Random-projector quantum diagnostics of Ramsey numbers and a prime-factor heuristic for $R(5,5)=45$

TL;DR

This paper presents a quantum-inspired statistical framework for estimating Ramsey numbers using randomized spectral diagnostics, significantly reducing quantum resource requirements and introducing a heuristic linking R(5,5)=45 to prime sequences.

Contribution

It introduces a novel quantum-inspired spectral diagnostic method for estimating Ramsey numbers that reduces qubit requirements and connects these estimates to prime sequences.

Findings

Successfully identified R(5,5)=45 using quantum diagnostics.

Reduced quantum resource requirements compared to direct edge encoding.

Proposed a prime-sequence heuristic linking Ramsey numbers to prime properties.

Abstract

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 $P_{exp}(\alpha)$, 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 Erd\H{o}s'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.