Non-Abelian qLDPC: TQFT Formalism, Addressable Gauging Measurement and Application to Magic State Fountain on 2D Product Codes

I. INTRODUCTION

• The work addresses the universality-connectivity tradeoff by showing that native logical non-Clifford gates do not require 3D product codes when gates need not be transversal or stabilizers Pauli.The introduction identifies addressable and parallelizable logical gates as a further challenge because many logical qubits occupy one qLDPC code block.Source: §I · p. 2

A. Summary of results

• The authors generalize combinatorial TQFT from manifold triangulations to Poincaré CW complexes and certain chain complexes, constructing topological-invariant path integrals and twisted Clifford-stabilizer 2D hypergraph-product codes.The path integral is invariant under gauge transformations and CW-complex subdivision; its structure can also be pulled back to skeleton codes, where it can be expressed through chain-cochain pairing.Source: §I · pp. 2–3, A. Summary of results · p. 4
• Gauging measurements factor a transversal symmetry into locally measurable dressed-X-stabilizers, and the spacetime path integral derives the protocol’s logical action on qLDPC codes.For random measurement outcomes, error correction is represented by defects that a successful decoder closes homologically trivially; the protocol is also adapted to skeleton codes.Source: §I · p. 3, A. Summary of results · p. 5
• Many inequivalent top-dimensional 2-cycles in general 2D chain complexes support 0-form subcomplex symmetries that enable addressable transversal Clifford gates, unlike the unique top-dimensional cycle of a manifold.When lifted to a higher-dimensional Poincaré CW complex, these subcomplex symmetries become higher-form symmetries; the framework includes addressable non-Clifford gates as well.Source: §I · p. 4, A. Summary of results · p. 5
• Θ(√n) disjoint logical CZ magic states with distance Ω(√n) can be prepared in parallel using O(d) rounds of error correction.The protocol combines selected dressed-X-stabilizer measurement outcomes to project onto independently addressable symmetries, without ancillas or additional scaling overhead.Source: §I · p. 4, A. Summary of results · p. 5
• The twisted qLDPC codes exhibit non-Abelian topological order, with non-Abelian fusion rules and Borromean-ring braiding statistics derived from the spacetime path integral.The Borromean statistics is related to triple intersections of magnetic-excitation world-sheets, the mechanism underlying the logical action.Source: §I · p. 4, A. Summary of results · p. 5
• The thickened 2D hypergraph-product codes have constant encoding rate k = Θ(n) and polynomial subsystem-code distance d = Ω(√n), and the skeleton codes retain the same parameter scaling.The subsystem distance treats spurious short cycles as gauge-qubit logical-operator support; pulling the construction back to skeleton codes reduces constant overhead.Source: A. Summary of results · p. 5

A. Mapping classical codes to manifolds

• The reviewed construction maps a skeleton classical code to a higher-dimensional manifold using a lifted chain complex and handle construction, with r ≥ 8 chosen to separate logical from spurious cycles.The parity-check matrix is lifted over the integers, with the naive lift given as a straightforward choice.Source: A. Mapping classical codes to manifolds · p. 6
• Each check is represented by a dressed 2-handle, while each bit becomes a dressed 3-handle attached to the checks it couples to.A bit coupled to f(i) checks has a dressed 3-handle consisting of one 3-handle and (f(i) −1) 1-handles.Source: A. Mapping classical codes to manifolds · pp. 7–8
• Attaching the dressed 3-handles to the dressed 2-handles according to the lifted boundary map constructs a 3-handlebody H from the classical code’s Tanner graph.The corresponding CW complex need not discretize a manifold: a single 2-cell can attach to more than two 3-cells.Source: A. Mapping classical codes to manifolds · p. 8, Figure 7 · p. 9
• Doubling H with an upside-down copy along their common boundary constructs a closed 8-manifold M8, whose triangulation carries a classical code with bits on 3-simplices and checks on 2-simplices.The two copies are glued along ∂H by the identity map.Source: A. Mapping classical codes to manifolds · pp. 8–9

B. Deformation retraction to the hidden Poincar´e CW complex

• The construction maps the Tanner graph to a manifold, then retracts dressed handles to a CW complex of cells that encode the checks and bits.The dressed 2-handles become 2-cells, while dressed 3-handles become 3-cells with attached 1-cells.Source: B. Deformation retraction to the… · pp. 9–10
• The thickened code uses 2-cells and 3-cells for checks and bits, while the hidden CW structure supplies cup products for defining path-integral invariants.The compact realization is described as practical for near-term implementation with minimal overhead.Source: B. Deformation retraction to the… · p. 10
• The deformation-retracted CW complex preserves Poincaré duality, making it a Poincaré complex suited to defining path integrals as topological invariants.The text notes this advantage relative to general CW or simplicial complexes.Source: B. Deformation retraction to the… · p. 10
• The thickened classical code has constant rate k = Θ(n) and linear distance, and is therefore asymptotically good.The construction transfers the classical code dimension and distance through the chain-complex mapping.Source: B. Deformation retraction to the… · p. 11
• If the transposed skeleton code is asymptotically good, the CW complex also defines a good classical code on its second cohomology group.The corresponding code space is H2(Lc; Z2).Source: B. Deformation retraction to the… · p. 11
• Spurious cycles and cocycles can have O(1) weight and produce short logical cycles after tensor products, so the construction treats them as gauge qubits.The subsystem-code approach ignores these short logical cycles.Source: B. Deformation retraction to the… · p. 11

C. Define cup product

• A p-cochain cup-producted with a q-cochain yields a (p + q)-cochain, evaluated on simplicial complexes using an ordered-vertex formula and extended to CW complexes by subdivision.Subdivision can refine the qubit model with a constant number of extra qubits, or be used only to evaluate cup products on the original complex.Source: C. Define cup product · p. 11

III. NON-ABELIAN QLDPC CODES AS TWISTED GAUGE THEORIES

• The section constructs non-Abelian qLDPC codes and corresponding combinatorial TQFTs, and studies their stabilizers, logical operators, symmetries, rate, and distance.Figure 8 illustrates Pachner moves and subdivision of CW cells.Source: §III · pp. 11–12, Figure 8 · p. 12

A. Feynman path integral in spacetime

• The Poincaré CW-complex path integral is invariant under gauge transformations and subdivision, but a general CW complex with nontrivial boundary need not preserve this topological invariance.Re-triangulation invariance of manifold TQFTs does not directly permit defining them on general chain complexes such as product complexes from expander graphs.Source: A. Feynman path integral in spacetime · pp. 12–13

B. Explicit construction of models with non-trivial triple intersections

• A hypergraph-product construction uses factor complexes and the time complex to produce nontrivial triple cup products and intersections.The example has 8D factor CW complexes, a 16D space complex, and a 17D spacetime complex; its spatial cocycles also have nontrivial triple intersections.Source: B. Explicit construction of models… · pp. 13–14

1. Construction of the codes

• The codes are Clifford-stabilizer codes whose construction is derived from gauging a higher-form symmetry-protected topological phase.The associated stabilizer group is a subgroup of the n-qubit Clifford group.Source: §1 · p. 14, C. Clifford stabilizer codes on… · p. 14

3. Stabilizer commutation relations and logical operators

• The twisted code is noncommuting in the full Hilbert space, but its Clifford stabilizers commute within the +1 subspace of the Z-stabilizer generators.The code is frustration free.Source: §3 · p. 15

2. Constraint of the code space from TQFT

• The TQFT-derived stabilizer relations constrain the code space, while the stabilizers commute on the stabilized subspace where the B-type stabilizers equal 1.The commutator of the dressed A and B stabilizers generates the B stabilizers, so commutation holds on the zero-flux subspace.Source: §3 · pp. 15–16

3. Stabilizer commutation relations and logical operators

• Logical-Z operators are electric Wilson operators, while magnetic operators on contractible cycles act trivially and the magnetic logical operators commute with stabilizers on the code space.The magnetic logical operators are distinguished by support on non-contractible cycles.Source: §3 · pp. 16–17

Charge parity operators as higher-form and subcomplex symmetries

• Charge parity operators reduce to transversal CZ operators supported on the corresponding cocycle or dual cycle.The charge-parity constraints have +1 eigenvalues in the code space, and their violations indicate charge excitations.Source: §4 · p. 18
• Unlike a 2-manifold's single global charge parity, a higher-dimensional complex supports transversal CZ symmetries on multiple cycle-class subcomplexes.The corresponding charges are extended objects on non-contractible cycles, though compactified dimensions can make them effectively point-like at large scale.Source: §4 · p. 18
• Dual cycle bases give independent total charge parities, allowing addressable and parallelizable measurement of individual transversal CZ operators.The independence follows from the intersection pairing between the dual basis cycles.Source: §4 · p. 19
• On Σg × S1, O(g) charge classes can be paired with corresponding CZ operators for independent measurement.For each charge class [b(i)], the parity is detected by the operator supported on a(i) × c.Source: §4 · pp. 19–20, Figure 10 · p. 19

• Both untwisted code C and twisted code ˜C have constant rate, with k=Θ(n) and ˜k=Θ(˜n)=Θ(n).Source: §1 · pp. 20–21
• The primary r copy has Ω(n) logical qubits, whereas the secondary b copy has only Ω(√n), because its zeroth Betti number is 1.Source: §1 · p. 21
• Gauging imposes Θ(√n) independent constraints, yet the twisted code dimension remains Θ(n).Source: §1 · pp. 21–22

• Subsystem encoding treats short-cycle logical degrees of freedom as gauge qubits, so spurious O(1)-weight cycles do not set the encoded subsystem distance.Source: §2 · p. 22
• The untwisted thickened hypergraph-product code has subsystem-code distance d=Ω(√n).Source: §2 · p. 22
• The twisted code has subsystem-code distance ˜d=Ω(√n), assuming Statement 1 holds for ˜C.Source: §2 · p. 24
• In the twisted auxiliary g copy, magnetic logical operators have minimum support Ω(√n), while electric logical operators are only lower-bounded by Ω(1).Source: §2 · p. 25

1. Truncating the CW complex and pullback of the cup product

• Truncation removes cell structure irrelevant to the needed cup products while preserving the path-integral invariant and Clifford stabilizer construction.The required products include b2 ∪ b*6 and b2 ∪ b*6 ∪ c0, as well as a3 ∪ a*5 and a3 ∪ a*5 ∪ c0.Source: §1 · p. 25

2. Path integral on the skeleton 2D general chain complex

• The skeleton spacetime path integral uses Z2 1-cocycles as gauge fields on three truncated chain complexes, with a nontrivial triple-intersection structure.Source: §2 · p. 27

3. Explicit evaluation of the cup product on the skeleton chain complex and the CW complex

• Cycle-cocycle duality and the repetition-code word allow cup products and path-integral invariants to be evaluated directly on the 2D general chain complex, without pulling them back from CW complexes.Source: §3 · pp. 27–28

4. Twisted skeleton hypergraph-product codes and the code parameter scaling

• Untwisted and twisted skeleton codes have constant rate and subsystem-code distance Ω(√n), assuming Statement 1 holds for the twisted skeleton code.The twisted skeleton stabilizers are pulled back from the thickened code; spurious short (co)cycles remain, so subsystem distance ignores them.Source: §4 · pp. 28–29

5. The 0-form subcomplex symmetries

• A 0-form subcomplex symmetry is supported on a top-dimensional subcomplex and enables addressable, parallel gauging measurements of logical Clifford operators.Source: §5 · p. 29
• These symmetries are absent in manifold TQFTs with a unique top-dimensional cycle, while higher-form symmetries are special cases of the more general subcomplex mechanism.Source: §5 · p. 29

1. Gauging measurement of higher-form symmetries

• The protocol gauges transversal CZ symmetries in two untwisted code copies by promoting them to local dressed X-stabilizer measurements in a twisted code.Combining measurement outcomes over selected basis cycles recovers the corresponding symmetry eigenvalues.Source: §1 · p. 30

A. Addressable and parallel gauging measurement of logical CZ

• Selected homology-class products of dressed X-stabilizer outcomes give subcomplex symmetries, enabling addressable, highly parallelizable gauging measurements.The protocol obtains all transversal CZ measurements without additional scaling overhead and in an ancilla-free way.Source: §1 · p. 30

1. Gauging measurement of higher-form symmetries

• The gauging path integral introduces a third Z2 gauge field coupled to the CZ charge, yielding the twisted-code path integral.The added field acts as a Lagrange multiplier fixing the charge parity on the relevant support.Source: §1 · p. 30
• Gauging and ungauging form spacetime domain walls between the untwisted code and a twisted Z2 × Z2 × Z2 gauge-theory code.In the spacetime picture, the measurement detects total charge parity on a cycle through the parity of an intersecting charge worldsheet.Source: §1 · p. 30, Figure 12 · p. 31

2. The gauging measurement protocol

• The product of dressed X-stabilizer outcomes measures the transversal CZ operator and its charge parity on the corresponding cycle.The gauged code is held for O(d) error-correction rounds to collect reliable syndromes despite measurement errors.Source: §2 · p. 31, §1 · p. 31

3. Instantiation: preparing disjoint magic states via the magic state fountain

• Θ(√n) disjoint logical CZ magic states with distance Ω(√n) are prepared by the gauging-measurement fountain.The theorem states this result for the 2D thickened hypergraph-product code, and the protocol uses O(d) rounds of error correction.Source: §3 · pp. 32–33
• Initializing one red logical qubit per cluster in |+⟩ and the other red qubits in |0⟩, with all blue qubits in |+⟩, selects unique nontrivial triple intersections.This choice yields Θ(√n) logical qubits initialized in |+⟩.Source: §3 · p. 32
• For a +1 logical CZ measurement outcome, each selected disjoint pair is left in a logical CZ magic state.The resulting resources can be swapped into memory and converted to parallelizable logical non-Clifford gates by gate teleportation.Source: §3 · pp. 32–33
• The gauging measurement yields an average of Θ(√n) +1 outcomes per execution, leading to Θ(√n) disjoint CZ magic states.The fountain scheme is motivated by the goal of preparing many logical magic states in parallel.Source: §3 · pp. 32–33

1. General formalism

• A spacetime path integral on a cylinder with state boundaries defines an operator whose matrix elements determine the logical action between input and output states.Boundary amplitudes define the input and output states, and the framework associates them with spacetime Poincaré complexes.Source: §1 · p. 34

2. Imaginary-time vs real-time evolution: relation to quantum error correction

• Replacing post-selected stabilizer projectors with syndrome measurements gives the real-time error-correction process; decoder recovery closes defects homologically trivially when successful.The defect-free imaginary-time path integral gives the same logical action as the actual error-corrected process.Source: §2 · p. 35
• The path integral generalizes the spacetime-code framework to Clifford-stabilizer codes and can describe non-Clifford circuits as well as error correction.The paper notes that the stabilizer measurement circuit is derived through gauging here, rather than directly from the path integral.Source: §2 · p. 35

3. Derivation of the logical action from cellular path integral

• The derived logical action is diagonal and is a product of projectors onto logical CZ eigenspaces selected by the triple-intersection condition.The spacetime derivation identifies these projectors with eigenstate projectors of the transversal CZ operator.Source: §3 · pp. 36–37

C. Pullback to the skeleton 2D chain complex and gauging measurement of the 0-form subcomplex symmetries

• Pulling back the cocycles and cup product adapts the gauging protocol to the skeleton chain complex’s 0-form subcomplex symmetry.The resulting logical action is translated from the corresponding CW-complex construction.Source: C. Pullback to the skeleton 2D chain… · p. 37

A. Non-Abelian fusion rules

• The projected magnetic-operator fusion rule can produce a sum of electric operators supported on cycles.This generalizes the cited fusion rule for magnetic flux excitations in non-Abelian D4 topological order in (2+1)D.Source: A. Non-Abelian fusion rules · pp. 37–38

B. Borromean ring braiding

• The generalized Borromean configuration has three operators that are linked as a whole, although eliminating any one leaves the other two unlinked.The paper presents Borromean braiding as detectable even with ambient trivial topology.Source: B. Borromean ring braiding · p. 38, §1 · pp. 38–39

1. Path integral

• The three magnetic operators are supported on contractible cycles whose membranes are perturbed to resolve their overlaps and form the link.Their gauge-field sources are the Poincaré duals of the corresponding bounding hypercubes.Source: §1 · pp. 38–39
• The linked operators’ path-integral correlation function is an invariant of the generalized Borromean ring.The construction uses three hypercubes with nontrivial triple intersection.Source: §1 · p. 39
• The generalized link is not restricted to the chosen hypercubes: it can be defined on a general CW complex by deformation preserving the correlation function.The paper reports the magnetic-defect correlation on the Borromean link as (−1).Source: §1 · pp. 38–39

2. Operator-based derivation

• The sequence of magnetic and symmetry operators diagnoses the Borromean ring invariant on the stabilizer code.The operators are nontrivially linked as a whole, while removing one leaves the other two unlinked.Source: §2 · pp. 39–40

VI. DISCUSSION AND OUTLOOK

• The twisted HGP construction is expected to extend to twisted balanced product codes, with distance beyond Ω(√n).The passage describes this extension as straightforward.Source: §VI · p. 40
• The twisted 2D HGP construction uses a spacetime path integral for non-Clifford logical operations rather than transversal non-Clifford gates.The codes have no boundaries; finding transversal non-Clifford gates for them remains an open direction.Source: §VI · p. 40
• Twisted N-dimensional HGP codes generalize to higher Clifford-hierarchy stabilizer codes with (N + 1)-fold intersections and N-body Borromean-like braiding.The corresponding theory has non-Abelian excitations with non-Abelian fusion and braiding.Source: §VI · p. 40
• In twisted 2D HGP codes, the gauging measurement implements addressable logical C_N−1Z gates and a magic-state fountain through an N-fold intersection.The gauging booster targets the (N + 1)th Clifford-hierarchy level, beyond the Nth-level transversal gates of untwisted N-dimensional HGP codes.Source: §VI · p. 40
• The 2D gauging measurement requires O(d) rounds of error correction, creating a space-time tradeoff.For twisted HGP codes of dimension N ≥ 3, single-shot state preparation is expected; the gauging booster may then avoid that time overhead.Source: §VI · pp. 40–41
• The authors conjecture that gauging boosters could implement native logical gates in arbitrary Clifford hierarchies using twisted 2D product codes.The proposed scope includes HGP and balanced product codes.Source: §VI · p. 41

Appendix A: Derivation of the Clifford stabilizer models by gauging a higher-form SPT

• The derivation starts from a higher-form SPT Hamiltonian and gauges its symmetries to obtain Clifford-stabilizer models for non-Abelian qLDPC codes.The construction generalizes higher-form gauge theory to the CW-complex setting.Source: Appendix A · p. 41

1. SPT Hamiltonian

• The construction uses higher-form symmetries with (p − 1)-, (q − 1)-, and (s − 1)-form components, constrained by p + q + s = D.The three cell types host the red, blue, and green copies, respectively.Source: §1 · p. 41
• The SPT entangler applies CCZ gates to cell triples satisfying a triple-intersection condition.The cup-product sum determines whether each CCZ is applied.Source: §1 · p. 42
• The SPT Hamiltonian is invariant under generalized symmetries acting on dual-complex subcomplexes rather than the entire system.These symmetries act on cycles of specified codimensions.Source: §1 · p. 43

2. Constructing twisted qLDPC codes via gauging

• The appendix distinguishes promoting background fields to dynamical fields from promoting global symmetries to local gauge symmetries, and pursues the latter.The selected procedure adds Z2 gauge fields and imposes Gauss law.Source: §2 · p. 43
• Gauging introduces Z2 gauge fields and imposes Gauss-law constraints to define the gauge-invariant physical Hilbert space.The fields are placed on the p-, q-, and s-cells in the three copies.Source: §2 · pp. 43–44
• The gauged Hamiltonian imposes flatness of the operator-valued gauge fields and yields the Clifford-stabilizer generators of the twisted code.The flatness conditions are d a = d b = d c = 0; the untwisted code instead comes from gauging the trivial paramagnetic Hamiltonian.Source: §2 · p. 44

Appendix B: Subsystem-code distance

• A logical Z on a k-cycle anticommutes with the conjugate logical X on its k-cocycle, and vice versa.An X error flips Z only if its support wraps nontrivially around the conjugate cocycle; a Z error must lie on the cycle to flip X.Source: Appendix B · pp. 44–45
• The conjugate-cycle pairing also constrains X-type errors: they must wrap nontrivially around the conjugate cocycle to flip logical Z.The dual support condition links logical-error susceptibility to nontrivial representatives.Source: Appendix B · pp. 44–45
• The subsystem code's Z-distance is the length of the shortest nontrivial representative among the basis k-cycles.This follows from the support condition for Z-type errors that flip logical X.Source: Appendix B · p. 45