PaperlayerView exact revision on arXiv

Source-linked AI summary

Quantum Low-Density Parity-Check Codes

Nikolas P. Breuckmann, Jens Niklas Eberhardt

arXiv:2103.06309v2quant-ph

TL;DR

Quantum LDPC codes are being developed to provide scalable quantum error correction with sparse checks and improved parameters over surface-code-style families. This Perspective surveys their mathematical constructions, product methods, hardware and decoding challenges, and applications. Recent distance records suggest that constant-rate, linear-distance codes may soon be within reach, though important construction and implementation questions remain.

  • Problem

    Quantum LDPC codes seek sparse-check quantum codes with strong scaling, because constant-rate constructions could reduce fault-tolerance overhead to a constant.

    Source: §I · p. 1

  • Method

    The Perspective surveys quantum LDPC constructions and analysis, including geometry, expander codes, product constructions, decoding, hardware implementation, and applications.

    Source: §I · p. 2, D. Hardware Implementation · p. 12, §VII · p. 13

  • Results

    Recent distance records suggest that quantum LDPC codes with constant encoding rate k/n > const. > 0 and linear distance d ∝n may soon be within reach.

    Source: §VIII · p. 13

  • Takeaways & Limitations

    Quantum LDPC codes offer promising new approaches to quantum error correction and draw on geometry, homological algebra, expander codes, and algebraic topology.

    Source: §VIII · p. 13

Abstract

from arXiv · show

Quantum error correction is an indispensable ingredient for scalable quantum computing. In this Perspective we discuss a particular class of quantum codes called low-density parity-check (LDPC) quantum codes. The codes we discuss are alternatives to the surface code, which is the currently leading candidate to implement quantum fault-tolerance. We introduce the zoo of LDPC quantum codes and discuss their potential for making quantum computers robust against noise. In particular, we explain recent advances in the theory of LDPC quantum codes related to certain product constructions and discuss open problems in the field.

I. INTRODUCTION

Quantum LDPC codes address the challenge of protecting fragile quantum information with sparse checks while seeking better scaling than surface codes. This Perspective surveys their constructions, mathematical foundations, implementation challenges, and open problems.

I. INTRODUCTION

  • Quantum error correction remains essential for scalable quantum computation because quantum information is fragile and fault-tolerant memories remain difficult to realize.Source: §I · p. 1
  • Surface codes offer high thresholds and planar layouts, but their distance scales as √n and they compare poorly with classical codes whose encoded bits and distance can scale linearly with n.Source: §I · p. 1
  • Quantum LDPC codes keep check and qubit participation bounded, and constant-rate families could reduce fault-tolerance overhead to a constant.Source: §I · p. 1
  • Quantum LDPC codes remain harder to construct than classical LDPC codes, and whether they can achieve both linear encoding rate and linear distance is still open.Source: §I · pp. 1–2
  • The Perspective surveys stabilizer and CSS codes, homological and geometric viewpoints, manifold and product constructions, non-LDPC high-rate codes, and applications beyond fault tolerance.Source: §I · p. 2

Notation and Conventions

  • The manuscript uses [n, k, d] for classical codes and [[n, k, d]] for quantum codes, with asymptotic notation defined for comparing code families.Source: Notation and Conventions · p. 2

A. Quantum codes

The paper introduces stabilizer, CSS, and quantum LDPC codes, then recasts CSS codes through chain complexes and homology. These perspectives connect logical operators and distances to algebraic and topological structures.

1. Stabilizer and CSS codes

  • Stabilizer codes encode k logical qubits in the common +1 eigenspace of commuting Pauli checks, with distance given by the smallest support of a nontrivial logical operator.Source: §1 · pp. 2–3
  • CSS codes use two classical binary codes and parity-check matrices whose orthogonality ensures commuting X- and Z-type stabilizers; any stabilizer code maps to a CSS code with parameters [[4n, 2k, 2d]].Source: §1 · p. 3

2. Quantum LDPC codes

  • Quantum LDPC families bound both check weight and the number of checks incident on each qubit by constants, while good codes require k ∈Θ(n) and d ∈Θ(n).Source: §2 · p. 3

B. Perspectives on CSS codes

1. Chain complexes

  • In a chain complex, logical Z operators are homology classes and logical X operators are cohomology classes, while k and the distances derive from their dimensions and minimum weights.Source: §1 · p. 3

2. Tanner graphs

Tanner graphs represent code incidences between checks and bits or qubits, while manifold tessellations provide a geometric route to quantum codes. The latter connects logical operators and distances to topology.

2. Tanner graphs

  • A classical Tanner graph is bipartite, with bits and checks on opposite sides connected when a bit participates in a check; quantum CSS graphs add X-check, qubit, and Z-check layers.Source: §2 · pp. 3–4
  • In quantum Tanner graphs, incidence matrices give HX and HZ, and commutativity requires every X-check/Z-check neighborhood intersection to contain an even number of qubits.Source: §2 · p. 4

3. Manifolds and Cell complexes

  • Surface tessellations associate X-checks, qubits, and Z-checks with cells of adjacent dimensions; for the Shor code, vertices, edges, and faces yield three X-checks, nine qubits, and seven Z-checks [7, 10].Source: Figure 3 · p. 4, §3 · p. 4
  • The construction generalizes to higher-dimensional manifolds, where logical operators arise from homology and cohomology classes and finer reasonable tessellations produce LDPC code families.Source: §3 · p. 4

III. GEOMETRICAL CONSTRUCTIONS

Geometrical constructions use manifolds and related structures to build quantum LDPC codes with finite encoding rates and improved distance scaling, while exposing important decoding and geometric limitations. The section covers hyperbolic surface and higher-dimensional codes, Freedman–Meyer–Luo codes, and open parameter constraints.

A. Hyperbolic quantum codes

  • Hyperbolic manifolds yield quantum codes with constant encoding rate because middle-dimensional homology grows linearly with manifold volume.Source: A. Hyperbolic quantum codes · p. 5

1. Hyperbolic Surface Codes

  • Hyperbolic surface codes trade check weight against encoding rate through k = (1 − 2/r − 2/s)n + 2, while retaining logarithmic distance.They support minimum-weight perfect matching and union-find decoding, but below-threshold logical-error suppression scales only polynomially with system size.Source: §1 · p. 5

2. Higher-Dimensional Hyperbolic Codes

  • Higher-dimensional hyperbolic codes achieve distance d ∈ Θ(n^α) for some α > 0, including 4D families with α > 0.1.For arithmetic 4D manifolds, an upper bound α < 0.3 is established, while its extension to general 4D manifolds remains open.Source: §2 · p. 5
  • Efficient local decoding for 4D hyperbolic codes has been proposed, but it is proven to correct errors only up to size log(n).Source: §2 · p. 5

B. Freedman–Meyer–Luo codes

  • Freedman–Meyer–Luo codes form 4-manifolds by twisting Σg × [0,1], removing short non-contractible loops, and taking a product with a loop of length g/log(g).Tessellating the resulting manifolds produces homological codes whose logical operators are non-contractible surfaces with area Θ(g).Source: B. Freedman–Meyer–Luo codes · p. 6, Figure 5 · p. 6

C. Haah’s code

Haah’s cubic code illustrates both the distinctive structure and unresolved distance behavior of geometric quantum LDPC codes. The section then places these limitations alongside product constructions, which have produced major distance-scaling advances despite incomplete asymptotic properties.

C. Haah’s code

  • Haah’s cubic code has fractal logical operators and known distance bounds Ω(n^(1/3)) ≤ d ≤ O(n^(2/3)), but its asymptotic distance scaling remains open.Source: C. Haah’s code · p. 6

D. Bounds on parameters

  • Geometric quantum LDPC codes obey restrictions such as kd^α ≤ O(n) on D-dimensional Euclidean lattices, while extending these bounds to non-Euclidean lattices is unresolved.For D = 2, hyperbolic surface codes violate the Euclidean bound, motivating the open higher-dimensional question.Source: D. Bounds on parameters · p. 6

IV. PRODUCT CONSTRUCTIONS

  • Product constructions transfer tensor-product ideas from classical and quantum codes into quantum LDPC code design, forming the basis of recent breakthrough results.Source: §IV · pp. 6–7
  • Hypergraph, homological, distance-balancing, fiber-bundle, lifted, and balanced products modify tensor constructions to improve code parameters, especially distance.Fiber bundles introduce a twist, while several related constructions helped break the earlier polylog(n)√n distance barrier in 2020.Source: §IV · p. 7

A. Tensor/hypergraph products

Product constructions generate quantum LDPC codes from classical codes or chain complexes, with tensor-based, fiber-bundle, and lifted-product variants improving code parameters. Recent constructions progressively increase achievable distance while retaining useful logical-qubit scaling.

A. Tensor/hypergraph products

  • Hypergraph products combine two classical codes into quantum LDPC codes, achieving constant encoding rate and distance d ∈Θ(√n) for suitable classical LDPC inputs.The construction produces a [[n1n2+r1r2, k1k2, min{d1,d2}]] code.Source: A. Tensor/hypergraph products · p. 7
  • Tensor-product distance balancing yields quantum codes with d ∈O(√n log(n)), improving highly asymmetric distances from Ramanujan complexes.The procedure combines a quantum code with a classical code and multiplies one distance component by the classical distance.Source: A. Tensor/hypergraph products · p. 7
  • Iterated tensor products of Ramanujan complexes achieve d ∈O(√n log(n)^m) for arbitrary positive integers m.Kaufman–Tessler reported this record four months after the distance-balancing construction.Source: A. Tensor/hypergraph products · p. 7

B. Fiber Bundles

  • Fiber-bundle codes introduce twists that permute fiber qubits, potentially increasing distance while preserving the tensor product’s counts of physical qubits, logical qubits, and checks.The twists are specified by fiber-code automorphisms associated with base-code bit-check incidences.Source: B. Fiber Bundles · p. 8
  • Random-base fiber-bundle codes achieve k ∈Θ(n^3/5/polylog(n)) logical qubits and d ∈Ω(n^3/5/polylog(n)).The construction uses a random classical base code, a cyclic repetition fiber, and random twists.Source: B. Fiber Bundles · p. 8

C. Lifted Products

  • Lifted products collapse commuting ℓ^2 × ℓ^2 block matrices to ℓ × ℓ matrices, reducing physical-qubit count by a factor of ℓ relative to hypergraph or tensor products.The commutativity assumption ensures that the resulting matrices define a CSS code.Source: C. Lifted Products · p. 8
  • Lifted-product codes achieve logical bits in Θ(n^α log(n)) and distance d ∈Ω(n^(1−α/2) log(n)) for any 0 ≤α < 1, giving the first almost-linear-distance quantum LDPC codes.The bounds rely on (co-)expansion properties of the associated classical expander code.Source: C. Lifted Products · p. 8

D. Balanced Products

Balanced products reduce symmetry in tensor products to construct shorter quantum codes, while related product and non-LDPC constructions probe the possibility of good quantum LDPC parameters. These approaches establish strong rate-distance trade-offs but leave key distance and check-weight questions open.

D. Balanced Products

  • Balanced products quotient a tensor product by a common group action on its physical qubits and checks, reducing code length while generalizing to quantum codes and chain complexes.The construction is defined for classical codes with a shared symmetry group.Source: D. Balanced Products · pp. 8–9
  • Applying balanced products to expander and repetition codes yields non-random codes with k ∈Θ(n^4/5) and d ∈Ω(n^3/5).The expander codes derive from Cayley graphs of PSL(2,q).Source: D. Balanced Products · p. 9

E. The Possibility of Good Quantum LDPC Codes

  • Cyclic or abelian twists appear insufficient for good quantum LDPC codes because existing bounds are in some sense optimal in those cases.A proposed route around these bounds is to use non-abelian twists, although their distance is not yet clear.Source: E. The Possibility of Good Quantum… · p. 9
  • Balanced products can produce codes isomorphic to their duals, avoiding the need for distance balancing.Whether balanced products of Cayley-graph expander codes yield good LDPC codes remains open.Source: E. The Possibility of Good Quantum… · p. 9

F. XYZ Products

  • XYZ product codes use three classical codes and non-CSS X, Y, and Z checks, but their general minimum-distance bounds remain unknown.For the Chamon code, the number of encoded qubits is 4 gcd(n1,n2,n3).Source: F. XYZ Products · p. 9

A. Bravyi–Hastings Codes

  • Bravyi–Hastings tensor products of random non-LDPC CSS codes yield [[n, Θ(n), Θ(n)]] codes with high probability, but their check weights are Θ(√n).The construction therefore gives good quantum codes without satisfying the strict LDPC condition.Source: A. Bravyi–Hastings Codes · pp. 9–10
  • A related Hastings construction achieves d ∈Ω(n^(1−ϵ)) for arbitrary ϵ > 0 with logarithmic stabilizer weight, under a conjecture in geometry.Whether iterated products can make stabilizer check weight arbitrarily low remains open.Source: A. Bravyi–Hastings Codes · p. 10

B. Bravyi–Bacon–Shor Codes

Bravyi–Bacon–Shor codes derive subsystem codes from binary matrices or pairs of classical codes, while circuit-based and approximate constructions relax strict stabilizer-code requirements. These alternatives offer favorable parameters but have important locality, check-weight, decoding, or fault-tolerance limitations.

B. Bravyi–Bacon–Shor Codes

  • Bravyi–Bacon–Shor codes place qubits on the nonzero entries of a binary matrix, with k = rk(A) and distance equal to the minimum weight of nonzero row- or column-span vectors.Their gauge operators are generated by adjacent XX interactions in columns and ZZ interactions in rows.Source: B. Bravyi–Bacon–Shor Codes · p. 10
  • Using two constant-rate, linear-distance classical codes gives Bravyi–Bacon–Shor codes with k logical qubits, distance min{d1,d2}, and optimal scaling for 2D subsystem codes.The construction has between min{n1d2,d1n2} and n1n2 physical qubits and inherits a decoder from the input codes.Source: B. Bravyi–Bacon–Shor Codes · p. 10

C. Subsystem Codes from Quantum Circuits

  • Circuit-derived subsystem codes can have physical checks of weight O(1) and distance Θ(n^(1−ϵ)), but their actual stabilizer checks have unbounded weight and are therefore not LDPC.Restricting the code to D-dimensional Euclidean space gives distance Θ(n^(1−ϵ−1/D)).Source: C. Subsystem Codes from Quantum Circuits · p. 10
  • The circuit construction maps gates to gauge operators on intervening physical qubits, preserving logical operators up to gauge multiplication.Its stated parameters use concatenated quantum codes with parameters [[n0,1,Θ(n0)]].Source: C. Subsystem Codes from Quantum Circuits · p. 10
  • These circuit-derived codes have logarithmically distributed, non-uniform stabilizer weights, and efficient decoding remains an open problem.The text also questions whether such codes can have a threshold.Source: C. Subsystem Codes from Quantum Circuits · p. 10

D. Approximate Codes from Spacetime Circuit Hamiltonians

  • Approximate non-stabilizer codes from spacetime circuit Hamiltonians have parameters [[n, Ω(n/polylog(n)), Ω(n/polylog(n))]], with nine-qubit local terms and polylogarithmic participation per qubit.Their recovery restores the initial state with high fidelity, but the codes use non-Pauli Hamiltonian terms and approximate rather than exact encoding.Source: D. Approximate Codes from Spacetime… · p. 10
  • Whether non-stabilizer and approximate codes can yield practical, competitive fault-tolerance schemes remains open.Source: D. Approximate Codes from Spacetime… · pp. 10–11

VI. CHALLENGES AND OPPORTUNITIES

Quantum LDPC codes may reduce fault-tolerance overhead and improve logical operations or decoding, but these benefits depend on code assumptions, suitable decoders, and unresolved implementation questions. The section frames constant overhead, gate depth, and noisy-measurement robustness as major opportunities and challenges.

A. Reduction in Overhead

  • Gottesman’s theorem gives constant physical-qubit overhead when an LDPC family has constant rate, polynomially spaced sizes, and efficient error suppression below threshold.Under these assumptions, a circuit on k qubits uses at most ηk/R physical qubits.Source: A. Reduction in Overhead · p. 11
  • Fawzi–Grospellier–Leverrier satisfy Gottesman’s assumptions using hypergraph-product expander codes and a small-set-flip decoder.Other LDPC families may also qualify if sufficiently simple decoders can prove the required error suppression.Source: A. Reduction in Overhead · p. 11

B. Logical Operations

  • Gottesman’s constant-overhead protocol performs logical operations sequentially with ancilla states and requires a minimum number of logical qubits whose value is not yet known.This unresolved threshold may affect practical comparisons with alternative schemes.Source: B. Logical Operations · p. 11
  • Spatially local two-dimensional codes support only constant-depth Clifford logical gates, so surface and two-dimensional color codes require circuit depth scaling with code size for other logical gates.This motivates studying nonlocal LDPC codes, although their logical-gate capabilities remain poorly understood.Source: B. Logical Operations · p. 11

C. Decoding

  • Optimal decoding for a general stabilizer code is #P-complete, making practical sub-optimal decoders such as minimum-weight perfect matching important.Source: C. Decoding · p. 11
  • LDPC codes may enable faster and simpler decoding through better encoding rates, simpler algorithms, and reduced classical-processing load compared with surface-code schemes.The text connects these properties to lower heat dissipation and closer placement of classical control hardware.Source: C. Decoding · p. 11
  • Belief propagation is attractive for hardware implementations but struggles with short loops and equivalent stabilizer-related solutions in quantum LDPC Tanner graphs.Combining it with renormalization or ordered-statistics decoding has shown good performance on several quantum LDPC codes.Source: C. Decoding · pp. 11–12
  • Single-shot decoding can make some LDPC codes robust to noisy stabilizer measurements without repeating those measurements, with simulations reported for tensor-product and 4D hyperbolic codes.Source: C. Decoding · p. 12

D. Hardware Implementation

Quantum LDPC codes face unresolved hardware challenges involving connectivity, stabilizer-check scheduling, decoding, and scalable architecture. The perspective also connects these codes to quantum complexity theory and geometry, while identifying progress toward strong code parameters.

D. Hardware Implementation

  • Nonplanar Tanner graphs and expanding connectivity complicate physical layouts, although book embeddings and modular or nonplanar hardware architectures may provide alternatives.Most discussed Tanner graphs are nonplanar, and the minimum page number for their book embeddings is unknown.Source: D. Hardware Implementation · p. 12

VIII. CONCLUSION AND OUTLOOK

  • Hardware viability depends on future developments in low-latency control, fast decoding, interconnectivity, and wiring for scalable fault-tolerant quantum computation.The authors note that modular architectures could relax spatial constraints, while other proposals permit direct nonplanar interactions.Source: D. Hardware Implementation · p. 12, §VIII · p. 13

D. Hardware Implementation

  • Fault-tolerant stabilizer measurement requires efficient gate schedules that avoid spreading errors, but such schedules are known only for hyperbolic surface codes among finite-rate examples.The perspective identifies good measurement schedules for the discussed LDPC codes as an open problem.Source: D. Hardware Implementation · pp. 12–13

A. Quantum Complexity Theory

  • Quantum LDPC codes connect to QMA, the local Hamiltonian problem, and the NLTS and quantum PCP conjectures in quantum complexity theory.The NLTS discussion links quantum LDPC codes with linear rate and polynomial distance to the absence of sufficiently low-energy trivial states.Source: A. Quantum Complexity Theory · p. 13

B. Geometry

  • Quantum LDPC codes also provide a reverse connection from coding theory to geometry: code properties can determine geometric properties of constructed manifolds.Freedman–Hastings construct dimension-11 manifolds from quantum LDPC codes, suggesting applications to systolic geometry.Source: B. Geometry · p. 13

VIII. CONCLUSION AND OUTLOOK

  • The perspective concludes that constant-rate, linear-distance quantum LDPC codes may soon be within reach, although scalable implementation problems remain unresolved.The authors suggest these codes may eventually be better suited than surface codes for quantum computers in the mid- to long-term.Source: §VIII · p. 13

APPENDIX

The appendix reviews chain complexes and their homological structures, then constructs total complexes from double complexes. These constructions provide the algebraic framework used for product-based quantum codes.

1. Chain complexes

  • Chain complexes associate cycles, boundaries, and homology through kernels, images, and quotient spaces of boundary operators.The appendix also introduces cochains, cocycles, coboundaries, and cohomology using dual vector spaces and their pairing.Source: §1 · p. 14
  • Quantum codes can be extracted from cell complexes by taking parity-check matrices from boundary operators, with logical operators represented by nontrivial homology classes.The X-checks, qubits, and Z-checks correspond to adjacent cell dimensions.Source: §1 · p. 14

2. Total Complex of Double Complexes

  • A double complex is a grid of vector spaces with commuting squares and direction-wise compositions that vanish.Its vertical and horizontal maps define the two directions of the construction.Source: §2 · p. 14, Figure 9 · p. 14
  • The total complex collects entries along diagonals, and its boundary operators sum maps between successive diagonals.The boundary operators square to zero because the vertical and horizontal operators commute and each squares to zero.Source: §2 · p. 14
  • The total-complex construction generalizes naturally from double complexes to higher-dimensional complexes.A triple complex provides an example of this higher-dimensional extension.Source: §2 · p. 14, Figure 10

3. Tensor product of chain complexes and hypergraph products

Tensor products of chain complexes are represented as total complexes of double complexes and support the construction of hypergraph product quantum codes. Homological identities make their logical-qubit counts tractable.

3. Tensor product of chain complexes and hypergraph products

  • The tensor product of complexes of lengths n and m is a chain complex of length n+m−1, formed by combining the complexes across degrees.For two length-2 complexes, the result has length 3.Source: §3 · pp. 14–15
  • Tensor products can be defined through a double complex with vertical and horizontal boundary operators, whose total complex recovers the tensor product.After choosing bases, tensor products of maps correspond to Kronecker products of matrices.Source: §3 · pp. 14–15
  • Hypergraph product codes arise by associating length-2 chain complexes to two classical codes and taking their tensor product.The resulting quantum parity-check matrices are obtained from the classical check matrices and their transposes.Source: §3 · p. 15
  • The Künneth formula relates the homology of a tensor product to tensor products of the input homologies.This relation enables computation of the number of logical qubits in tensor-product and hypergraph-product quantum codes.Source: §3 · p. 15
  • Iterated tensor products produce higher-dimensional complexes, extending the construction beyond two input complexes.A tensor product of three complexes is illustrated as a triple complex.Source: §3 · p. 15, Figure 11 · p. 15

4. Fiber Bundle Codes

Fiber bundle and lifted product codes generalize tensor-product constructions through twists and algebra-valued matrices. These constructions preserve useful structural control while enabling improved code parameters, including almost linear distance in lifted products.

4. Fiber Bundle Codes

  • Fiber bundle codes were introduced to break the √n polylog(n) distance barrier using a random base code, a cyclic repetition-code fiber, and a random twist.The construction is attributed to Hastings–Haah-O’Donnell [68].Source: §4 · p. 15
  • Fiber bundle codes introduce a twist into tensor-product boundary operators to increase the distance of the resulting quantum code.When the twist is trivial, the fiber bundle specializes to the ordinary tensor product.Source: §4 · p. 15
  • The fiber bundle construction uses a base complex, a fiber complex, and automorphisms assigned to incident basis-vector pairs as twists.Under additional technical conditions, its number of logical qubits coincides with the encoded-bit count of the base code.Source: §4 · p. 15

5. Lifted Product Codes

  • Lifted product codes extend tensor products from vector spaces to modules over algebras, reducing checks and logical qubits by a factor of ℓ relative to corresponding tensor or hypergraph products.The construction can be expressed using chain complexes over an algebra R.Source: §5 · pp. 15–16
  • Lifted products use algebra-valued matrices and Kronecker products over R to define the quantum check matrices.Equivalent constructions interpret block matrices over F2 and collapse grouped submatrices into ℓ×ℓ blocks.Source: §5 · p. 16
  • Using circulant-matrix algebras and expander-based inputs, lifted products yield quantum LDPC codes with almost linear distance.The cited construction combines an Sipser–Spielman code on a random cyclic covering with the matrix 1 + x from a repetition code.Source: §5 · p. 16

6. Balanced Product Codes

Balanced product codes extend balanced products and tensor products over a group to chain complexes, enabling quantum LDPC constructions. These constructions are closely related to lifted-product and fiber-bundle codes through equivariant topological realizations.

6. Balanced Product Codes

  • Balanced products quotient Cartesian products by a group action, and the analogous tensor product over G identifies v·g ⊗ w with v ⊗ g·w.Source: §6 · p. 16
  • Chain complexes with compatible right and left G-actions form a balanced-product complex as the total complex of a double complex.Source: §6 · p. 16
  • Under the assumption that G is a finite group of odd order, the balanced-product construction admits a Künneth formula.Source: §6 · p. 16
  • Breuckmann–Eberhardt [70] used balanced products of highly symmetrical Sipser–Spielman codes and a cyclic repetition code to construct quantum LDPC codes.Their Sipser–Spielman codes used Cayley graphs of PGL(2, q), whose automorphism group is PGL(2, q).Source: §6 · p. 16

7. Relation of Fiber Bundle and Balanced Product Codes

  • Balanced-product, lifted-product, and fiber-bundle codes can describe the same constructions when equivariant tessellations of the underlying spaces yield quantum codes.For example, a Z2 action on two circles produces a Klein bottle as a balanced product and a fiber bundle over a circle.Source: §7 · p. 16

Related papers

Loading 2103.06309v2…