Random-projector quantum diagnostics of Ramsey numbers and a prime-factor heuristic for $R(5,5)=45$
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909780868595712 |
|---|---|
| author | Tamburini, Fabrizio |
| author_facet | Tamburini, Fabrizio |
| contents | We introduce a statistical framework for estimating Ramsey numbers by embedding two-color Ramsey instances into a $Z_2 \times Z_2$-graded Majorana algebra. This approach replaces brute-force enumeration with two randomized spectral diagnostics applied to operators of a given dimension d associated with Ramsey numbers: a linear projector $P_{lin}$ and an exponential map $P_{exp}(α)$, suitable for both classical and quantum computation. In the diagonal case, both diagnostics identify R(5,5) at n=45. The quantum realizations act on a reduced module and therefore require only five data qubits plus a few ancillas via block-encoding/qubitization for R(5,5)=45, in stark contrast to the $\binom{n}{2} \approx 10^3$ logical qubits demanded by direct edge encodings. We also provide few-qubit estimates for R(6,6) and R(7,7), and propose a simple "prime-sequence" consistency heuristic that connects R(5,5)=45 to constrained diagonal growth. Our method echoes Erdős's probabilistic paradigm, emphasizing randomized arguments rather than explicit colorings, and parallels the classical coin-flip approach to Ramsey bounds. Finally, we discuss potential applications of this framework to machine learning with a limited number of qubits. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16699 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random-projector quantum diagnostics of Ramsey numbers and a prime-factor heuristic for $R(5,5)=45$ Tamburini, Fabrizio Quantum Physics Combinatorics We introduce a statistical framework for estimating Ramsey numbers by embedding two-color Ramsey instances into a $Z_2 \times Z_2$-graded Majorana algebra. This approach replaces brute-force enumeration with two randomized spectral diagnostics applied to operators of a given dimension d associated with Ramsey numbers: a linear projector $P_{lin}$ and an exponential map $P_{exp}(α)$, suitable for both classical and quantum computation. In the diagonal case, both diagnostics identify R(5,5) at n=45. The quantum realizations act on a reduced module and therefore require only five data qubits plus a few ancillas via block-encoding/qubitization for R(5,5)=45, in stark contrast to the $\binom{n}{2} \approx 10^3$ logical qubits demanded by direct edge encodings. We also provide few-qubit estimates for R(6,6) and R(7,7), and propose a simple "prime-sequence" consistency heuristic that connects R(5,5)=45 to constrained diagonal growth. Our method echoes Erdős's probabilistic paradigm, emphasizing randomized arguments rather than explicit colorings, and parallels the classical coin-flip approach to Ramsey bounds. Finally, we discuss potential applications of this framework to machine learning with a limited number of qubits. |
| title | Random-projector quantum diagnostics of Ramsey numbers and a prime-factor heuristic for $R(5,5)=45$ |
| topic | Quantum Physics Combinatorics |
| url | https://arxiv.org/abs/2508.16699 |