Arithmetic Depth, Minimal Gate Alphabets, and Exact Program-State Resources for Single-Qubit Dyadic Programs

Abstract

Resource analyses for quantum programs commonly provide safe upper bounds on qubits, gates, or non-Clifford operations. We study a single-qubit dyadic fragment in which the deepest exact resource layer can instead be inferred exactly. Starting from the normalized-midpoint misty-state syntax, we define dyadic program states and the real cyclotomic tower K_D=ℚ(2cos(π/2^D)). For the projective single-qubit Clifford-cyclotomic tower Ĝ_D⊆ SO(3), we prove the relative intersection law Ĝ_D∩ SO(3,K_d)=Ĝ_d for d≤ D. Combining this descent theorem with an arithmetic-closure invariant for adaptive stabilizer processing yields Fdepth(U)=Adepth(U)=Rdepth(U): the smallest real cyclotomic field containing the Bloch matrix of an exactly synthesizable channel, the shallowest dyadic gate alphabet realizing it up to global phase, and the shallowest dyadic program-state layer sufficient for exact stabilizer-assisted implementation coincide. At that common minimum, known fixed-alphabet synthesis gives minimum primitive-rotation count. We package the result as an exact static resource effect and outline a certificate-producing compiler.

Rewire and reset are all you need: Minimal resources for manipulating non-Abelions
Chiu Fan Bowen Lo, Rohith Sajith, Anasuya Lyons +2
arXiv5 Oct 2026PreprintarXiv licenceFull textCondensed matter Quantum physics & quantum informationarXiv
Form Factor Alphabets and Antipodal Duality from Cluster Algebras
Song He, Jiahao Liu
arXiv4 Sep 2026PreprintarXiv licenceFull textHigh-energy & particle physicsarXiv
Scalable Postselection of Quantum Resources
J. Wilson Staples, Winston Fu, Jeff D. Thompson
PRX Quantum13 Oct 2026Peer-reviewedPublisher termsQuantum physics & quantum informationCrossref
Symmetry-resolved parent Hamiltonians for entangled bosonic cat resources
Šimon Bräuer, Klaus Mølmer
arXiv1 Oct 2026PreprintarXiv licenceFull textQuantum physics & quantum informationarXiv
Inequivalent Quantum Resources from Multipartite State Discrimination
Srijani Das, Tuhin Paul, Manasi Patra +2
arXiv7 Oct 2026PreprintCC BY 4.0Full textQuantum physics & quantum informationarXiv

Cite

BibTeX RIS

BibTeX

@misc{newton2026arithmetic,
  title = {{Arithmetic Depth, Minimal Gate Alphabets, and Exact Program-State Resources for Single-Qubit Dyadic Programs}},
  author = {Tony Newton and Dan-Adrian German},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06909},
  eprint = {2611.06909},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06909},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Arithmetic Depth, Minimal Gate Alphabets, and Exact Program-State Resources for Single-Qubit Dyadic Programs
AU  - Tony Newton
AU  - Dan-Adrian German
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06909
UR  - https://arxiv.org/abs/2611.06909
AB  - Resource analyses for quantum programs commonly provide safe upper bounds on qubits, gates, or non-Clifford operations. We study a single-qubit dyadic fragment in which the deepest exact resource layer can instead be inferred exactly. Starting from the normalized-midpoint misty-state syntax, we define dyadic program states and the real cyclotomic tower K_D=ℚ(2cos(π/2^D)). For the projective single-qubit Clifford-cyclotomic tower Ĝ_D⊆ SO(3), we prove the relative intersection law Ĝ_D∩ SO(3,K_d)=Ĝ_d for d≤ D. Combining this descent theorem with an arithmetic-closure invariant for adaptive stabilizer processing yields Fdepth(U)=Adepth(U)=Rdepth(U): the smallest real cyclotomic field containing the Bloch matrix of an exactly synthesizable channel, the shallowest dyadic gate alphabet realizing it up to global phase, and the shallowest dyadic program-state layer sufficient for exact stabilizer-assisted implementation coincide. At that common minimum, known fixed-alphabet synthesis gives minimum primitive-rotation count. We package the result as an exact static resource effect and outline a certificate-producing compiler.
ER  -

Abstract

Related

Cite