{ "_generated": "2026-04-09T15:13:48.408447+00:00", "_description": "Site-wide index of all papers in the RQM Technologies technical paper series.", "_source": "Generated by scripts/generate_index.py", "total": 4, "papers": [ { "paper_id": "qqc-001-foundations", "title": "Foundations of Quaternionic Quantum Computing: Qubits, Spinors, and SU(2) Geometry", "slug": "foundations", "version": "0.1.0", "status": "draft", "publication_date": null, "last_updated": "2026-04-09", "authors": [ { "name": "John G. Van Geem", "affiliation": "RQM Technologies", "orcid": "0009-0002-4003-8452", "email": null, "corresponding": true } ], "abstract": "This paper assembles a conservative mathematical foundation for the use of quaternions in single-qubit quantum computing. The goal is not to replace standard quantum mechanics, but to organize familiar structures in a way that makes their geometry more explicit. We distinguish carefully between normalized state vectors in C^2 ≅ R^4, the unit sphere S^3 inside that space, pure states modulo global phase, the projective space CP^1 ≅ S^2, the single-qubit gate group U(2), and SU(2) representatives obtained after removing global phase. Within that framework, we exhibit an explicit isomorphism between the unit quaternions and SU(2), explain why unit quaternions are therefore a natural model for single-qubit gates modulo global phase, and show how quaternion multiplication exposes the axis-angle and composition structure of rotations. Standard facts are separated from the organizational choices adopted in the RQM Technologies series. The paper is restricted to single-qubit structure and makes no claim about new physics, multi-qubit quaternionic state evolution, or computational advantage.", "keywords": [ "qubit geometry", "spinors", "SU(2)", "unit quaternions", "Bloch sphere", "single-qubit gates" ], "series": "RQM Quaternionic Quantum Computing Series", "series_number": 1, "depends_on": [], "canonical_url": null, "pdf_url": null, "html_url": null, "jats_url": null, "doi": null, "journal": null, "volume": null, "issue": null, "pages": null, "citations": [ "adler1995", "bloch1946", "hall2015", "kuipers1999", "nielsen2010", "penrose1984" ], "related_artifacts": [ "main.tex", "artifacts/figures/fig-spinor-quotient.svg", "artifacts/figures/fig-su2-quaternion.svg", "artifacts/figures/fig-axis-angle.svg", "artifacts/notebooks/README.md", "artifacts/code/README.md" ], "license": "CC-BY-4.0", "notes": "Draft foundation paper for the series. This metadata mirrors the current authored source and remains explicitly limited to standard single-qubit mathematics plus project-specific framing choices.", "_path": "papers/qqc-001-foundations/", "_package_index": "papers/qqc-001-foundations/", "_html": "papers/qqc-001-foundations/paper.html", "_pdf": "papers/qqc-001-foundations/paper.pdf", "_metadata": "papers/qqc-001-foundations/metadata.json", "_claims": "papers/qqc-001-foundations/claims.json", "_notation": "papers/qqc-001-foundations/notation.json", "_glossary": "papers/qqc-001-foundations/glossary.json", "_jats": "papers/qqc-001-foundations/paper.jats.xml", "_bib": "papers/qqc-001-foundations/references.bib" }, { "paper_id": "qqc-002-canonical-ir", "title": "A Canonical Quaternionic Intermediate Representation for Single-Qubit Quantum Circuits", "slug": "canonical-ir", "version": "0.1.0", "status": "draft", "publication_date": null, "last_updated": "2026-04-09", "authors": [ { "name": "John G. Van Geem", "affiliation": "RQM Technologies", "orcid": "0009-0002-4003-8452", "email": null, "corresponding": true } ], "abstract": "This paper defines a canonical quaternionic intermediate representation for single-qubit circuit segments. Building on the fixed quaternion-SU(2) correspondence established in qqc-001-foundations, it treats the physically relevant action of a single-qubit gate modulo global phase by first choosing an SU(2) representative and then representing that representative by a unit quaternion q = w + x i + y j + z k. The paper introduces a compiler-facing IR object u1q(w, x, y, z), formalizes its unit-norm and sign-ambiguity constraints, and defines deterministic normalization and canonicalization rules. The canonical representative is chosen from {q, -q} by a closed-hemisphere convention w > 0, together with an explicit equatorial tie-break when w = 0. This yields a shortest-geodesic, deterministic internal form suitable for fusion and rewrite pipelines. Explicit translations are given for the identity, Pauli gates, H, S, T and their adjoints, and axis rotations Rx(theta), Ry(theta), Rz(theta). Composition of adjacent single-qubit gates is shown to correspond to quaternion multiplication followed by re-canonicalization, and worked examples illustrate the relation between named gate syntax, canonical IR, and single-qubit segments extracted from larger circuits. The scope is limited to single-qubit gate action modulo global phase; no multi-qubit IR, optimization guarantee, or hardware claim is made.", "keywords": [ "quaternionic intermediate representation", "single-qubit circuits", "compiler IR", "SU(2)", "unit quaternions", "circuit canonicalization" ], "series": "RQM Quaternionic Quantum Computing Series", "series_number": 2, "depends_on": [ "qqc-001-foundations" ], "canonical_url": null, "pdf_url": null, "html_url": null, "jats_url": null, "doi": null, "journal": null, "volume": null, "issue": null, "pages": null, "citations": [ "cross2022", "hall2015", "kuipers1999", "nielsen2010", "smith2016" ], "related_artifacts": [ "main.tex", "artifacts/figures/fig-ir-pipeline.svg", "artifacts/figures/fig-hemisphere-canonicalization.svg", "artifacts/figures/fig-segment-extraction.svg", "artifacts/notebooks/README.md", "artifacts/code/README.md" ], "license": "CC-BY-4.0", "notes": "Draft representation paper defining a canonical quaternionic IR for single-qubit circuit segments. This paper depends conceptually on qqc-001-foundations for the fixed quaternion-SU(2) convention and stays explicitly within the single-qubit scope.", "_path": "papers/qqc-002-canonical-ir/", "_package_index": "papers/qqc-002-canonical-ir/", "_html": "papers/qqc-002-canonical-ir/paper.html", "_pdf": "papers/qqc-002-canonical-ir/paper.pdf", "_metadata": "papers/qqc-002-canonical-ir/metadata.json", "_claims": "papers/qqc-002-canonical-ir/claims.json", "_notation": "papers/qqc-002-canonical-ir/notation.json", "_glossary": "papers/qqc-002-canonical-ir/glossary.json", "_jats": "papers/qqc-002-canonical-ir/paper.jats.xml", "_bib": "papers/qqc-002-canonical-ir/references.bib" }, { "paper_id": "qqc-003-gate-fusion", "title": "Quaternionic Gate Fusion on S^3: Geometry-Native Optimization for Single-Qubit Circuit Segments", "slug": "gate-fusion", "version": "0.1.0", "status": "draft", "publication_date": null, "last_updated": "2026-04-09", "authors": [ { "name": "John G. Van Geem", "affiliation": "RQM Technologies", "orcid": "0009-0002-4003-8452", "email": null, "corresponding": true } ], "abstract": "This paper studies a local optimization pass for single-qubit circuit segments using the canonical quaternionic intermediate representation defined in qqc-002-canonical-ir. A maximal run of adjacent single-qubit gates on one wire is translated into unit quaternions, multiplied in circuit order, re-normalized, and sign-canonicalized on the quotient q ~ -q induced by global-phase removal. The result is a unique canonical fused representative in S^3 for the segment action modulo global phase. A layered reconstruction strategy then emits either no gate, an exact named gate, an exact axis-aligned rotation, or a generic single-qubit primitive when no simpler interpretable output is available. The paper proves exact equivalence preservation up to global phase, determinism of the fused representative under the fixed canonicalization rule, and minimality of the fully fused segment in the quaternionic IR itself. A small reference benchmark corpus shipped with the paper package is evaluated by an included plain-Python harness. On 13 measured segments containing 36 input single-qubit gates, the local fusion pass reconstructs to 11 output gates while preserving exact segment action under the authoritative canonical-quaternion equivalence check. Two segments eliminate to identity, and 8 of 13 outputs are recovered as either named gates or axis-aligned rotations. These measurements are limited to the included single-qubit reference corpus and are not presented as a claim of universal compiler superiority.", "keywords": [ "quaternionic gate fusion", "single-qubit optimization", "compiler method", "SU(2)", "unit quaternions", "local circuit simplification" ], "series": "RQM Quaternionic Quantum Computing Series", "series_number": 3, "depends_on": [ "qqc-001-foundations", "qqc-002-canonical-ir" ], "canonical_url": null, "pdf_url": null, "html_url": null, "jats_url": null, "doi": null, "journal": null, "volume": null, "issue": null, "pages": null, "citations": [ "cross2022", "hall2015", "kuipers1999", "nielsen2010", "smith2016" ], "related_artifacts": [ "main.tex", "artifacts/figures/fig-fusion-pipeline.svg", "artifacts/figures/fig-canonical-hemisphere.svg", "artifacts/figures/fig-segment-boundaries.svg", "artifacts/figures/fig-before-after.svg", "artifacts/code/benchmark_gate_fusion.py", "artifacts/code/benchmark_results.json", "artifacts/notebooks/README.md", "artifacts/code/README.md" ], "license": "CC-BY-4.0", "notes": "Draft optimization-method paper defining and evaluating a local quaternionic gate-fusion pass for single-qubit circuit segments. The benchmark numbers reported in the paper are computed by the shipped plain-Python harness on a small reference corpus rather than inferred or fabricated.", "_path": "papers/qqc-003-gate-fusion/", "_package_index": "papers/qqc-003-gate-fusion/", "_html": "papers/qqc-003-gate-fusion/paper.html", "_pdf": "papers/qqc-003-gate-fusion/paper.pdf", "_metadata": "papers/qqc-003-gate-fusion/metadata.json", "_claims": "papers/qqc-003-gate-fusion/claims.json", "_notation": "papers/qqc-003-gate-fusion/notation.json", "_glossary": "papers/qqc-003-gate-fusion/glossary.json", "_jats": "papers/qqc-003-gate-fusion/paper.jats.xml", "_bib": "papers/qqc-003-gate-fusion/references.bib" }, { "paper_id": "qqc-004-verification-equivalence", "title": "Verification, Canonicalization, and Equivalence in Quaternionic Quantum Compilation", "slug": "verification-equivalence", "version": "0.1.0", "status": "draft", "publication_date": null, "last_updated": "2026-04-09", "authors": [ { "name": "John G. Van Geem", "affiliation": "RQM Technologies", "orcid": "0009-0002-4003-8452", "email": null, "corresponding": true } ], "abstract": "This paper addresses the trust problem for quaternionic optimization in single-qubit quantum compilation. Building on qqc-001-foundations, qqc-002-canonical-ir, and qqc-003-gate-fusion, it defines verification criteria for quaternionic compilation outputs, including exact SU(2) equivalence modulo global phase, canonical sign conventions on the quotient q ~ -q, invariants preserved by translation and fusion, and reproducible output forms for deterministic comparison. The paper identifies canonical quaternion equality after normalization and sign-canonicalization as the authoritative equivalence check for single-qubit segment actions, then relates that check to interoperable matrix-based diagnostics such as phase-aware residuals, trace overlap, and process fidelity under the fixed quaternion-SU(2) correspondence. A shipped reference verifier evaluates positive and negative segment pairs, including named-gate recoveries, identity elimination, generic u1q outputs, wrong-axis substitutions, and perturbed near-matches. On the included seven-case corpus, all four positive cases pass and all three negative cases are rejected by the authoritative canonical check, while the auxiliary diagnostics provide interpretable matrix-space evidence for external toolchains. The paper is limited to single-qubit compilation segments and does not claim full-circuit verification completeness or universal compiler correctness.", "keywords": [ "quantum compilation verification", "canonical equivalence", "global-phase-aware equivalence", "SU(2)", "unit quaternions", "compiler validation" ], "series": "RQM Quaternionic Quantum Computing Series", "series_number": 4, "depends_on": [ "qqc-001-foundations", "qqc-002-canonical-ir", "qqc-003-gate-fusion" ], "canonical_url": null, "pdf_url": null, "html_url": null, "jats_url": null, "doi": null, "journal": null, "volume": null, "issue": null, "pages": null, "citations": [ "cross2022", "hall2015", "kuipers1999", "nielsen2010", "smith2016" ], "related_artifacts": [ "main.tex", "artifacts/figures/fig-verification-pipeline.svg", "artifacts/figures/fig-canonical-hemisphere.svg", "artifacts/figures/fig-matrix-interoperability.svg", "artifacts/figures/fig-positive-negative-cases.svg", "artifacts/code/verify_canonical_equivalence.py", "artifacts/code/verification_results.json", "artifacts/notebooks/README.md", "artifacts/code/README.md" ], "license": "CC-BY-4.0", "notes": "Draft trust-layer paper defining verification and canonical equivalence procedures for quaternionic single-qubit compilation. The shipped verifier computes both authoritative canonical-quaternion checks and matrix-based diagnostics on a small reference corpus.", "_path": "papers/qqc-004-verification-equivalence/", "_package_index": "papers/qqc-004-verification-equivalence/", "_html": "papers/qqc-004-verification-equivalence/paper.html", "_pdf": "papers/qqc-004-verification-equivalence/paper.pdf", "_metadata": "papers/qqc-004-verification-equivalence/metadata.json", "_claims": "papers/qqc-004-verification-equivalence/claims.json", "_notation": "papers/qqc-004-verification-equivalence/notation.json", "_glossary": "papers/qqc-004-verification-equivalence/glossary.json", "_jats": "papers/qqc-004-verification-equivalence/paper.jats.xml", "_bib": "papers/qqc-004-verification-equivalence/references.bib" } ] }