PaperlayerView exact revision on arXiv

Source-linked AI summary

Non-Abelian Quantum Low-Density Parity Check Codes and Non-Clifford Operations from Gauging Logical Gates via Measurements

Maine Christos, Chiu Fan Bowen Lo, Vedika Khemani, Rahul Sahay

arXiv:2602.12228v1quant-phcond-mat.str-el

TL;DR

The paper addresses open questions about qLDPC phases and their excitations, including whether braiding has an analogue in this setting. It gauges transversal Clifford gates using two approaches, obtaining non-Abelian qLDPC behavior and a measurement-based route to magic-state preparation.

  • Problem

    Basic questions about qLDPC phases, excitations, and whether exchange and braiding statistics can be defined remain unanswered.

    Source: §I · p. 1

  • Method

    The paper gauges transversal Clifford gates using homological gauging for codes with generalized Poincaré-duality structure and graph gauging for more general qLDPC codes.

    Source: §I · p. 2

  • Results

    Repeated logical-operator action can trap a qLDPC analog of a single anyon on a torus, despite the absence of an underlying spatial manifold.

    Source: §I · p. 2

  • Takeaways & Limitations

    Measuring logical Clifford gates enables magic-state preparation and provides a route to non-Clifford operations on qLDPC codes.

    Source: §I · p. 2

Abstract

from arXiv · show

In this work, we introduce constructions for non-Abelian qLDPC codes obtained by gauging transversal Clifford gates using measurement and feedback. In particular, we identify two qualitatively different approaches to gauging qLDPC codes to obtain their non-Abelian counterparts. The first approach applies to codes that exhibit a generalized form of Poincar\'e duality and leads to a qLDPC non-Abelian Clifford stabilizer code, whose stabilizers are reminiscent of the action of a Type-III twisted quantum double. Our second approach applies to general qLDPC codes, and uses a graph of ancilla qubits which may be tailored to properties of the input codes to gauge a single transversal gate. For both constructions, the resulting gauged codes are shown to have properties analogous to 2D non-Abelian topological order -- e.g. the analog of a single anyon on a torus. We conclude by demonstrating that our gauging procedures enable magic state preparation via the measurement of logical Clifford gates. Consequently, our gauging constructions offer a protocol for performing non-Clifford operations on any qLDPC code.

I. INTRODUCTION

The paper addresses gaps in understanding qLDPC physics and systematically manipulating their logical information by constructing non-Abelian qLDPC codes through gauging. It relates those constructions to non-Abelian anyon behavior and measurement-based preparation of non-Clifford resources.

I. INTRODUCTION

  • qLDPC codes lack a systematic account of logical manipulation, while basic questions about their phases, excitations, and braiding remain open.The paper frames these as challenges for both quantum computation and understanding qLDPC physics.Source: §I · p. 1
  • Homological gauging uses generalized Poincaré-duality structure, while graph gauging applies more generally and can produce hybrid codes with Abelian and non-Abelian sectors.The constructions gauge transversal Clifford gates; the paper also uses measurement of logical Clifford gates to prepare magic states.Source: §I · p. 2
  • In qLDPC codes, repeated logical-operator action can trap an analog of a single anyon on a torus despite the absence of an underlying spatial manifold.The authors derive the codes’ ground states and logical operators and identify this as non-Abelian braiding behavior.Source: §I · p. 2

A. Non-Abelian Anyons from Gauging

  • Gauging a symmetry that acts on Abelian-code anyons intertwines those anyons with gauge fields, yielding non-Abelian anyons and braiding statistics.In the D(D4) example, braiding gauge-field anyons with original-code anyons implements symmetry automorphisms.Source: A. Non-Abelian Anyons from Gauging · p. 3
  • Winding two pairs of non-Abelian anyons around the torus cycles can leave a single unpaired anyon, a hallmark of non-Abelian topological order.The paper notes this phenomenon has also been observed experimentally.Source: A. Non-Abelian Anyons from Gauging · p. 4

B. Gauging on the Lattice

  • The lattice gauging procedure prepares ancillas in a cluster state, applies symmetry enrichment, and projects vertex ancillas to obtain the gauged code.For the D(D4) example, the enrichment uses a CCZ pattern and the resulting X-stabilizers commute in the ground-state subspace.Source: B. Gauging on the Lattice · pp. 4–5

C. Preparation Procedure from Measurement and Feedback

The paper describes measurement and feedback for preparing gauged codes, then contrasts two choices of ancilla gauge fields. A toy model shows how those choices yield different gauged theories and different scope for the construction.

C. Preparation Procedure from Measurement and Feedback

  • Measurement and feedforward implement the gauging protocol; random outcomes create Abelian eA anyons that can be paired by a finite-depth decoder, with another procedure requiring one round.The projectors for nondeterministic outcomes can be corrected back to the |+⟩ state.Source: C. Preparation Procedure from… · p. 5

III. GAUGING CONSTRUCTION OF NON-ABELIAN QLDPC CODES

A. Toy Model for the Two Approaches to Gauging

  • For stacked toric codes, layer-matched ancilla clusters provide independent symmetries for gauging each layer’s CZ gate.This construction repeats the two-dimensional gauging procedure independently across the stack.Source: A. Toy Model for the Two Approaches to… · p. 6
  • A single global symmetry on a distinct three-dimensional ancilla graph yields 3D gauge fields with non-Abelian mutual statistics with restricted-mobility 2D gauge fields.This graph-based approach does not require independently addressable symmetries inherited from the stacked layers.Source: A. Toy Model for the Two Approaches to… · p. 6

B. Generalizing Gauging to qLDPC Codes

The homological formalism expresses CSS qLDPC codes through chain complexes, while the gauging framework links ancilla symmetries to transversal Clifford gates. The section distinguishes the logical symmetries of a code from the transversal gates the constructions aim to gauge.

1. Homological Formalism of qLDPC Codes

  • CSS-code stabilizers and logical operators map to boundaries, coboundaries, and homology classes in a chain complex.The LDPC condition corresponds to sparse boundary maps, and stabilizer commutation follows from the chain-complex relation ∂1 ◦ ∂2 = 0.Source: §1 · pp. 7–8

B. Generalizing Gauging to qLDPC Codes

  • The section frames the generalization as a choice of which transversal symmetry to gauge and what ancilla structure can support it.Those choices are intertwined and differ between homological and graph gauging.Source: §2 · p. 8

2. Symmetries and Gauging in qLDPC Codes

  • Logical 1-cycle and 1-cocycle symmetries represent logical operators, but the construction instead gauges transversal Clifford gates that act as emergent code-space symmetries.These gates can map stabilizers to products of stabilizers and may act globally or on a subset of code qubits.Source: §2 · p. 8
  • The gauging protocol uses an ancilla cluster state and a symmetry-enrichment operator to tie its symmetries to transversal Clifford gates on the input codes.The ancilla complex need not copy the input complex, provided the symmetry-enrichment condition holds for every subsystem symmetry; projecting the vertex ancillas into |+⟩ completes the procedure.Source: §2 · pp. 8–9

3. Two Approaches to Gauging qLDPC Codes

The paper presents homological gauging for codes with a generalized Poincaré duality structure and graph gauging for cases that do not require that structure. In the toric-code setting, Poincaré duality pairs logical X and Z operators through a cup product and a fundamental class.

3. Two Approaches to Gauging qLDPC Codes

  • Homological gauging requires a Poincaré-duality-like structure, while graph gauging uses a graph ancilla and does not require that structure.For graph gauging, the graph choice depends on the input chain complex; the approach is intended for a single transversal gate.Source: §3 · p. 9

1. Poincar´e Duality in the Toric Code

  • In the toric code, Poincaré duality pairs logical X and Z operators associated with the same handle of the torus.The pairing is expressed as an isomorphism between cohomology H^α and homology H^(2−α), built from the cup product and fundamental class.Source: §1 · p. 10
  • The cup product and fundamental class define the cap-product map that sends cochains to chains, providing the homological form of this duality.For a cocycle, the resulting chain is a cycle.Source: §1 · p. 10

2. Poincar´e Duality in qLDPC Codes

Generalized cup products and Poincaré duality can connect qLDPC-code redundancies to transversal CZ gates. The resulting homological gauging construction specifies ancilla and enrichment choices and yields non-Abelian qLDPC stabilizers.

2. Poincar´e Duality in qLDPC Codes

  • Generalized cup products exist for several qLDPC code families, but arbitrary qLDPC codes need not admit a cup product or Poincaré duality.The cited forms include sheaf-code products and “cups and gates” products; the latter may obey an integrated Leibniz rule rather than the ordinary Leibniz rule.Source: §2 · p. 11
  • A cup product satisfying the stated Leibniz conditions defines transversal CZ gates indexed by fundamental classes, which can be multiple for a generic qLDPC code.For the non-Abelian construction, the “cups and gates” form requires Λ ≥ 3.Source: §2 · p. 11

3. Homological Expression for non-Abelian D4 Codes

  • The homological construction chooses an ancilla cluster state and symmetry-enrichment operator to gauge the transversal CZ symmetries, producing specified diagonal and off-diagonal stabilizers.With the toric-code cup product, the formulas reproduce D(D4) stabilizers; the procedure simultaneously gauges all symmetries of the form U_CZ[M ⌢ µ].Source: §3 · pp. 11–12

4. Homological Gauging of transversal SWAP Gates

Homological gauging can address transversal SWAP gates under additional structural conditions, while graph gauging offers a route for codes without Poincaré duality. Both approaches require ancilla and enrichment choices that preserve the LDPC property.

4. Homological Gauging of transversal SWAP Gates

  • When the cup product and Poincaré-duality map meet the stated LDPC conditions, each indexed transversal SWAP is either the identity or a symmetry of the ground-state codespace.The construction assumes a bijective LDPC map from H2 and uses commutators with the input stabilizers to establish the symmetry condition.Source: §4 · p. 12
  • For homological SWAP gauging, the protocol follows the CZ procedure but uses a symmetry-enrichment operator chosen for the SWAP symmetries.The vertex and plaquette stabilizers are given as the resulting code’s stabilizer families.Source: §4 · p. 13

D. Non-Abelian Codes from Gauging with Graph States

  • The graph-based approach can gauge a single uniform transversal gate without Poincaré duality, using a graph ancilla whose connectedness gives one global 0-cocycle symmetry.The graph and enrichment operator must tie that ancilla symmetry to the gate and keep the resulting stabilizers LDPC.Source: D. Non-Abelian Codes from Gauging with… · p. 13, §1 · p. 13

2. Example: Gauging transversal SWAP gate

The graph-gauging example gauges transversal SWAP using an ancilla graph, with graph and symmetry choices constrained by the resulting code’s LDPC properties. The section also discusses other symmetries and deterministic measurement-based preparation.

2. Example: Gauging transversal SWAP gate

  • For transversal SWAP, the symmetry map φ is taken to assign each edge a single vertex, with a sparse matrix required to preserve the LDPC property.The example starts from two copies of a qLDPC code and assumes a connected ancilla graph, without further graph assumptions.Source: §2 · pp. 13–14
  • The ancilla graph and symmetry enrichment must preserve LDPC stabilizers; generic graphs may make some stabilizer strings nonlocal, so graph choice depends on the code and desired properties.One proposed construction adds edges according to the symmetry map, but the resulting graph must also be checked for closed loops, distance, and encoding rate.Source: §2 · p. 14

3. Gauging Other Symmetries

  • Graph gauging can also gauge Z_N transversal symmetries, including Z_2 charge conjugation of a Z_3 code to obtain an S_3 qLDPC code.A graph state supported only on a subsystem symmetry can instead produce a non-Abelian slice within an Abelian qLDPC code.Source: §3 · p. 15

E. Measurement Preparation of Non-Abelian qLDPC Codes

  • A decoder for the ancilla complex can correct random measurement outcomes with Pauli Z operators, enabling deterministic preparation of the gauged non-Abelian code.The authors state that sequentially gauged codes can be prepared with finite-depth adaptive circuits, and stabilizer-measurement corrections can be performed in software.Source: E. Measurement Preparation of… · p. 15
  • The deterministic preparation argument also applies to graph gauging and controlled-symmetry enrichment operators.The correction uses Pauli Z operators that commute with the other stabilizers.Source: E. Measurement Preparation of… · p. 15

IV. EXAMPLES OF NON-ABELIAN QLDPC CODES

The examples apply graph and homological gauging to fracton and bivariate bicycle qLDPC codes. They include global and addressable CZ gates, with the addressable construction producing a non-Abelian slab in an otherwise Abelian code.

IV. EXAMPLES OF NON-ABELIAN QLDPC CODES

  • The examples include graph-gauged non-Abelian fractons from transversal CZ and non-Abelian bivariate bicycle codes from a non-local, but qLDPC, transversal CZ.The homological fracton construction produces a system in which only part of the code is non-Abelian.Source: §IV · pp. 15–16

A. Non-Abelian Fractons by Gauging transversal CZ Gates

  • For the ALP model, graph gauging uses ancilla qubits on cubic-lattice vertices and edges to gauge a transversal CZ between two code copies.The example specifies a local, surjective symmetry map and derives the resulting stabilizers.Source: A. Non-Abelian Fractons by Gauging… · p. 16

B. Non-Abelian Fractons by Gauging Addressable Gates

  • Homological gauging of addressable CZ gates yields a qLDPC code with a non-Abelian slab in its xz plane and an otherwise Abelian remainder.The gate acts on a fixed xz plane and implements logical CZ on O(1) logical qubits.Source: B. Non-Abelian Fractons by Gauging… · p. 17
  • The ALP addressable CZ acts on a single fixed-y xz plane, preserves the stabilizer group, and implements logical CZ on O(1) logical qubits.An identical gate is available on yz planes at fixed x.Source: B. Non-Abelian Fractons by Gauging… · p. 17

1. A Cup Product for the ALP Code

  • The ALP cup product is built from the Ising and plaquette-Ising components of its hypergraph product, and the stated construction suffices for addressable CZ gates in the xz plane.The cup product is chosen translationally invariant and satisfies the Leibniz rule; the passage notes that a more complete construction is forthcoming.Source: §1 · p. 17

2. Gauging the Addressable CZ Gates

The section gauges addressable CZ gates in the ALP model and a non-local transversal CZ in bivariate bicycle codes. The ALP construction produces a non-Abelian slab, while the bicycle example constructs its ancilla graph and transformed stabilizers.

2. Gauging the Addressable CZ Gates

  • Gauging the addressable ALP CZ gates produces an ordinary Abelian qLDPC code with a non-Abelian slab in its xz plane.The symmetry enrichment acts only on part of the pair of ALP codes, and the resulting stabilizers change mainly near the plane.Source: §2 · p. 18

C. Non-Abelian Bivariate Bicycle Codes from Gauging Non-Local CZs

  • The bivariate bicycle example gauges a non-local transversal CZ using an ancilla graph, with the input symmetry preserving the original code’s ground-state subspace.The example gives transformed X stabilizers and states that the diagonal plaquette stabilizers remain unchanged.Source: C. Non-Abelian Bivariate Bicycle Codes… · pp. 19–20

V. PROPERTIES OF NON-ABELIAN QLDPC CODES

The section derives ground states and logical operators for the gauged codes, then shows that their logical actions reproduce key features of non-Abelian braiding. In the homological construction, single gauge charges are tied to subsystem symmetries.

A. Ground States and Logical Operators

  • Homological gauged ground states resemble those of D(D4) topological order on a torus, but their expressions sum over H0 representatives; logical eZ operators remain unchanged.The gauged logical operators are obtained by tracking the input logical operators through the gauging procedure.Source: A. Ground States and Logical Operators · pp. 20–21
  • For SWAP gauging, both ˜X and ˜Z logicals transform, and only SWAP-invariant products remain invariant; eAeB and mBmC are Abelian sectors, while eB and mC are non-Abelian.This contrasts with CZ gauging, where the ˜Z logicals are unchanged.Source: A. Ground States and Logical Operators · p. 21

B. Non-Abelian Braiding: Single “Gauge Charge” on a Subsystem Symmetry

1. Homological case

  • In homological gauging, an unpaired charge is associated with a subsystem-symmetry sector µ, and distinct global parity operators detect the charges for different µ.Unlike the 2D toric code's single global charge-parity operator, qLDPC codes have distinct operators for each subsystem symmetry.Source: §1 · p. 22
  • An Abelian qLDPC code cannot host the same single unpaired charge: flipping one qubit upsets an even number of vertex stabilizers within each µ.Consequently, each Aµ measures even parity in the Abelian case.Source: §1 · p. 22

2. Graph case

The graph construction supports an unpaired gauge charge when the input codes contain a nontrivial cycle. More broadly, gauging followed by ungauging measures a logical Clifford symmetry, and the resulting measurement can support magic-state preparation in the stated qLDPC setting.

2. Graph case

  • For graph gauging, a nontrivial cycle in the B and C codes permits a state with a single unpaired gauge charge.The connected graph ancilla has a single H0 generator, so this charge is associated with a unique generator.Source: §2 · pp. 22–23

VI. QUANTUM COMPUTATION ON QLDPC CODES VIA GAUGING

  • Gauging a Clifford symmetry and then ungauging measures its logical action on the input codes; depending on the logical initial states, this provides a route to magic-state preparation for level-(2 + 1)-dimensional qLDPC codes.The section describes this as an interpretation of the gauging procedure through quantum computation.Source: §VI · p. 23

A. Logical Clifford Measurement from Gauging and Ungauging

  • The measurement outcomes are those of the transversal CZ gate in the example, and the same gauging-and-ungauging result holds for any Clifford gate.The measured symmetry is UCZ[µ] on the B and C codes.Source: A. Logical Clifford Measurement from… · p. 24

VII. DISCUSSION AND OUTLOOK

The authors discuss how their gauging constructions connect non-Abelian qLDPC physics with logical-information manipulation, and outline unresolved questions about fault tolerance and the codes’ broader uses. They identify decoder development and characterization of the codes’ excitations and landscape as future work.

Fault tolerance and logical operations

  • Gauging measures a logical Clifford gate and can prepare a magic state from any code with such a gate.The authors identify fault tolerance, including decoding errors during gauging, as the largest unaddressed question for practical use of the logical CZ measurement protocol.Source: §VII · pp. 24–25
  • A decoder for graph gauging may require carefully engineered expansion properties, while a just-in-time decoder offers a possible route without them.The cited graph-gauging decoder relies on an expander ancilla graph; the authors suggest homological gauging may be fault tolerant only for expander codes, while noting a 2D just-in-time precedent.Source: §VII · p. 25

Future quantum-computation directions

  • Using the constructed codes directly for fault-tolerant computation leaves open how to encode information in excitations, braid them, and obtain a universal gate set.The authors also raise fault tolerance and possible advantages over conventional 2D non-Abelian codes as unresolved questions.Source: §VII · p. 25

Future physics directions

  • The constructed codes’ quasi-particle content, excitation statistics, fusion rules, and place in the landscape of non-Abelian qLDPC codes remain to be characterized.The authors identify UV/IR mixing as a challenge to efficiently describing the data used to organize that landscape.Source: §VII · p. 25

Appendix A: Derivation for Gauging using Cup Product to D4 qLDPC code

The derivation uses cup-product structure to gauge a transversal CZ gate and obtain stabilizers for non-Abelian D4 qLDPC codes. It also derives stabilizers for a D4 fracton model using symmetry enrichment and projection.

Appendix A: Derivation for Gauging using Cup Product to D4 qLDPC code

  • A well-defined cup product supplies a transversal gate, treated as an emergent ground-state symmetry and gauged using the same chain complex for the ancillas.The derivation concerns the projectors of the non-Abelian D4 qLDPC code obtained by gauging transversal CZ.Source: Appendix A · p. 29

1. Gauging the transversal CZ gate

  • The transversal CZ is symmetry-enriched, then its stabilizers are simplified using the Leibniz rule, Stokes’ theorem, and terms trivial on the original code’s ground state.The resulting expressions give the final vertex stabilizers of the D4 model.Source: Appendix A · pp. 29–30, §1 · pp. 30–31

1. Derivations for Non-Abelian D4 Fractons by Gauging transversal CZ Gates

  • For the non-Abelian D4 fracton model, the derivation tracks symmetry-enriched stabilizers and projects the a sublattice to Xa = +1 before obtaining the resulting stabilizers.The resulting stabilizers are summarized after the derivation.Source: §1 · pp. 31–32

Appendix C: Non-Abelian S3 Fractons by Gauging Charge Conjugation

This appendix derives a non-Abelian S3 fracton model by gauging the charge-conjugation symmetry of the Z3 anisotropic lineon planeon model. It constructs commuting projectors after symmetry enrichment and projection.

Appendix C: Non-Abelian S3 Fractons by Gauging Charge Conjugation

  • The construction gauges the Z2 charge-conjugation symmetry of the Z3 anisotropic lineon planeon model to obtain an S3 fracton model.The model’s charge conjugation maps qutrit Z and X operators to their Hermitian conjugates.Source: Appendix C · p. 33
  • A qubit graph cluster state and controlled charge-conjugation symmetry enrichment are used to gauge the transversal charge-conjugation gate.The cluster-state stabilizers are defined on a cubic lattice, with the pattern repeated at each vertex.Source: Appendix C · p. 33
  • After projecting the a sublattice to X = +1, the precursor stabilizers are symmetrized into commuting projectors for the non-Abelian S3 ALP code.The appendix states that these projectors complete the derivation of the model.Source: Appendix C · p. 34

Appendix D: Ground States of Non-Abelian qLDPC codes

The appendix follows logical operators through homological gauging and constructs ground states for graph gauging. The derivations use projection-compatible operator forms and, for the graph example, a SWAP-symmetric ground-state assumption.

a. Non-Abelian Logical Operators from Homological Gauging

  • Homological gauging updates a logical X operator by rewriting it to commute with projection onto the ancilla |+⟩ state.The derivation uses the initialized B and C codes’ Z logical operators and rewrites part of the operator as a logical Z factor.Source: a. Non-Abelian Logical Operators from… · pp. 34–35
  • The homological derivation decomposes a chain into H0 and an image-boundary term to rewrite the logical operator after gauging.The resulting expression identifies a logical Z operator of the C code within the transformed operator.Source: a. Non-Abelian Logical Operators from… · p. 35

b. Ground States for Graph State Construction

  • For graph gauging of a transversal SWAP, the ancilla ground state differs in form from those on the B and C sublattices.The derivation assumes the original qLDPC codes are initialized in a SWAP-symmetric case.Source: b. Ground States for Graph State… · pp. 35–36

Related papers

Loading 2602.12228v1…