Improve the normalization bound for tilted operators

Improve the upper bound on the normalization factor E_b=\lambda_{\max}(D+bG) for quantum tilted walks, potentially by exploiting problem-specific spectral structure, especially for slowly mixing global chains.

Background

The amplitude analysis requires an upper bound on the largest eigenvalue E_b of the tilted operator D+bG, because normalization can offset the positive path contribution responsible for amplitude gain. The paper currently derives this bound from a log-Sobolev inequality and one-step quadratic variation. It explicitly leaves open whether stronger, problem-specific spectral information can produce better normalization certificates, particularly when the underlying chain mixes slowly.

References

The second is to improve the upper bound on $E_b$. Our present bound uses a log-Sobolev inequality and one-step quadratic variation; problem-specific spectral structure may yield a stronger normalization certificate, especially when the global chain mixes slowly.

— Super-Quadratic Quantum Speedups for Combinatorial Optimization via Tilted Walks  (2609.40088 - Ozgul et al., 30 Sep 2026) in Section 2, Discussion, paragraph “Future Work”

Determining the value of the exponent with this ideal choice of $b$ requires a significant numerical study, which we defer to future work.

— Super-Quadratic Quantum Speedups for Combinatorial Optimization via Tilted Walks  (2609.40088 - Ozgul et al., 30 Sep 2026) in Section 2, Discussion, paragraph “Comparison to Classical Algorithms and the Possibility of Further Dequantization”