Exact Chebyshev QFF does not extend to the α-perturbed n-cycle for α ≠ 0 due to eigenvalues outside [-1,1], but a truncated-Chebyshev LCU approximation achieves degree O(|α|t + √(t log(t/η))) that recovers the reversible √t scaling only when |α| = O(t^{-1/2}).
A unified framework of quantum walk search
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Quantum Fast-Forwarding Beyond Reversibility: The $\alpha$-Perturbed $n$-Cycle
Exact Chebyshev QFF does not extend to the α-perturbed n-cycle for α ≠ 0 due to eigenvalues outside [-1,1], but a truncated-Chebyshev LCU approximation achieves degree O(|α|t + √(t log(t/η))) that recovers the reversible √t scaling only when |α| = O(t^{-1/2}).