Automorphism-Weighted Ensembles: Theory & Applications

• Automorphism-weighted ensembles are frameworks that assign weights inversely to the size of the automorphism group, integrating symmetry into the inference process.
• In coding theory, these ensembles improve error correction by running parallel decoders on permuted inputs and fusing outputs using groupoid measures.
• In TQFT and holography, automorphism weighting underpins ensemble averaging over boundary theories, linking discrete symmetry measures to gravitational path integrals.

An automorphism-weighted ensemble is a class of collective inference or averaging frameworks in which each object (typically a code or a boundary theory) is assigned a weight inversely proportional to the size of its automorphism group, with ensemble outputs reflecting integration over underlying symmetries. In information theory and quantum coding, automorphism-weighted ensemble decoders (notably AutDEC) harness the automorphism group of the code to construct multiple parallel decoding branches, fusing their outputs via a weighting scheme. In topological quantum field theory (TQFT), the automorphism-weighted ensemble over boundary conditions or conformal field theories (CFTs) is defined via the groupoid measure, i.e., uniform up to isomorphism, with each class weighted by $1/|\mathrm{Aut}(\cdot)|$. The automorphism-weighted ensemble paradigm generalizes permutation ensemble methods, yielding systematically improved inference or decoding performance, provided the symmetry group is sufficiently large.

1. Mathematical Structure of Automorphism-Weighted Ensembles

Let $\mathcal{C}$ denote a set of objects (e.g., codes, algebras, or boundary theories) equipped with automorphism groups $\operatorname{Aut}(x)$. Formally, an automorphism-weighted ensemble average of a quantity $f$ is

$$\langle f \rangle = \frac{\sum_{x\in \mathcal{C}} f(x)/|\operatorname{Aut}(x)|}{\sum_{x\in \mathcal{C}} 1/|\operatorname{Aut}(x)|}$$

This measure is canonical for any finite groupoid, and is referred to as the "uniform up-to-isomorphism" or groupoid measure. In this context, the unnormalized weight assigned to $x$ is $w(x) = 1/|\operatorname{Aut}(x)|$ (Barbar, 6 Nov 2025).

Applications in Coding Theory

In automorphism-ensemble (AE) decoding for classical and quantum codes, one selects a subset of code automorphisms $G = \{g_1,\dots,g_K\}$, runs parallel decoders on codewords (or syndromes) permuted by these automorphisms, then combines outputs using weights $w_g$ satisfying $\sum_g w_g = 1$ (Koutsioumpas et al., 3 Mar 2025, Bioglio et al., 2022).

Interpretation in TQFT and Holography

In the holographic context, the automorphism-weighted sum extends to boundary CFTs classified by Lagrangian algebras in a modular tensor category. The ensemble sum

$\mathcal{C}$0

gives the natural notion of ensemble averaging of boundary data in TQFT gravity (Barbar, 6 Nov 2025).

2. Automorphism Groups and Their Action

Automorphism groups encode the structural symmetries of the object under consideration. For an $\mathcal{C}$1 linear code with parity-check matrix $\mathcal{C}$2, the automorphism group $\mathcal{C}$3 is the subgroup of permutations $\mathcal{C}$4 such that the rowspace of $\mathcal{C}$5 is preserved: $\mathcal{C}$6 where $\mathcal{C}$7 is the $\mathcal{C}$8 permutation matrix representing $\mathcal{C}$9 (Koutsioumpas et al., 3 Mar 2025). The group action on syndromes is $\operatorname{Aut}(x)$0.

In TQFT, automorphism groups are realized as invertible algebra automorphisms of Lagrangian algebras $\operatorname{Aut}(x)$1 in a modular tensor category $\operatorname{Aut}(x)$2: $\operatorname{Aut}(x)$3 This determines the groupoid structure and weighting in boundary theory ensemble averages (Barbar, 6 Nov 2025).

In polar and Reed–Muller codes, the automorphism group can be fully characterized in terms of affine (or more generally, block-affine) groups and analyzed via their action on monomial orbits (Ma et al., 2024, Bioglio et al., 2022).

3. Ensemble Decoder Construction and Weight Assignment

For quantum and classical code decoders:

1. Selection: Choose $\operatorname{Aut}(x)$4 automorphisms $\operatorname{Aut}(x)$5.
2. Offline Analysis: For each $\operatorname{Aut}(x)$6, compute its action on the code—permuted parity-check matrix $\operatorname{Aut}(x)$7 and corresponding syndrome update $\operatorname{Aut}(x)$8.
3. Parallel Decoding: For a given received syndrome $\operatorname{Aut}(x)$9, each branch decodes $f$0 using BP or SC algorithms.
4. Weight Assignment: Each automorphism $f$1 is assigned a weight $f$2 (uniformly $f$3, or by empirical success/failure probabilities, or inverse error rate).
5. Output Fusion: Either aggregate log-likelihoods

$f$4

or select the best candidate based on minimum-weight or maximum-likelihood under the noise prior (Koutsioumpas et al., 3 Mar 2025, Bioglio et al., 2022, Bioglio et al., 2022).

In TQFT, the boundary ensemble sum is defined with

$f$5

and ensemble averages are computed analogously (Barbar, 6 Nov 2025).

Weighting Schemes in AE decoders for coding:

Scheme	Weight Expression	Context
Uniform	$f$6	Default/naive
Empirical performance	$f$7	Blast error rate
Soft-likelihood	$f$8	Gaussian channel
Groupoid measure	$f$9	TQFT, groupoids

4. Complexity and Implementation

For automorphism-ensemble decoders in quantum LDPC codes (Koutsioumpas et al., 3 Mar 2025):

• BP Complexity: Each BP run on the code of length $$\langle f \rangle = \frac{\sum_{x\in \mathcal{C}} f(x)/|\operatorname{Aut}(x)|}{\sum_{x\in \mathcal{C}} 1/|\operatorname{Aut}(x)|}$$0 with check degree $$\langle f \rangle = \frac{\sum_{x\in \mathcal{C}} f(x)/|\operatorname{Aut}(x)|}{\sum_{x\in \mathcal{C}} 1/|\operatorname{Aut}(x)|}$$1 and $$\langle f \rangle = \frac{\sum_{x\in \mathcal{C}} f(x)/|\operatorname{Aut}(x)|}{\sum_{x\in \mathcal{C}} 1/|\operatorname{Aut}(x)|}$$2 iterations costs $$\langle f \rangle = \frac{\sum_{x\in \mathcal{C}} f(x)/|\operatorname{Aut}(x)|}{\sum_{x\in \mathcal{C}} 1/|\operatorname{Aut}(x)|}$$3.
• Branch Parallelism: Full parallelization retains $$\langle f \rangle = \frac{\sum_{x\in \mathcal{C}} f(x)/|\operatorname{Aut}(x)|}{\sum_{x\in \mathcal{C}} 1/|\operatorname{Aut}(x)|}$$4 wall-clock time; serial evaluation scales as $$\langle f \rangle = \frac{\sum_{x\in \mathcal{C}} f(x)/|\operatorname{Aut}(x)|}{\sum_{x\in \mathcal{C}} 1/|\operatorname{Aut}(x)|}$$5.
• Offline Enumeration: Tanner graph automorphism computation via BLISS or Leon's method scales as $$\langle f \rangle = \frac{\sum_{x\in \mathcal{C}} f(x)/|\operatorname{Aut}(x)|}{\sum_{x\in \mathcal{C}} 1/|\operatorname{Aut}(x)|}$$6 (practically linear).
• Ensemble Output Fusion: Candidate combination stage is $$\langle f \rangle = \frac{\sum_{x\in \mathcal{C}} f(x)/|\operatorname{Aut}(x)|}{\sum_{x\in \mathcal{C}} 1/|\operatorname{Aut}(x)|}$$7.
• TQFT Ensemble: Evaluation reduces to combinatorial sums over Heegaard splittings or groupoid objects, with computational bottleneck in classification and automorphism enumeration (Barbar, 6 Nov 2025).

For classical codes, specifically polar codes (Bioglio et al., 2022, Pillet et al., 2021):

• AE decoder has total complexity $$\langle f \rangle = \frac{\sum_{x\in \mathcal{C}} f(x)/|\operatorname{Aut}(x)|}{\sum_{x\in \mathcal{C}} 1/|\operatorname{Aut}(x)|}$$8 for $$\langle f \rangle = \frac{\sum_{x\in \mathcal{C}} f(x)/|\operatorname{Aut}(x)|}{\sum_{x\in \mathcal{C}} 1/|\operatorname{Aut}(x)|}$$9 distinct automorphisms and code length $x$0.
• Number of Nonredundant Branches: Determined by group/coset enumeration over block-affine automorphism classes.

5. Performance and Scaling Results

Quantum Codes (AutDEC) (Koutsioumpas et al., 3 Mar 2025):

• For $x$1 Quantum Reed–Muller code: AutBP-5 (ensemble size $x$2) achieves a pseudo-threshold $x$3, matching BP+OSD-0 accuracy; plain BP exhibits zero threshold due to short cycles.
• Bivariate bicycle codes under circuit-level noise: AutBP ensembles (full graph automorphism) achieve performance at or above BP+OSD-0, significantly outperforming standard BP as blocklength increases.

Classical Polar Codes (Bioglio et al., 2022, Pillet et al., 2021, Ma et al., 2024):

• AE-SC decoders with nonredundant automorphisms (distinct under SC) achieve near-ML performance for moderate $x$4 and outperform random draws.
• AE–BP based on upper-diagonal (UTL) design produces stronger gains, outperforming SCL decoders at a comparable or smaller latency.
• Numerical results confirm that enumeration and weighting based on automorphism orbit structure uniformly improve error rates relative to fixed permutation schemes.

6. Extensions, Limitations, and Relation to Groupoid Theory

Limitations:

• The effectiveness of automorphism-weighted ensemble methods is contingent on the code (or underlying object) possessing a nontrivial automorphism group; codes with trivial symmetry offer no benefit (Koutsioumpas et al., 3 Mar 2025, Bioglio et al., 2022, Ma et al., 2024).
• For large codes, automorphism enumeration can become a computational bottleneck, requiring either partial enumeration or restriction to specific subgroups.
• Hardware implementation of fully parallelized AE decoders is required to eliminate the $x$5-fold serial cost present in some reference implementations.

Extensions:

• Enlarging the search space to endomorphisms or generalized code symmetries (e.g., incorporating endomorphism semigroups) (Koutsioumpas et al., 3 Mar 2025).
• Analytical investigation into subgroup structure to identify most beneficial symmetries for ensemble improvement.
• In TQFT, automorphism-weighted groupoid measures generalize to cases with continuous families (e.g., conformal manifolds for noncompact TQFTs) with the ensemble measure integrating both automorphism weights and geometric measures (e.g., Zamolodchikov measures in CFT moduli space) (Barbar, 6 Nov 2025).
• Ensemble approaches conjecturally extend to baby universe Hilbert space averages and to statistical models of ensemble duality in quantum gravity.

7. Connections to the Siegel–Weil Formula and TQFT Gravity

Automorphism-weighted ensemble sums in TQFT, specifically for boundary CFTs classified by Lagrangian algebras, are the nonabelian generalization of the Siegel–Weil formula: $x$6 where $x$7, connecting boundary theory averages to bulk gravitational path integrals via automorphism group measures (Barbar, 6 Nov 2025). In the large-genus limit, groupoid sums over automorphism classes precisely match gravitational ensemble averages, providing a holographic perspective on ensemble duality in quantum field theory and quantum gravity.

Summary Table: Representative Domains and Automorphism-Weighted Ensemble Realizations

Domain	Objects	Weight Definition	Key Role
Classical & Quantum LDPC Codes	Codes, syndromes	$x$8 from symmetry/order	Improved decoding by fusion of BP/SC outputs
Topological QFT/Holography	Lagrangian algebras	$x$9	Groupoid averaging, holographic dualities, baby universes

Automorphism-weighted ensembles, by exploiting and integrating over symmetry, enable systematic and principled improvement in inference across both discrete and field-theoretic settings, tightly linking performance gains to group-theoretic structure and groupoid measures (Koutsioumpas et al., 3 Mar 2025, Barbar, 6 Nov 2025, Ma et al., 2024, Bioglio et al., 2022, Pillet et al., 2021).