In quantum phase estimation, the risk-minimizing input state for any loss function is the minimum eigenvector of a Toeplitz matrix of the loss's Fourier coefficients, and cosine states approximate it well.
Shor, Algorithms for quantum computation: discrete logarithms and factoring, in Proc
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Risk-minimizing states for the quantum-phase-estimation algorithm
In quantum phase estimation, the risk-minimizing input state for any loss function is the minimum eigenvector of a Toeplitz matrix of the loss's Fourier coefficients, and cosine states approximate it well.