the sheaf cluster-state SPT phase or directly from the untwisted code; gapped interfaces form a topological defect network.Source: §I · pp. 3–4, A. Summary of results · p. 6

I. INTRODUCTION

  • High-rate qLDPC codes need addressable, parallel non-Clifford operations to avoid magic-state-distillation overhead; this work dispenses with the third dimension by using non-Abelian topological order.The introduction motivates pairing that order with the low space overhead and large distance of expander-based constructions.Source: §I · p. 3

II. GENERALIZED SHEAF-THEORETICAL APPROACH FOR CLASSICAL CODES

  • Sipser–Spielman Tanner codes are identified with sheaf homology, and subdivision lets arbitrary classical codes enter a sheaf framework where cup and cap products are defined.The graph-based approach expresses local constraints through stalk spaces and incidence maps; products of graphs yield square complexes.Source: §I · p. 4, A. Summary of results · p. 5, §II · p. 7

A. Summary of results

  • For every sheaf 0-cocycle, a depth-one transversal CZ circuit acts as addressable logical CZ gates in the untwisted code and as a charge-parity operator in the twisted code.The number of independently addressable gates is dim H0(L, Fg), which grows with system size; no transversal CCZ is used.Source: A. Summary of results · p. 6
  • [[n, Θ(n), Ω(√n)]] codes prepare Θ(√n) logical CZ magic states on disjoint pairs in one run, protected by distance Ω(√n).The protocol measures commuting weight-O(1) dressed X-stabilizers for O(d) rounds, then ungauges.Source: A. Summary of results · p. 6
  • Over any finite field Fq, the twisted path integral remains a cohomology invariant, and the gauge group is (Zp)^u per colour rather than Zq.The Frobenius tensor Tijk = Tr(θiθjθk) selects which component CZ gates appear.Source: A. Summary of results · p. 7
  • For every ϵ > 0, a family over Fq with q = p^Oϵ(1) has k ≥ n^(1−ϵ), d ≥ n^((1−ϵ)/2), and prepares s ≥ n^(1−ϵ) disjoint CZ magic states, giving magic rate ϱ ≥ n^−ϵ.The disjoint gates arise from a perfect matching, and the construction descends to qubits with the rate changed by only a constant factor.Source: A. Summary of results · p. 7

A. Sipser–Spielman code

  • Bits on graph edges satisfy a local code at each vertex, defining the Sipser–Spielman Tanner code.For constant degree, the code is LDPC; a rank bound gives positive rate when k0 > ∆/2.Source: A. Sipser–Spielman code · p. 8
  • Sipser–Spielman codes are computed by sheaf (co)homology.The vertex-local codes form the local system, or sheaf.Source: A. Sipser–Spielman code · p. 8

1. Relation to sheaf homology

  • Allowing different local codes at each vertex, the sheaf code is H1(G, F): its global 1-cycle condition holds exactly when each vertex's local codeword condition holds.The local codes need not agree across vertices.Source: §1 · pp. 8–9

2. Relation to sheaf cohomology

The paper relates classical codes to sheaf cohomology and subdivides arbitrary classical codes onto graphs with local coefficients. The construction preserves the code dimensions through explicit cycle-to-code isomorphisms and is chain-homotopy equivalent to the original complex.

2. Relation to sheaf cohomology

  • The transposed code is the zeroth sheaf cohomology H0(G, F), with edge conditions requiring adjacent vertex co-restrictions to agree.The paper identifies this as the 0-cocycle code.Source: §2 · p. 9

B. Subdividing Classical Codes into Sheaf Codes

  • Subdivision replaces each classical-code bit’s hyperedge with incident graph edges whose common value recovers that bit, while original check vertices enforce the parity checks.The resulting first homology is isomorphic to the original code; the transposed code is handled on the same graph with dual local codes.Source: B. Subdividing Classical Codes into… · pp. 11–12
  • The subdivision complex is explicitly chain-homotopy equivalent to the original complex.The inclusion, projection, and homotopy maps provide the equivalence.Source: B. Subdividing Classical Codes into… · pp. 12–13

2. Relation to sheaf cohomology

  • H0(G, F⊥) is isomorphic to H1(G, F), and H0(G, F) is isomorphic to H1(G, F⊥).These identifications connect the two dual sheaves’ cocycle spaces to the other sheaf’s cycle space.Source: B. Subdividing Classical Codes into… · p. 13

III. SHEAVES AND CUP PRODUCTS

This section defines cochain complexes with local coefficients on graded incidence posets and develops cup and cap products. The products obey cochain identities and induce maps on cohomology.

A. Graded incidence posets and cochain complexes

  • An even-incidence condition on a graded poset ensures that successive coboundary maps compose to zero; the construction assumes a finite field of characteristic 2.The cochain groups are indexed by cells of each grade.Source: A. Graded incidence posets and cochain… · p. 13

B. Presheaves and sheaf codes

  • A presheaf assigns stalks and compatible maps to cells, producing a block-matrix coboundary whose successive maps compose to zero.The quantum sheaf code is defined as a CSS code from three consecutive terms of this cochain complex.Source: B. Presheaves and sheaf codes · p. 14

C. The cup product

  • The cup product combines cochains into a cochain of summed degree and induces a well-defined product on cohomology.Its definition uses front and back faces of an ordered simplex.Source: C. The cup product · p. 14

D. The cap product

  • The cap product pairs cochains to produce a chain of the remaining degree and likewise descends to a well-defined map on (co)homology.The cap product is compatible with the cup product when degrees match.Source: D. The cap product · p. 15

E. The pairing isomorphism

The paper uses sheaf cohomology and cochain maps to define logical operations and construct untwisted quantum sheaf codes on product complexes. Their stabilizers arise from coboundary maps, with bounded weights under bounded local degree and stalk dimension.

E. The pairing isomorphism

  • The induced pairing on H_i(X, F) is non-degenerate, and an invariant form descends to cohomology to define a logical multi-controlled-Z gate when coboundary operators are sparse.The untwisted code is set up as an explicit Pauli stabilizer model before introducing the twist.Source: E. The pairing isomorphism · p. 15, §IV · p. 15

IV. UNTWISTED QLDPC CODES ON SHEAF COMPLEX

  • Extracting three consecutive cochain-complex terms gives a CSS code with qubits on 1-cells, X-checks on 0-cells, and Z-checks on 2-cells.The corresponding check matrices are HT X = d0 and HZ = d1.Source: §IV · p. 15

A. Hypergraph Product of Sheaf Codes

  • For bounded-row- and bounded-column-weight inputs, the sheaf hypergraph product preserves encoded dimension, changes X- and Z-distances by at most constant factors, and changes block length by a constant factor.These parameter conclusions follow from explicit chain maps.Source: A. Hypergraph Product of Sheaf Codes · p. 16

B. Operator-valued sheaf cochains

  • Operator-valued sheaf cochains assign commuting binary-valued operators to cell stalks, with the edge Z2 gauge field represented by an edge 1-cochain.Physical qubits are labelled by an edge and a component of its stalk.Source: B. Operator-valued sheaf cochains · p. 16

C. X-stabilizers from d0

  • Each X-stabilizer is indexed by a vertex and stalk component, implements a coboundary gauge shift, and has bounded weight when vertex degree and stalk dimension are bounded.Its parity-check row is taken from HX = (d0)T.Source: C. X-stabilizers from d0 · p. 18

D. Z-stabilizers from d1

  • Each face has one Z-stabilizer because its top-cell stalk is scalar; the stabilizer measures flux and has bounded weight under bounded local stalk dimensions.The Z-stabilizer matrix is HZ = d1.Source: D. Z-stabilizers from d1 · p. 18

E. The untwisted stabilizer Hamiltonian

  • The X- and Z-stabilizers commute because d1◦d0 = 0, while flat gauge fields modulo coboundaries leave logical qubits counted by sheaf cohomology.The twisted code keeps Z-stabilizers and dresses each X-stabilizer with CZ gates on two further code copies.Source: E. The untwisted stabilizer Hamiltonian · p. 19

F. Sheaf gauge theory as a topological defect network

The paper interprets sheaf gauge theories as networks of TQFT patches joined by gapped interfaces, and builds their twisted response from sheaf cohomology on a selected cycle. The resulting sheaf cluster-state construction supports a 0-form subcomplex symmetry and a gauge-invariant triple-cup-product twist.

F. Sheaf gauge theory as a topological defect network

  • A shared edge between TQFT patches acts as a gapped interface: it condenses electric-anyon pairs and the magnetic-anyon triple, with dual interfaces exchanging e and m behavior.For the constant sheaf, the interface data is generated by e1e2, e2e3, and m1m2m3; the non-constant-sheaf dual has L′ = ⟨m1m2, m2m3, e1e2e3⟩.Source: F. Sheaf gauge theory as a topological… · pp. 19–20

A. Spacetime path integral as a cohomology invariant

  • The path integral is gauge invariant, and its triple-intersection term supplies the mechanism for non-Abelian braiding and the gauging protocol’s non-trivial logical action.The twist is integrated over a spacetime cycle; the sheaf gauge fields are 1-cochains, and the first three action terms generalize BF theory.Source: A. Spacetime path integral as a… · pp. 20–22

B. Sheaf cluster state and sub-complex Z3 2 SPT

  • The sheaf cluster state is supported on the codimension-0 subcomplex selected by a tensor-sheaf 2-cycle, rather than necessarily on the whole spatial complex.The cycle condition ∂η2 = 0 ensures gauge invariance; the supported region can be a proper subcomplex.Source: B. Sheaf cluster state and sub-complex… · p. 22

1. Operator-valued sheaf cochains

  • Operator-valued sheaf cochains assign commuting, binary-valued matter fields to cells, with stalk components corresponding to physical qubits.The elementary sheaf cochains correspond one-to-one with physical qubits, extending scalar-coefficient operator-valued cochains to local coefficient systems.Source: §1 · p. 23

2. Topological response action

  • The topological response is a triple cup product of three background gauge fields integrated over the spacetime 3-cycle, generalizing the type-III cocycle SPT.The cell sum is restricted to the subcomplex where η2 is nonzero.Source: §2 · pp. 23–24

3. The CCZ Entangler

The construction uses a sheaf-weighted CCZ entangler to form a cluster-state SPT whose symmetry generators are labelled by sheaf 0-cocycles. Gauging that subcomplex symmetry produces a qLDPC code whose twist is localized by the chosen 2-cycle.

3. The CCZ Entangler

  • The CCZ entangler applies gates only to mutually incident qubit triples whose common 2-cell lies in Lη2, yielding a constant-depth, geometrically local circuit.The resulting state is a cluster state on Lη2 and a product of |+⟩ states elsewhere.Source: §3 · p. 24

4. The Sheaf cluster-state and SPT Hamiltonian

  • The sheaf cluster state is the unique ground state of a Clifford-stabilizer Hamiltonian whose single-site X terms are dressed by products of CZ gates.The entangler conjugates the trivial paramagnet to this Hamiltonian.Source: §4 · pp. 24–25

5. Contrast with the ordinary cluster state

  • Unlike the ordinary cluster state, the sheaf version uses local-code-weighted incidences and can occupy a proper codimension-0 subcomplex, leaving the rest in a trivial product state.For a hypergraph product, η2 may be chosen as z ⊗ z′, selecting a sparse sub-grid.Source: §5 · p. 25

6. The 0-form subcomplex symmetry of SPT

  • The SPT’s 0-form symmetry generators correspond to sheaf 0-cocycles and act on the selected subcomplex rather than necessarily on all of space.The cycle condition on η2 makes that subcomplex a valid symmetry support and ensures response-action gauge invariance.Source: §6 · pp. 25–26

C. Constructing twisted qLDPC codes by gauging the 0-form subcomplex symmetry

  • Gauging the subcomplex symmetry produces an untwisted sheaf code outside Lη2 and CZ-dressed, non-Abelian stabilizers on it; different homology classes [η2] give inequivalent twisted codes.The Gauss law is imposed throughout the complex, while the twist is localized to the selected subcomplex.Source: C. Constructing twisted qLDPC codes by… · p. 26

1. Gauge fields on edges

The construction gauges three sheaf-valued fields on edges, then minimally couples them to the sheaf SPT to obtain a twisted Clifford stabilizer code. Its CZ dressing is localized by the triple cup product and twist cycle, preserving qLDPC structure while coupling the three copies.

1. Gauge fields on edges

  • Gauge fields are operator-valued sheaf 1-cochains on edge stalks, with local gauge transformations shifting them by sheaf coboundaries.Each edge may carry multiple gauge qubits, indexed by stalk components.Source: §1 · p. 26

2. Gauss’s law

  • Elementary matter-site cochains generate Gauss’s law operators whose X parts are rows of the sheaf coboundary matrix, weighted by co-restriction maps.Imposing these operators as identity recovers cocycle-labeled 0-form subcomplex symmetries as transformations acting trivially on gauge fields.Source: §2 · pp. 26–27

3. Minimal coupling and the gauged Hamiltonian

  • Minimal coupling replaces matter fields with gauge fields, and the flux terms enforce flatness on 2-cells while remaining untouched by the twist.On the physical Hilbert space, the gauged SPT Hamiltonian is expressed through Gauss’s law and flux terms.Source: §3 · pp. 27–28

4. The twisted sheaf code

  • The twisted code leaves Z-type generators unchanged and dresses X-type generators with CZ gates on the two other colors, only within Lη2.The triple cup product and 2-cycle η2 determine the pairs; their bounded-size support preserves qLDPC structure while coupling the three copies into non-Abelian topological order.Source: §4 · pp. 28–29

D. Clifford stabilizer codes on sheaf complexes

  • On a sheaf complex, logical Z operators are labeled by sheaf 1-chains and logical X operators by sheaf 1-cochains, without invoking a dual complex or Poincaré duality.Cell labels carry local coefficient indices, and incidences are weighted by co-restriction maps.Source: D. Clifford stabilizer codes on sheaf… · p. 29

1. Construction of the codes

  • The twisted stabilizer group is a Clifford stabilizer group; setting all CZ dressings to identity recovers three decoupled untwisted sheaf codes.The twist changes only the X stabilizers; the Z stabilizers are identical in the twisted and untwisted groups.Source: §1 · p. 29

2. Constraints on the code space

  • Each sheaf 0-cocycle labels a code-space constraint under which the corresponding transversal CZ operator acts trivially.The constraints follow from products of twisted stabilizers and apply to the three color copies.Source: §2 · pp. 29–30

3. Stabilizer commutation relations

The twisted stabilizers fail to commute on the full Hilbert space but commute on the zero-flux subspace. The same sheaf framework describes their logical operators and cocycle-labeled transversal CZ symmetries.

3. Stabilizer commutation relations

  • Twisted X-stabilizers of different colors have commutators equal to products of third-color Z-stabilizers, so all generators commute on the zero-flux subspace.Other generator pairs commute exactly; the cycle condition ∂η2 = 0 ensures both gauge invariance of the response and consistency of the twisted stabilizer group.Source: §3 · p. 30

4. Logical operators

  • Logical Z operators are labeled by sheaf 1-homology classes, while nontrivial sheaf 1-cohomology classes label genuine dressed-X logical operators.The twisted magnetic dressing is well-defined, and the resulting logical operators underlie the code’s non-Abelian fusion rules.Source: §4 · p. 31

VI. 0-FORM SUBCOMPLEX SYMMETRY IN UNTWISTED AND TWISTED QLDPC CODES

  • Each sheaf 0-cocycle labels a transversal CZ symmetry, which is a logical gate in the untwisted code and a charge-parity stabilizer product in the twisted code.These are 0-form symmetries supported on a proper subcomplex rather than the entire complex.Source: §VI · p. 31

A. Transversal CZ operators and their support

  • Capping the twist cycle with a symmetry cocycle gives a closed, codimension-0 subcomplex supporting a depth-one circuit of two-qubit CZ gates.Distinct cocycle classes yield distinct subcomplexes, providing addressability.Source: A. Transversal CZ operators and their… · p. 32

B. Transversal logical CZ in the untwisted code

In the untwisted code, capped-subcomplex symmetries implement addressable logical CZ gates; in the twisted code, the same operators become charge-parity constraints. The paper connects this change to the sheaf code’s extensive symmetry space and describes an alternative gauging route from untwisted qLDPC codes.

B. Transversal logical CZ in the untwisted code

  • A depth-one physical CZ circuit implements logical CZ gates whose pattern is determined by the triple cup product.The logical action is well defined on cohomology classes.Source: B. Transversal logical CZ in the… · p. 33
  • The independently addressable logical CZ gates grow linearly with system size because their labels lie in the 0-cocycle sheaf code.Each label selects its own capped codimension-0 subcomplex.Source: B. Transversal logical CZ in the… · p. 33

C. Charge parity operators in the twisted code

  • In the twisted code, the same transversal CZ operator is a product of twisted stabilizers and acts as the identity.Gauging absorbs the logical gate into the stabilizer group.Source: C. Charge parity operators in the… · p. 34
  • The twisted stabilizers are non-Pauli and fail to commute off the zero-flux subspace, while commuting on that subspace.The stabilizer commutator of different colours is a product of third-colour Z-stabilizers.Source: C. Charge parity operators in the… · p. 34

D. Charge interpretation and contrast with manifolds

  • Charge-parity operators measure weighted g-charge parity and are unaffected by Pauli-Z errors, whose violations indicate a different super-selection sector.The measured parity is the total charge contained in the labelled subcomplex.Source: D. Charge interpretation and contrast… · p. 35
  • Unlike a closed manifold’s single global CZ operator, the sheaf complex has extensively many independent operators, making their logical CZ patterns separately addressable.The number of independent classes grows linearly with system size.Source: D. Charge interpretation and contrast… · p. 35

E. Alternative route: gauging the subcomplex symmetry from the untwisted qLDPC codes

  • The symmetry labels form a space of dimension kg that grows linearly with system size; gauging projects onto invariant sectors, removing one logical qubit per projection.The rank-kg projections determine the logical-dimension reduction.Source: E. Alternative route: gauging the… · p. 36
  • An alternative construction gauges the depth-one transversal CZ symmetry between two untwisted qLDPC codes, producing the twisted sheaf code by minimal coupling.This route starts from a standard CSS qLDPC code rather than a cluster state.Source: E. Alternative route: gauging the… · pp. 36–37

VII. GENERALIZATION TO Fq: QUDIT PATH INTEGRAL AND CLIFFORD STABILIZER CODES

The section lifts the spacetime path integral, untwisted qLDPC code, and twisted Clifford stabilizer code from F2 to Fq. The field structure supplies the twist, while the gauge group remains determined by the additive group of Fq.

A. Qudits, generalized Paulis, and the trace pairing

B. The Fq path integral

  • The Fq path integral is a cohomology invariant, and its exponent is the triple cup-product invariant valued in Fq.Summing over magnetic Lagrange multipliers imposes the cocycle equations.Source: B. The Fq path integral · p. 38
  • The twisted term combines type-III Dijkgraaf–Witten cocycles using Fq multiplication, but the gauge group is (Zp)^u, not cyclic Zq unless u = 1.The field’s multiplication supplies the twist; the gauge group comes from the additive group of Fq.Source: B. The Fq path integral · p. 38
  • The cochain and twisted-stabilizer constructions also work over commutative rings, but the algebraic-geometry coding theory used later requires a field.A cyclic Z_{p^u} gauge theory would use Z/p^uZ rather than Fq.Source: B. The Fq path integral · p. 39

C. The untwisted qudit sheaf code

  • The untwisted qudit sheaf code uses CSS check matrices over Fq, with X- and Z-type logical operators labeled by H1(L, F).The X- and Z-type logical spaces are paired non-degenerately.Source: C. The untwisted qudit sheaf code · p. 39

D. The twisted Clifford stabilizer code over Fq

The Fq twisted stabilizer generators retain their Clifford structure and commute on the zero-flux subspace. The section then extends subcomplex-symmetry gauging over Fq and derives when the hypergraph-product cohomology invariant can be non-zero.

D. The twisted Clifford stabilizer code over Fq

  • Same-colour twisted generators commute exactly and form a group isomorphic to C0(L, Fr), while the twisted stabilizer group is a subgroup of the Fq Clifford group rather than the Pauli group.Other pairs commute exactly, and the stated cross-colour commutation condition holds on the zero-flux subspace.Source: D. The twisted Clifford stabilizer… · p. 39
  • Transversal Fq CZ operators act as the identity on the twisted code space under the code-space constraints.The argument uses the non-degeneracy of the trace pairing to promote a phase condition to an Fq-valued equation.Source: D. The twisted Clifford stabilizer… · p. 40

E. Subcomplex symmetry and gauging over Fq

  • Gauging an Fq-linear subcomplex symmetry yields Fq-valued labels and measurement outcomes; the protocol’s other stated properties remain unchanged.Inserting a g Wilson operator of charge w shifts the constraint to w.Source: E. Subcomplex symmetry and gauging… · p. 40

VIII. CONDITION FOR NON-TRIVIAL COHOMOLOGY INVARIANTS IN HYPERGRAPH-PRODUCT CONSTRUCTIONS

  • For the hypergraph-product setting, the temporal contribution to the invariant is automatically non-trivial, leaving the spatial factors to determine whether the full invariant is non-zero.The invariant factorizes into pairings on the spatial tensor factors, and the spacetime invariant equals the spatial one.Source: §VIII · p. 40, B. Factorization of the invariant by… · p. 42

B. Factorization of the invariant by the K¨unneth theorem

  • The invariant can be non-zero only if the tensor sheaf on each spatial factor has a non-zero 1-cycle; either factor lacking one makes the twisted theory trivial.No condition arises from the temporal factor.Source: B. Factorization of the invariant by… · p. 42

C. The local cycle condition

The section translates non-triviality of the twist into local cycle constraints on the tensor sheaf, then relates the fundamental-class solution to a multiplication property of local codes. It uses this criterion to describe sheaf-complex constructions of twisted 2D hypergraph-product codes.

C. The local cycle condition

  • At each vertex, a tensor-sheaf 1-cycle must satisfy the local cycle condition, and the resulting number of constraints can make non-zero solutions absent for generic local codes.The condition imposes a product of the three local check numbers of scalar equations on the same edge variables.Source: C. The local cycle condition · p. 43

D. The fundamental class and the multiplication property

  • Choosing the fundamental class as the cycle makes the local cycle condition equivalent to a multiplication property of the dual local codes.This converts a spacetime homological requirement into a check on local codes of length Δ = O(1).Source: D. The fundamental class and the… · pp. 43–44

E. Multiplication property of classical codes

  • Reed–Solomon and Reed–Muller code families satisfy multiplication properties usable for the fundamental-class cycle, but this imposes a rate penalty.The passage attributes the penalty to the product code having strictly larger dimension than its factors.Source: E. Multiplication property of… · p. 44
  • A multiplication property in the local codes is the stated design principle for obtaining a non-trivial twist and hence a genuinely non-Abelian qLDPC code from a hypergraph product.The local cycle condition is equivalent to the existence of the required tensor-sheaf cycles on the spatial factors.Source: E. Multiplication property of… · p. 44

A. Code construction

  • The construction subdivides arbitrary asymptotically good input codes into sheaf codes and defines twisted 2D HGP codes on the resulting sheaf complexes.The twisted codes are specified by the general Clifford stabilizer construction and have the corresponding path-integral description.Source: A. Code construction · p. 45
  • For standard Sipser-Spielman codes with uniform local codes, the construction can instead proceed without subdivision, using three sheaf-complex copies.The local codes need not follow the alternating SPC and repetition pattern used for subdivision.Source: A. Code construction · p. 46

B. Subcomplex symmtery and logical CZ gate structure via Poincar´e duality of sheaf

The section uses sheaf Poincaré duality to establish a non-degenerate pairing, derives the logical CZ structure, and shows that the resulting codes retain constant rate and square-root distance.

B. Subcomplex symmtery and logical CZ gate structure via Poincaré duality of sheaf

1. From Poincaré duality to the non-degenerate pairing

  • Composing Poincaré duality with the canonical homology–cohomology pairing gives a non-degenerate bilinear form, with dual bases satisfying P([a1_i], [b0_j]) = δij.Source: §1 · pp. 46–47

2. Logical CZ structure

  • Triple-cup-product evaluation with Poincaré duality determines the logical CZ gate structure, which in turn determines the magic-state-fountain rate.Source: §2 · p. 47

C. Encoding rate and distance

  • The untwisted and twisted Construction IX.1 codes have constant rate k = Θ(n) and subsystem-code distance d = Ω(√n), matching the corresponding skeleton codes [1].Source: C. Encoding rate and distance · p. 47
  • Direct subdivision lets the sheaf carry the local structure otherwise supplied by extra dimensions, so code parameters can be read from the underlying classical code without an intermediate manifold.Source: C. Encoding rate and distance · pp. 47–48

1. Inputs from the classical code

The construction subdivides an asymptotically good classical LDPC code into a sheaf code and takes homological products to obtain untwisted and twisted codes. Both retain constant encoding rate and subsystem-code distance Ω(√n), with the twisted-code distance conditional on an additional assumption.

1. Inputs from the classical code

  • Subdividing the classical code and taking homological products yields the sheaf-code construction; the transposed code is also assumed to be good.The constant-sheaf factor contributes a single homology class, making the b and g copies sub-extensive while the r copy carries the constant-rate logical information.Source: §1 · p. 48

2. Encoding rate

  • Both untwisted and twisted codes have constant encoding rate, k = Θ(n) and ˜k = Θ(n), respectively.The twist removes only a sub-extensive number of logical qubits.Source: §2 · pp. 48–49

3. Subsystem-code distance

  • The untwisted code has subsystem-code distance d = Ω(√n), and the twisted code has the same bound under Assumption IX.12.The subsystem-code restriction demotes short classes to gauge qubits; Assumption IX.12 bounds condensation descendants by the weights of the basic operators.Source: §3 · pp. 49–50, §4 · p. 50

4. Summary

  • The sheaf derivation reproduces the parameter scaling of Ref. [1] without an intermediate high-dimensional complex, though subdivision still creates short classes requiring a subsystem-code restriction.The sheaf quantities follow directly from Künneth products of single-graph quantities.Source: §4 · p. 50

X. LOGICAL NON-CLIFFORD OPERATION VIA GAUGING MEASUREMENT AND SPACETIME PATH INTEGRAL

The paper implements addressable logical CZ measurements by gauging between untwisted and twisted sheaf codes, then ungauging. In the Construction IX.1 instantiation, this protocol prepares Θ(√n) disjoint CZ magic states protected by subsystem-code distance Ω(√n), subject to the stated distance assumption.

1. The symmetry operator and its logical action

  • Measuring commuting local twisted X-stabilizers yields one outcome per g-type basis symmetry class, realizing addressable logical CZ measurements in parallel.In the twisted code, each transversal CZ coincides with a g-type charge-parity operator built from local stabilizer generators.Source: §1 · p. 51

2. The gauging measurement protocol

  • The protocol gauges at ti, performs O(d) rounds of error correction in the twisted code, then ungauges by measuring every g qubit in the Z basis.The slab boundaries are spacetime domain walls between the untwisted Z2 × Z2 theory and the twisted Z2 × Z2 × Z2 theory.Source: §1 · p. 51, Figure 14 · p. 52, §2 · p. 52

B. Instantiation: preparing Θ(√n) disjoint magic states via the magic state fountain

  • Θ(√n) disjoint logical CZ magic states are prepared in one run, protected by subsystem-code distance Ω(√n) under Assumption IX.12.The outcomes occur with constant probability, giving Θ(√n) states on average.Source: B. Instantiation: preparing Θ(√n)… · p. 53
  • Initializing one r logical qubit per cluster in |+⟩ and the remaining r qubits in |0⟩ makes the measurement project Θ(√n) disjoint r–b pairs onto CZ magic states.Distinct symmetry labels give distinct active pairs, so the projectors commute and factorize over pairs.Source: B. Instantiation: preparing Θ(√n)… · pp. 53–54

C. Spacetime path integral as logical action

The spacetime path integral expresses the gauging protocol’s logical action in terms of boundary states and measurement outcomes. The paper also uses local codes with a multiplication property to obtain an almost-constant magic rate.

1. General formalism

  • On a spacetime cylinder, boundary amplitudes define states and the bulk path integral defines an operator between them; relative g-charge worldlines make the protocol a measurement channel.The operator depends on the measurement-outcome vector associated with those relative classes.Source: §1 · p. 55

2. Imaginary-time versus real-time evolution

  • When error-correction decoding closes defects homologically trivially, topological invariance equates the defect-decorated path integral with the defect-free logical action.This connects the Euclidean projector description to the real-time error-corrected process.Source: §2 · p. 55

3. Derivation of the logical action

  • The path-integral derivation reproduces the protocol’s logical action, with each factor in the Construction IX.1 instantiation projecting onto a CZ on a disjoint pair.The spacetime and operator derivations are controlled by the same coefficient matrix.Source: §3 · pp. 55–56

XI. ALMOST-CONSTANT MAGIC RATE FROM LOCAL CODES WITH A MULTIPLICATION PROPERTY

  • The earlier Θ(√n) yield has a rate that decays with system size because the constant-sheaf factor has dim H0 = 1; removing it requires a genuine multiplication property.The stated earlier rate is ϱ = Θ(n^−1/2).Source: §XI · p. 56
  • n^(1−ϵ) independently addressable transversal logical CZ gates on disjoint pairs yield an almost-constant magic rate ϱ ≥ n^−ϵ at essentially unchanged distance.The construction uses the Golowich–Tamo–Zhu codes over a constant-size extension field.Source: §XI · p. 56

A. Sheaves and systems of local codes: a dictionary

B. The classical input

  • The classical input provides local codes with a multiplication property, large dimension and distance, and constant check weight; over F2, the stated rate and distance requirements force the use of a larger field.The input construction uses constant-sized algebraic-geometry codes and punctured tensor powers arranged on a t-partite graph.Source: B. The classical input · p. 57

C. The construction

The construction places twisted sheaf qLDPC codes on 2D hypergraph-product complexes and builds addressable logical CZ gates into a perfect matching. Gauging these gates yields parallel magic-state preparation, while the discussion identifies parameter and decoding limits and open extensions.

C. The construction

  • Taking the fundamental class [L] as the twist makes it a 2-cycle and gives Construction XI.5 a nontrivial cohomology invariant.The construction uses L = Γ × Γ and sets η2 = [Γ] ⊗ [Γ] = [L].Source: C. The construction · p. 57

D. Logical CZ structure: a perfect matching

  • Each g symmetry label selects one r–b pair, yielding |M|² independent disjoint logical CZ gates that commute and can be applied or measured in parallel.The matching is built into the code; gauge-fixing the r or b logical qudits is unnecessary for these pairs.Source: D. Logical CZ structure: a perfect… · p. 58

E. Parameters

  • Construction XI.5 gives [[n, n^(1-ε), n^((1-ε)/2)]]_q with ϱ ≥ n^-ε, versus Construction IX.1’s ϱ = Θ(n^-1/2).The paper characterizes the magic-rate improvement as quadratic, with distance unchanged up to the n^ε factor and encoding rate reduced by n^-ε.Source: E. Parameters · p. 59

F. Gauging measurement at almost-constant magic rate

  • s ≥ n^(1-ε) logical CZ magic states are prepared in one run at magic rate ϱ ≥ n^-ε, with distance Ω(n^((1-ε)/2)) under Assumption IX.12.The protocol acts on disjoint pairs of logical qudits.Source: F. Gauging measurement at… · p. 59

XII. DISCUSSION AND OUTLOOK

  • The framework supplies non-Abelian Clifford-stabilizer qLDPC codes on 2D hypergraph-product complexes, with addressable transversal CZ gates and a gauging-based magic-state fountain.The paper describes these as consequences of the sheaf-complex framework.Source: §XII · p. 60

Code parameters

  • Distance scales as n^1/2 up to the n^-ε loss, while q grows as ε approaches zero and fixed-alphabet multiplication-property codes face a Singleton-type rate-loss bound.Linear distance and a genuinely constant magic rate are presented as open parameter questions.Source: Code parameters · p. 60

Decoding

  • Decoding remains largely untreated: the syndrome graph is unchanged, but CZ-dressed non-Pauli stabilizers require tracking error propagation, and a phenomenological threshold remains future work.The proposed threshold analysis must include measurement errors during O(d) rounds of dressed X-stabilizer measurement.Source: Decoding · p. 60

Beyond CZ, and beyond Z2 / Physics of non-Abelian qLDPC codes

  • Other twists, gauge groups, and state-sum invariants are possible directions, but useful codes and universal gate sets are unestablished; sheaf gauge theories also lack a settled anyon or fusion-category description.The paper additionally identifies the image of field structures in group cohomology and binary multiplication-friendly codes as open problems.Source: Beyond CZ, and beyond Z2 · p. 61, Physics of non-Abelian qLDPC codes · p. 61

Appendix A: Local acyclicity of the subdivided sheaf

The appendix establishes local acyclicity for the subdivided sheaf and its dual, then explains how Fq codes and gates reduce to Fp components. It distinguishes straightforward bilinear CZ conversion from the harder trilinear CCZ case.

Appendix A: Local acyclicity of the subdivided sheaf

  • Full-row-rank local parity-check matrices imply graph-sheaf local acyclicity, and the subdivided sheaf and its dual meet this criterion.The criterion applies to the local parity-check matrices used in the subdivision.Source: Appendix A · pp. 61–62

1. Three layers of the problem

  • The reduction separates Hilbert-space and gate expansion: Fq-CZ follows a bilinear form, while trilinear CCZ expansion is the main difficulty.An Fq-qudit is represented by u qudits of dimension p.Source: §1 · p. 62

Appendix B: Alphabet reduction from Fq to F2

  • Alphabet reduction maps [n, k, d]_q to [un, uk, ≥d]_p, preserving rate and absolute distance while reducing relative distance by factor u.Check weights increase by at most u, so this layer has constant overhead when u is constant.Source: §1 · p. 62

2. The bilinear case: a change of basis

  • A dual basis under Tr(λxy) factors CZ(λ) into u ordinary Fp-qudit CZ gates in depth one; the same-basis circuit uses at most u(u + 1)/2 gates in O(u) depth.The dual basis is chosen so that Tr(λθ_iϕ_j) = δ_ij.Source: §2 · p. 63

3. The trilinear case

  • Because trilinear forms cannot generally be diagonalized, multiplication-friendly embeddings with m=O(u) address gate growth; the protocol's non-Clifford content resides in gauging measurement.A transversal Fq-CCZ does not automatically descend to disjoint binary CCZ gates; the gauging measurement supplies the non-Clifford resource instead.Source: §3 · pp. 63–64
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