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Theory of (Co)homological Invariants on Quantum LDPC Codes

Zimu Li, Yuguo Shao, Fuchuan Wei, Yiming Li, Zi-Wen Liu

arXiv:2603.25831v1quant-phcs.ITmath-phmath.ITmath.MP

TL;DR

The paper addresses the need for a systematic framework to analyze cohomological invariants and logical gates in sheaf codes. It develops methods for constructing logical representatives and computing cup products, establishing results on representative and gate counts while identifying open limits in the tensor analysis.

  • Problem

    A systematic framework for analyzing cohomological invariants and logical gates in sheaf codes is needed to address an important gap in their understanding and methodology.

    Source: §1.1 · p. 2, §1.2 · p. 3

  • Method

    The paper develops a framework for constructing explicit logical representatives and computing cup products, using algebraic and combinatorial descriptions of lifted sheaf-code coboundary operators.

    Source: §1.2 · p. 3, §1.3 · p. 7

  • Results

    Θ(N) inequivalent polarized logical representatives are identified, and the associated operation can consist of Θ(N/polylog N) logical Ct-1Z gates on disjoint tuples.

    Source: §1.2 · p. 4

  • Takeaways & Limitations

    The framework accommodates constructions from elementary examples to asymptotically good families, with numerical experiments described as feasible and fruitful for exploring and testing codes.

    Source: §1.4 · pp. 8–9

    The general behavior of Tξ as a t-tensor remains elusive even in the simplified setting where Ii ≡ I is a maximal ideal.

    Source: §6.4 · p. 62

Abstract

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With recent breakthroughs in the construction of good qLDPC codes and nearly good qLTCs, the study of (co)homological invariants of quantum code complexes, which fundamentally underlie their logical operations, has become evidently important. In this work, we establish a systematic framework for mathematically analyzing these invariants across a broad spectrum of constructions, from HGP codes to sheaf codes, by synthesizing advanced math tools. We generalize the notion of canonical logical representatives from HGP codes to the sheaf code setting, resolving a long-standing challenge in explicitly characterizing sheaf codewords. Building on this foundation, we present the first comprehensive computation of cup products within the intricate framework of sheaf codes. Given Artin's primitive root conjecture which holds under the generalized Riemann hypothesis, we prove that $\tilde{\Theta}(N)$ independent cup products can be supported on almost good qLDPC codes and qLTCs of length N, opening the possibility of achieving linearly many parallel, nontrivial, constant-depth multi-controlled-Z gates. Moreover, by interpreting sheaf codes as covering spaces of HGP codes via graph lifts, we propose a scheme that inductively generates families of both HGP and sheaf codes in an interlaced fashion from a constant-size HGP code. Notably, the induction preserves all (co)homological invariants of the initial code. This provides a general framework for lifting invariants or logical gates from small codes to infinite code families, and enables efficient verification of such features by checking on small instances. Our theory provides a substantive methodology for studying invariants in HGP codes and extends it to sheaf codes. In doing so, we reveal deep and unexpected connections between qLDPC codes and math, thereby laying the groundwork for future advances in quantum coding, fault tolerance, and physics.

1 Introduction

The introduction presents a framework for analyzing cohomological invariants in qLDPC codes and qLTCs in relation to fault-tolerant logical gates. It covers sheaf-code cup products, invariant-preserving lifts, logical representatives, and open algorithmic and algebraic directions for broader explicit constructions.

1.1 Background

  • The introduction motivates these results by the challenge of combining high qLDPC rates and distances with fault-tolerant non-Clifford gates, and identifies opaque sheaf-code representatives and cup products as key obstacles.Sparse complexes allow well-structured invariants to induce constant-depth circuits; higher-order invariants are relevant to non-Clifford gates and magic state distillation.Source: §1.1 · pp. 2–3

1.2 Main results

  • Under Artin’s primitive root conjecture, the construction yields Θ(N/polylog N) independent t-fold cup products and logical Ct−1Z gates on (nearly) good qLDPC codes.The conjecture’s validity is implied by GRH; the theorem also constructs Θ(N) inequivalent polarized logical representatives. The existence of the cycle needed for the gates requires a nontrivial tensor product sheaf, demonstrated for Reed–Solomon codes.Source: §1.2 · pp. 4–5
  • Theorem 1.2 shows that nontrivial invariants on a constant-size initial HGP code persist through inductive lifts to families of (nearly) good qLDPC codes.The result makes it possible to test for certain logical operators on the initial code and treats sheaf codes as covering spaces of HGP codes.Source: §1.2 · p. 5
  • For Cartesian products of double covers of connected regular graphs with trivial scalar coefficients, Theorem 1.3 gives addressable Ct−1Z gates and an invariant with subrank k/t.The X logical representatives divide into t groups, and each t-tuple containing one representative from each group has an addressable gate.Source: §1.2 · p. 5

1.3 Key techniques

  • For HGP codes, canonical logical representatives are simple tensor products, and their cup products can be computed algebraically, leaving construction of the cycle as the remaining task.With scalar coefficients, the graph structure and higher-dimensional products guarantee such a cycle, enabling a characterization of logical Cρ−1Z gates.Source: §1.3 · p. 7
  • For sheaf codes, the authors construct polarized representatives using combinatorial lifts and group-algebra module structure, making them suitable for cup-product computation.These representatives are tensor products of vertices with a fixed edge; the combinatorial view also supports analyses of coboundary ranks and inequivalent representatives.Source: §1.3 · p. 7
  • The cup-product construction isolates lifts of edges so induced vectors span the group algebra; Artin’s conjecture under GRH ensures this approach works for infinitely many primes.A counting argument gives Θ(l) independent induced vectors in the worst case when Θ(N) inequivalent representatives are available.Source: §1.3 · pp. 7–8
  • Linear algebra over flat modules evaluates local coefficients, while extremal graph theory selects compatible local-code tuples to symmetrize distinct sheaf-code blocks.The selection argument uses a hypergraph built from classical codes viewed as Grassmannian points and the existence of a favorable sub-hypergraph.Source: §1.3 · p. 8

1.4 Discussion and outlook

  • The authors establish a framework for analyzing and constructing cohomological invariants and fault-tolerant logical gates, while noting that numerical and algorithmic work and richer algebraic structures remain open directions.They describe their mathematical results as demonstrating the theoretical viability of this approach.Source: §1.4 · p. 8
  • The inductive construction needs a suitable sequence of lifts; sequential 2-lift algorithms [65, 66] exist, and logarithmically many steps mean even quasi-polynomial-time algorithms [46] yield sub-linear overall complexity.The authors propose implementing and benchmarking the lifting procedures to assess explicit code-family construction; they regard these algorithmic issues as largely orthogonal to the paper’s theoretical contributions.Source: §1.4 · p. 8
  • Local-code and tensor-product-sheaf conditions can make code searches difficult [6], but Reed–Solomon local codes and even plain HGP codes provide simpler examples with nontrivial cup-product or addressable multi-controlled-Z gates.The Reed–Solomon examples may not yield asymptotically good qLDPC codes; the authors also expect numerical experiments to be useful in the finite regime.Source: §1.4 · pp. 8–9
  • Alternative cup- and cap-product forms [25] may relax local-code constraints and expand non-Clifford gate constructions, with anticipated constant-depth gates on asymptotically good qLDPC families.The authors suggest these codes could enable constant-space, almost-logarithmic-time fault-tolerance overhead and may open paths toward qPCPs through fault tolerance.Source: §1.4 · p. 9

2 Preliminaries

This section develops mathematical and coding preliminaries for later analyses, covering cohomological and diagonal-gate concepts, expander graphs, presheaf-based cochain complexes and sheaf codes, graph lifts, Tanner codes and their distance guarantees, and quantum expander codes including the 3D construction.

2.1 CSS codes, invariant polynomials and diagonal logical gates

  • CSS codes are represented by 2-cochain complexes, with logical X and Z operators in cohomology and homology; surrounding terms relate to soundness and non-Clifford gates.The paper assumes fields of characteristic 2, with q = 2^s.Source: §2.1 · pp. 9–10
  • Invariant polynomials on consecutive CSS blocks induce diagonal multi-controlled-Z logical gates, which are constant-depth when each variable appears only constantly many times.The invariant form is unchanged by adding coboundaries to cocycles, so it induces a function on cohomology classes.Source: §2.1 · pp. 11–12
  • For constant-depth constructions, n_Cρ−1Z = Θ(N), while k_Cρ−1Z counts parallel logical gates and k_Cρ−1Z/n_Cρ−1Z characterizes resource overhead.The paper also defines the magic-state distillation overhead exponent using this ratio and code distance.Source: §2.1 · p. 12
  • For ρ = 3, tensor subrank captures nontrivial logical actions, and the authors make the first attempt to estimate it for gate formation on (almost) good qLDPC codes.Tensor subrank can be smaller than tensor rank, and higher-order tensor-rank computation is NP-complete over finite fields.Source: §2.1 · p. 12

2.2 Expander graph families and Ramanujan graphs

  • A constant spectral gap characterizes fixed-degree expander families and is a key condition in many LDPC distance proofs; Ramanujan graphs impose a nontrivial eigenvalue bound.For degree n, the stated Ramanujan bound is |λ| ≤ 2√(n−1).Source: §2.2 · p. 13

2.3 Presheaf and sheaf codes

  • A presheaf assigns vector spaces to poset elements and compatible linear maps, replacing scalar coboundary entries with those maps to form a sheaf cochain complex.The paper describes this block-matrix construction as a generalization of the scalar-coefficient cochain complex.Source: §2.3 · p. 14
  • A sheaf code is a CSS code obtained from any three consecutive terms of a presheaf cochain complex on a graded incidence poset, with characteristic-2 incidence ensuring the coboundary condition.The treatment uses the combinatorial structure without requiring a rigorous topology for the underlying space.Source: §2.3 · p. 15

2.4 Lifted graphs and their expansions

  • An l-lift replicates each base vertex l times and assigns edge permutations that bijectively connect fibers; cyclic-group lifts are called shift lifts.Lifting can also be viewed as replacing graph-incidence entries with permutation blocks, yielding a presheaf over the graph’s 1-chain complex.Source: §2.4 · pp. 16–17
  • An infinite sequence of n-regular bipartite Ramanujan graphs has 2iN0 vertices as i→∞ for some constant N0; every connected bipartite graph has a Ramanujan l-covering for every l [45].For any fixed base graph G, arbitrarily large lifts that are both abelian and Ramanujan are impossible; sequential abelian lifts are generally invalid for the cubical-complex construction.Source: §2.4 · pp. 17–18

2.5 Classical and quantum expander codes

  • Tanner codes place local parity-check constraints on graph incidences, expressing their parity-check matrix through a presheaf; unlike simple graph-boundary codes, expander codes can achieve linear distance.Simple graph-boundary codes cannot have constant rate and linear distance simultaneously.Source: §2.5 · pp. 19–20
  • When local-code distance d0 exceeds the graph’s nontrivial eigenvalue bound λ, Tanner codes have linear distance; with full-rank h and dual distance at least d0, the dual code also has linear distance [95].The result applies to the Tanner code defined by parity-check matrix H and gives separate distance bounds for ker H and ker H^T.Source: §2.5 · p. 20
  • Linear distances of ker H and ker H^T help control quantum-code parameters and provide a basic 1D example of Poincaré duality for sheaf codes and their duals [25].The proof also relies on the expander mixing lemma.Source: §2.5 · p. 20
  • Quantum expander codes are formed as hypergraph products of classical expander codes, with the 3D case outlined as a basis for comparison with higher-dimensional expanders.A consecutive 2-cochain complex from the 3D product can be cut to define a CSS code.Source: §2.5 · p. 20
  • Taking cycle coboundary operators as the classical inputs makes the 3D hypergraph product the 3D toric code.The paper later develops the upgrade from HGP codes to sheaf codes and studies cup-product gates on these codes.Source: §2.5 · p. 20

3 High dimensional expanders and cup products

This section develops cubical complexes from graph lifts and local codes and constructs cup and cap products to study cohomological invariants. It gives explicit 2D and 3D computations and shows that, under stated conditions, cup-product invariants define constant-depth multi-controlled-Z gates.

3 High dimensional expanders and cup products

  • High-dimensional expanders and local codes provide a framework for establishing almost good qLDPC codes.The section introduces basic notions for theoretical proofs in subsequent sections.Source: §3 · p. 21

3.1 Cubical complexes through products and lifts

  • Nontrivial graph lifts yield lifted products beyond ordinary HGP codes; non-Abelian lifts give good qLDPC codes in 2D, while cup-product gates require at least 3D structures.For the 3D-or-higher setting discussed, only almost good qLDPC codes are established in the cited construction.Source: §3.1 · pp. 22, 24

3.2 Local codes on cubical complexes

  • Adding t local codes to a t-dimensional cubical complex yields almost good qLDPC codes for t ≥3 and almost good qLTC codes for t ≥4.Full-rank local-code matrices, together with topological properties of the complex, make the presheaf a sheaf and support a Poincaré duality theorem used to bound code distance.Source: §3.2 · p. 24

3.3 Cup products and cap products

  • For sparse coboundary operators and constant dimension, cup-product invariants define constant-depth multi-controlled-Z gates.The invariant pairs a product of cohomology classes with a cycle; the cubical construction extends these products by subdividing into a simplicial complex and pulling them back.Source: §3.3 · p. 26
  • Subdivision maps and a cellular approximation transfer cochains between cubical and simplicial complexes, enabling cup products with local coefficients while preserving the Leibniz rule.When the complex dimension and subdivision size are constant, the transfer maps are sparse, supporting the constant-depth gate construction.Source: §3.3 · pp. 26–27
  • Under the theorem’s assumptions, pairing consecutive cup products with a cycle defines a cohomological invariant whose polynomial gives a constant-depth multi-controlled-Z gate.The degrees must sum to at most the cubical complex dimension, and the relevant maps and coboundary operators must be sparse.Source: §3.3 · pp. 27–28

3.4 Explicit computations in 2D and 3D

  • In 2D, [g; a1,0] ⌣c [a1·g;1,a2] and [g;0,a2] ⌣c [a2·g;a1,1] each equal [g;a1,a2]; in 3D, three edges have nontrivial cup product equal to π exactly when ordered along one of six paths.Source: §3.4 · pp. 29–30
  • For local-coefficient computations in 3D, the procedure applies cup products to three cochains and evaluates against a cycle with coefficients in the threefold tensor sheaf.The cycle lies in the kernel of the boundary map; the threefold tensor product enlarges the 2-chain coefficient space by a factor of m_i^3.Source: §3.4 · pp. 31–32

4 Induction scheme for qLDPC codes

This section presents an inductive lifting scheme for qLDPC code families that propagates codewords and preserves compatible nontrivial cup and cap products, so logical gates and other invariants of an initial constant-size code persist; the initial condition can be checked numerically.

4 Induction scheme for qLDPC codes

  • A nontrivial cup product gate in the initial constant-size code is inherited by every subsequent member of the constructed family.The initial code provides a finite instance on which the precondition can be tested numerically.Source: §4 · p. 32

4.1 Inductive lifting sequences

  • The inductive lifting sequence alternates exponential abelian lifts with ordinary HGP constructions, avoiding repeated fixed-base sequential lifts that are illegal for t ≥ 3.The sequence includes exponential-lift complexes and ordinary HGPs; the naive fixed-base approach is tantamount to a construction disallowed for high-dimensional expanders when t ≥ 3.Source: §4.1 · p. 32
  • The cochain maps for exponential lifts are not sparse, so composing them with an invariant polynomial does not yield a constant-depth circuit.Although these maps induce cohomology maps, their non-sparsity prevents this direct circuit construction.Source: §4.1 · p. 34
  • For an exponential lift by a group of odd order, canonical extension preserves nontrivial cohomology classes for both trivial coefficients and sheaves generated by local codes.For trivial coefficients, the projection composed with extension acts as the identity on cohomology; the result also holds for the corresponding boundary operators.Source: §4.1 · p. 34
  • For the sequential lift, an odd-order lifting group likewise preserves nontrivial cohomology classes under canonical extension, with analogous results for boundary operators.The stated condition applies to the lifted cocycle and its canonical extension.Source: §4.1 · p. 35
  • Any codeword from the initial code can be pushed forward to every subsequent cochain complex in the inductive lifting sequence.Source: §4.1 · p. 35

4.2 Preservation of (co)homological invariants

  • When two cubes share a containing t-cube, they have a unique smallest containing cube; their cup product is zero if that smallest cube does not exist.When it exists, the product is supported on that smallest cube.Source: §4.2 · pp. 36–37
  • Cup products are compatible with the inductive lift but not the covering map, and the preservation results extend to general invariant forms satisfying the same compatibility condition.Multiple cup products with sheaves generated by classical codes are included.Source: §4.2 · pp. 38–39
  • A nontrivial cup product gate on the initial complexes is inherited throughout the inductive lifting sequence.Theorem 4.13 uses canonical extensions to preserve the nonzero product pairing at every stage.Source: §4.2 · p. 39
  • Under the stated lifting and expansion assumptions, exponential lifts yield almost good qLDPC families with nontrivial gates of the same degree, and the result extends to qLTCs of length at least 5.The qLTC extension is stated equivalently for complexes with dim Xi = dim X′i.Source: §4.2 · p. 39

5 Logical action on hypergraph product codes

This section studies cup products and their logical actions on hypergraph product codes, constructing addressable invariant forms from explicit or canonical logical representatives for scalar and local coefficients. It also examines limits on these forms and code parameters, and uses graph lifts to produce structured cup products and, under the generalized Riemann hypothesis, families of independent kernel vectors.

5.1 Fundamental difficulties in computing the cup products

  • A cycle yields a nontrivial invariant form exactly when the cup-product map on cohomology is nontrivial; explicit HGP representatives address rank-computation and disjointness limitations.Source: §5.1 · p. 40

5.2 Hypergraph product with scalar coefficients

  • Scalar-coefficient cup products can yield an invariant form acting only on a selected triple of logical states, with the product supported on one 3-cube.Source: §5.2 · p. 42
  • For t-fold Cartesian products of double covers of connected n-regular graphs, each permutation and choice of one representative per group admits an addressable invariant form.Source: §5.2 · p. 43
  • For n ≥ 3 and an expander double cover, the invariant form has subrank k/t, although the resulting code distance is described as unsatisfactory.Source: §5.2 · p. 43

5.3 Hypergraph product with local codes

  • With local coefficients, canonical representatives retain their form but use redefined coefficient vectors chosen to make each edge pairing 1, so the cup product lies on a unique 3-cube.Source: §5.3 · p. 44
  • If mi/n is too large, ξ may have no solution; the Singleton bound on classical codes drastically diminishes hi’s code distance, so Theorem 2.20’s condition can never be fulfilled.Source: §5.3 · pp. 44–45
  • For any permutation α and t-tuple of logical representatives drawn from the groups, Theorem 5.3 provides an addressable invariant form.The representatives are defined from free variables of the local-code HGP construction.Source: §5.3 · pp. 45–46
  • The local-code HGP construction encodes k = Ω(N^(1/t)) logical qudits in the worst case, since a linear lower bound on ki is unavailable.The text notes that this may have the same scaling as the preceding construction without local codes.Source: §5.3 · p. 46

5.4 Hypergraph product with lifts

  • Choosing 1/2 n < mi < n makes ki increase with system size, and a large lift can magnify coding rate while facilitating well-behaved cup products.The Gilbert–Varshamov bound guarantees local codes with the desired code and dual distances in this range, but larger mi makes solving ξ more difficult.Source: §5.4 · p. 47
  • Under the generalized Riemann hypothesis, infinitely many primes l yield lifts with kernel vectors indexed by j and h forming a standard basis of F_l for a constant fraction of vi and ai.Theorem 5.4 states this for each lift of size l.Source: §5.4 · p. 47
  • In the lifted construction, the cup product vanishes unless j1 = j2 = j3, reducing dependence on ξ; when mi > n, finding ξ remains unclear.The lifted products are formed from logical representatives indexed by j; the text discusses the ξ difficulty further in Sections 6.4 and 6.5.Source: §5.4 · pp. 46, 48

6 Logical action on sheaf codes

This section develops methods for constructing and analyzing polarized logical representatives and cycles in sheaf codes and lifted graphs, including criteria and conditions for inequivalence and prescribed cup products. It uses these constructions to exhibit nontrivial constant-depth cup-product gates in HGP codes based on LPS graphs and Reed–Solomon local codes.

6.1 Polarized logical representatives

  • The construction rewrites lifted coboundary blocks and uses the group algebra Fq[H] to express their entries as operators on R-modules.Reorganizing rows and columns is needed to combine the lifted blocks with consistent indexing.Source: §6.1 · pp. 48–49
  • For j ≥ 2, the restricted coboundary operator has a nontrivial kernel that grows with |H| when local codes use (1−ν)n checks.The paper also bounds the restricted operator's kernel using local-code rates; for small ν and ν′, the stated lower bound is positive.Source: §6.1 · p. 50
  • For prime cyclic lifts, fixing [v1; a1] yields inequivalent first-direction polarized logical representatives, with their count scaling proportionally to n^(t−1)lV^t.The theorem setup assumes a prime l-shift-lift, small constant ν and ν′, and local codes and duals with distance at least 2.Source: §6.1 · pp. 51, 53
  • The abundance guarantee is specific to polarization in the first direction; the choice of local codes and Lemma 6.2 leave other directions without a guarantee of many inequivalent representatives.The authors identify polarized representatives with logical Pauli operators and note their expected use in forming cup products.Source: §6.1 · pp. 52–53
  • Polarized logical representatives persist for higher-level sheaf-code cochains and after changing the polarization direction, but converting them into cup products remains difficult because of local coefficients.The section identifies the next three subsections as addressing this difficulty.Source: §6.2 · p. 53, §6.1 · p. 53

6.2 Cyclic action on lifts and Artin’s conjecture

  • The cyclic group action on lifts preserves logical representatives and produces representatives that are not coboundaries, though distinct shifts can still be equivalent.The all-ones lift is an example equivalent to every group shift.Source: §6.2 · p. 54
  • If induced vectors are not contained in any maximal ideal of Fq[Cl], there are exactly l inequivalent logical representatives; assuming GRH, Artin’s conjecture supplies infinitely many primes l meeting this criterion.The argument uses q as a power of 2 with an odd exponent, so q is not a perfect square.Source: §6.2 · pp. 54–56

6.3 Proof of Theorem 6.7

  • A constant fraction of vertices have Θ(l) associated solution rows, and at least a d0/n fraction of a2 evaluations preserve Θ(l) independent vectors.These rank claims help select local data for the cyclic-lift criterion.Source: §6.3 · pp. 57–58
  • With Artin’s conjecture, a constant fraction of choices yield exactly l inequivalent polarized logical representatives whose induced vectors form the standard basis of Fq[Cl].This holds for any fixed [v1; a1].Source: §6.3 · p. 59

6.4 Hypergraph Turán problem and local codes

  • Using distinct sheaves to bypass the low-rank first restriction, the authors combine local-code selection with a Turán argument to obtain tuples of two-way ρ-product-expanding codes.The existence result applies for sufficiently large n and q=(n+1)^t.Source: §6.4 · pp. 59–60
  • Assuming Artin’s primitive-root conjecture, infinitely many primes l admit l-lifts with exactly l^t tuples of inequivalent logical representatives whose cup products equal the specified delta pairing.The base graph has size O(log l); the pairing holds for a constant fraction of base-graph vertices and at least d0 choices of each a_j.Source: §6.4 · pp. 60–61
  • Without assuming GRH and Artin’s conjecture, induced vectors generate a proper ideal; for t=3 (CCZ), bounding subrank is drastically difficult, with native lower bound floor(l/d) = O(1) from pairwise-disjoint translations of support f(x).Source: §6.4 · pp. 61–62

6.5 Solutions to the cycles

  • The boundary operator on the tensor-product sheaf has a nontrivial kernel if and only if the tensor product of the local codes has a nontrivial kernel.Source: §6.5 · p. 62
  • Solutions hz = 0 can be copied into the larger code space to solve Hx = 0, while Theorem 4.13 extends base-graph solutions through lifts to obtain ξ.Source: §6.5 · p. 63
  • The Vandermonde parity-check kernel is an RS code with parameters [[n, n − m, m + 1]], and the tensor-product parity-check construction yields a code with parameters [[n, k = 1, n]].The example chooses q = 2^s > n − 1 and n distinct elements of Fq.Source: §6.5 · pp. 63–64
  • The construction yields a unique ξ up to scalars by solving the component Tanner-code kernels, and ξ is nowhere vanishing.The passage states that ξ is obtained by solving each ker Ȟ_i^T as a classical Tanner code.Source: §6.5 · p. 64
  • The specified HGP codes support nontrivial constant-depth cup-product gates, although the construction gives no theoretical guarantee that dim ker H increases with V.The result applies to t-dimensional HGP codes based on LPS graph families with RS local codes defined by Vandermonde matrices; the stated corollary requires no extra assumptions.Source: §6.5 · p. 64
  • A conjectured dim ker H = Θ(V) would imply k = Θ(V^3) = Θ(N); in 3D this would yield Θ(N) invariant-form subrank and constant-overhead magic-state distillation.The invariant-form and distillation consequences are conditional on the conjecture.Source: §6.5 · p. 65
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