Pith Number

pith:CQZUPSJB

pith:2026:CQZUPSJBZLIT5WI674XLV4S5RM

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Graphical Algebraic Geometry: From Ideals and Varieties to Quantum Calculi

Aleks Kissinger, Dichuan Gao, Razin A. Shaikh

Graphical Algebraic Geometry supplies diagrammatic languages that are universal and complete for commutative algebras and affine varieties.

arxiv:2605.13993 v1 · 2026-05-13 · quant-ph · cs.LO · math.CT

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Record completeness

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3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We construct several languages within this family and prove that they are universal and complete for the corresponding (co)span semantics of commutative algebras and affine varieties.

C2weakest assumption

The assumption that the chosen diagrammatic generators and relations exactly capture the (co)span semantics of commutative algebras without introducing or losing algebraic information.

C3one line summary

Graphical Algebraic Geometry creates universal diagrammatic languages for commutative algebras and affine varieties that also characterize the qudit ZH calculus for quantum computation.

References

75 extracted · 75 resolved · 11 Pith anchors

[1] Geometry of 3D Environments and Sum of Squares Polynomials 2017 · arXiv:1611.07369

[2] Asymptotically Good Quantum Codes 2001 · doi:10.1103/physreva

[3] Miriam Backens and Aleks Kissinger. 2019. ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity.Electronic Proceed- ings in Theoretical Computer Science287 (Jan. 2019 · doi:10.4204/eptcs

[4] Miriam Backens, Aleks Kissinger, Hector Miller-Bakewell, John van de Wetering, and Sal Wolffs. 2023. Completeness of the ZH-calculus.Compositionality5 (July 2023), 5. doi:10.32408/compositionality-5-5 2023 · doi:10.32408/compositionality-5-5

[6] Categories in Control 2015 · doi:10.48550/arxiv

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-17T23:39:13.230344Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

143347c921cad13ed91eff2ebaf25d8b03d84635516aabe6d7c4a41b34c126a4

Aliases

arxiv: 2605.13993 · arxiv_version: 2605.13993v1 · doi: 10.48550/arxiv.2605.13993 · pith_short_12: CQZUPSJBZLIT · pith_short_16: CQZUPSJBZLIT5WI6 · pith_short_8: CQZUPSJB

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/CQZUPSJBZLIT5WI674XLV4S5RM \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 143347c921cad13ed91eff2ebaf25d8b03d84635516aabe6d7c4a41b34c126a4

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "14e4703cd6dc219f21c080a6ad6ff1315a0fd7c0fb4dd3d15dfb79013bacd867",
    "cross_cats_sorted": [
      "cs.LO",
      "math.CT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "quant-ph",
    "submitted_at": "2026-05-13T18:05:02Z",
    "title_canon_sha256": "1e1104235b398b0fca58e28760fefe6cdaa20cc2d5d4ac5c4d944c7c53f068ae"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13993",
    "kind": "arxiv",
    "version": 1
  }
}