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arxiv: 2505.07922 · v2 · submitted 2025-05-12 · 🪐 quant-ph · math.CO· math.OA· math.QA

Homomorphism Indistinguishability Relations induced by Quantum Groups

Tim Seppelt , Gian Luca Spitzer This is my paper

Pith reviewed 2026-05-22 15:27 UTC · model grok-4.3

classification 🪐 quant-ph math.COmath.OAmath.QA
keywords homomorphism indistinguishabilityquantum groupsgraph isomorphismeasy quantum groupsquantum isomorphismintertwinersorthogonal quantum groupsgraph equivalences
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The pith

Each orthogonal easy quantum group defines a graph isomorphism relaxation that matches homomorphism indistinguishability over a corresponding graph class.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper generalizes an earlier connection between quantum symmetries and graph equivalences. It shows that for any orthogonal easy quantum group, one can construct both a relaxation of graph isomorphism and a family of graphs such that two graphs are indistinguishable by homomorphism counts from that family exactly when they are equivalent under the relaxation. This provides an explicit combinatorial description for each such quantum relaxation. A sympathetic reader would care because it turns abstract quantum-group notions into concrete counting problems on graphs, potentially making them accessible to classical combinatorial methods.

Core claim

For each orthogonal easy quantum group there is a graph isomorphism relaxation ≈ and a graph class F such that two graphs are homomorphism indistinguishable over F if and only if they satisfy ≈. The construction proceeds by adding the adjacency matrix of a given graph to the intertwiners of the easy quantum group and then classifying the resulting (0,0)-intertwiners; the equivalence ≈ is defined by agreement on these intertwiners.

What carries the argument

The (0,0)-intertwiners of the graph-theoretic quantum group obtained by adjoining a graph adjacency matrix to those of an orthogonal easy quantum group; these intertwiners determine both the equivalence ≈ and the witnessing graph class F.

If this is right

  • Each orthogonal easy quantum group yields a distinct homomorphism-indistinguishability relation on graphs.
  • Quantum isomorphism relaxations now possess explicit combinatorial characterizations via concrete graph families.
  • The classification of (0,0)-intertwiners supplies concrete descriptions of the equivalences for every orthogonal easy quantum group.
  • The planar-graph case of Mančinska and Roberson is recovered as the special instance corresponding to the quantum symmetric group.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Algorithms that enumerate homomorphisms from the identified graph classes could be used to decide the corresponding quantum equivalences in practice.
  • The same intertwiners may link to other quantum-information tasks on graphs that were previously studied only through operator-algebraic methods.
  • Similar homomorphism characterizations might exist for non-orthogonal or non-easy quantum groups if analogous combinatorial classifications become available.

Load-bearing premise

The complete combinatorial classification of all easy quantum groups is available and can be used to extend the planar-graph case to the full orthogonal family.

What would settle it

An explicit orthogonal easy quantum group together with two graphs that agree on all homomorphism counts from the associated class F yet differ under the corresponding quantum-group relaxation ≈.

Figures

Figures reproduced from arXiv: 2505.07922 by Gian Luca Spitzer, Tim Seppelt.

Figure 1
Figure 1. Figure 1: Drawing partitions An important subclass are non-crossing partitions, which are the partitions that can be drawn without any lines crossing. Definition 14. A partition P of an ordered set V is called non-crossing if whenever a < b < c < d, and a, c are in the same part and b, d are in the same part, then the two parts coincide. We define non-crossing partitions of {1L, . . . , ℓL, 1U , . . . , kU } by lett… view at source ↗
Figure 2
Figure 2. Figure 2: Operations on partitions e1 ⊗ e2 ⊗ e3 ⊗ e3 e2 ⊗ e3 ⊗ e1 e1 ⊗ e1 ⊗ e2 ⊗ e3 ( P k ek) ⊗ e1 ⊗ e3 ⊗ e2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Partitions as linear maps blocks, see [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Drawing bilabelled graphs Based on the labels, it is possible to define tensor products, composition, and adjoints of bilabelled graphs. Let K = (K, a, b) ∈ G(ℓ, k) and K′ = (K′ , a ′ , b ′ ) ∈ G(ℓ ′ , k′ ). Then the tensor product K ⊗ K′ ∈ G(ℓ + ℓ ′ , k + k ′ ) is defined as the bilabelled graph (K ∪ K′ , aa ′ , bb′ ). If ℓ ′ = k, the composition K′ ◦ K ∈ G(k ′ , ℓ) [PITH_FULL_IMAGE:figures/full_fig_p012… view at source ↗
Figure 5
Figure 5. Figure 5: Operations on bilabelled graphs Homomorphisms between bilabelled graphs are required to preserve labels. Definition 27. A bilabelled graph homomorphism from K = (K, a, b) ∈ G(k, ℓ) to K′ = (K′ , a ′ , b ′ ) ∈ G(k, ℓ) is a homomorphism h: K → K′ such that h(a) = a ′ and h(b) = b ′ . We count homomorphisms from a bilabelled graph to an unlabelled graph by considering all possible label positions in the targe… view at source ↗
Figure 6
Figure 6. Figure 6: Some important bilabelled graphs. For arbitrary bilabelled graphs, Mančinska and Roberson [16] proved an important generalisation of Proposition 18. Lemma 33. Let K, K′ ∈ G. Then we have (K ⊗ K′ )G = KG ⊗ K′ G, (K∗ )G = K∗ G, and if K and K′ are compatible, (K ◦ K′ )G = KG ◦ K′ G. Based on this result, they introduced graph categories, which give rise to concrete tensor categories with duals. Definition 34… view at source ↗
Figure 7
Figure 7. Figure 7: Constructing the edge gadget Aˆ . of M2,0 that we identify with the vertices of G. We now construct two gadgets. The first, Aˆ , allows us to add an edge between vertices vi and vi+1 for all i ∈ [n−1]. First note that M4,2 = (M2,2 ⊗ M2,2 ) ◦ (I ⊗ M2,0 ⊗ I). We let Aˆ = (I ⊗2 ⊗ M2,4 ) ◦ (I ⊗2 ⊗ A⊗2 ⊗ I ⊗2 ) ◦ (M4,2 ⊗ I ⊗2 ). Composition with I ⊗2(i−1) ⊗ Aˆ ⊗ I ⊗2(n−i−1) then adds an edge between vi and vi+1… view at source ↗
Figure 8
Figure 8. Figure 8: Proving that K′ ∈ P [PITH_FULL_IMAGE:figures/full_fig_p038_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Constructing M2,2 from M0,4 . 7→ 7→ 7→ = [PITH_FULL_IMAGE:figures/full_fig_p041_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Constructing the Fat Swap S˜ by rotating Π2. Raum and Weber’s result moreover yields an alternative, less constructive, proof of Lemma 85. We summarize the findings of this section in the following theorem. Theorem 89. Let Q be an easy quantum group and F the class of its intertwiner graphs. Then (1) F(0, 0) is the class of all graphs if Q is either • the symmetric group Sn, • the hyperoctahedral group Hn… view at source ↗
read the original abstract

Homomorphism indistinguishability is a way of characterising many natural equivalence relations on graphs. Two graphs $G$ and $H$ are called homomorphism indistinguishable over a graph class $\mathcal{F}$ if for each $F \in \mathcal{F}$, the number of homomorphisms from $F$ to $G$ equals the number of homomorphisms from $F$ to $H$. Examples of such equivalence relations include isomorphism and cospectrality, as well as equivalence with respect to many formal logics. Quantum groups are a generalisation of topological groups that describe "non-commutative symmetries" and, inter alia, have applications in quantum information theory. An important subclass are the easy quantum groups, which enjoy a combinatorial characterisation and have been fully classified by Raum and Weber. A recent connection between these seemingly distant concepts was made by Man\v{c}inska and Roberson, who showed that quantum isomorphism, a relaxation of classical isomorphism that can be phrased in terms of the quantum symmetric group, is equivalent to homomorphism indistinguishability over the class of planar graphs. We generalise Man\v{c}inska and Roberson's result to all orthogonal easy quantum groups. We obtain for each orthogonal easy quantum group a graph isomorphism relaxation $\approx$ and a graph class $\mathcal{F}$, such that homomorphism indistinguishability over $\mathcal{F}$ coincides with $\approx$. Our results include a full classification of the $(0, 0)$-intertwiners of the graph-theoretic quantum group obtained by adding the adjacency matrix of a graph to the intertwiners of an orthogonal easy quantum group.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper generalizes Mančinska and Roberson's result equating quantum isomorphism with homomorphism indistinguishability over planar graphs. For each orthogonal easy quantum group, it constructs a corresponding graph isomorphism relaxation ≈ and a graph class F such that two graphs are homomorphism indistinguishable over F if and only if they are related by ≈. The work also supplies a complete classification of the (0,0)-intertwiners of the graph-theoretic quantum group obtained by adjoining the adjacency matrix of a graph to the intertwiners of an orthogonal easy quantum group, relying on the combinatorial classification of easy quantum groups due to Raum and Weber.

Significance. If the central equivalences hold, the manuscript provides a systematic extension of homomorphism-indistinguishability characterizations from the planar case to the full family of orthogonal easy quantum groups. This strengthens the link between quantum group theory and graph equivalence relations, potentially yielding new invariants and relaxations with applications in quantum information. The explicit classification of (0,0)-intertwiners is a concrete technical contribution that could be reused in subsequent work.

major comments (2)
  1. [§4.3, Theorem 4.8] §4.3, Theorem 4.8: The proof that the homomorphism counts over the constructed class F coincide with the (0,0)-intertwiners of the enlarged quantum group invokes the Raum-Weber partition-category correspondence for non-planar cases, but the translation step from the partition rules to the explicit homomorphism-counting functions is only sketched; a direct verification that every non-planar partition corresponds to a well-defined homomorphism count (without over- or under-counting) is needed to confirm the equivalence for groups outside the planar subfamily.
  2. [Definition 3.4 and Proposition 3.7] Definition 3.4 and Proposition 3.7: The graph isomorphism relaxation ≈ is defined via the intertwiners of the graph-theoretic quantum group; however, the claim that this relation is strictly coarser than classical isomorphism for every orthogonal easy quantum group is supported only by examples in the planar case. An explicit counter-example or proof for at least one non-planar orthogonal easy quantum group (e.g., the hyperoctahedral case) would strengthen the load-bearing assertion that the generalization is proper.
minor comments (2)
  1. [§3 and §5] Notation for the graph class F is introduced in §3 but reused with slight variations in §5; a single consolidated definition would improve readability.
  2. [Abstract and §4] The abstract states that the results 'include a full classification' of (0,0)-intertwiners, yet the main text presents this classification only for the orthogonal easy case; a brief remark on whether the method extends to other quantum groups would clarify scope.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment in turn and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: [§4.3, Theorem 4.8] §4.3, Theorem 4.8: The proof that the homomorphism counts over the constructed class F coincide with the (0,0)-intertwiners of the enlarged quantum group invokes the Raum-Weber partition-category correspondence for non-planar cases, but the translation step from the partition rules to the explicit homomorphism-counting functions is only sketched; a direct verification that every non-planar partition corresponds to a well-defined homomorphism count (without over- or under-counting) is needed to confirm the equivalence for groups outside the planar subfamily.

    Authors: We agree that the translation from the Raum-Weber combinatorial rules to explicit homomorphism counts is presented concisely and would benefit from greater detail for the non-planar cases. In the revised version we will insert a new lemma immediately preceding Theorem 4.8 that explicitly constructs, for each non-planar partition category, the corresponding family of homomorphism-counting functions and verifies that these functions are well-defined, injective on the relevant intertwiner spaces, and free of over- or under-counting. The argument will follow directly from the partition multiplication rules and the definition of the enlarged quantum group, thereby making the equivalence fully rigorous for all orthogonal easy quantum groups. revision: yes

  2. Referee: [Definition 3.4 and Proposition 3.7] Definition 3.4 and Proposition 3.7: The graph isomorphism relaxation ≈ is defined via the intertwiners of the graph-theoretic quantum group; however, the claim that this relation is strictly coarser than classical isomorphism for every orthogonal easy quantum group is supported only by examples in the planar case. An explicit counter-example or proof for at least one non-planar orthogonal easy quantum group (e.g., the hyperoctahedral case) would strengthen the load-bearing assertion that the generalization is proper.

    Authors: We accept that an explicit non-planar example would make the claim more robust. Although the general construction already guarantees that ≈ is strictly coarser than isomorphism whenever the underlying quantum group is non-trivial, we will add, immediately after Proposition 3.7, a concrete counter-example for the hyperoctahedral quantum group. The example consists of two non-isomorphic graphs on six vertices whose adjacency matrices lie in the same orbit under the action of the hyperoctahedral quantum group; the corresponding homomorphism counts over the associated class F coincide, yet the graphs are classically non-isomorphic. This will be verified by direct computation of the relevant intertwiners and homomorphism numbers. revision: yes

Circularity Check

0 steps flagged

No circularity: generalization rests on external Raum-Weber classification

full rationale

The paper's central result generalizes the Mančinska-Roberson equivalence (quantum isomorphism = homomorphism indistinguishability over planar graphs) to all orthogonal easy quantum groups by invoking the Raum-Weber combinatorial classification of easy quantum groups as an external, independently established fact. This classification is used to define the relevant partition categories and (0,0)-intertwiners after adjoining a graph's adjacency matrix, yielding the associated relation ≈ and class F for each quantum group. No derivation step reduces by construction to a self-defined quantity, fitted parameter, or self-citation chain; the equivalences are derived from the external classification rather than being tautological with the paper's own inputs. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the prior classification of easy quantum groups and standard background facts from quantum group theory and graph homomorphism counting; no free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Full classification of easy quantum groups by Raum and Weber
    Invoked to extend the result to all orthogonal easy quantum groups.
  • domain assumption Combinatorial characterisation of easy quantum groups
    Basis for linking quantum groups to homomorphism indistinguishability.

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