Simplifying a classical-quantum algorithm interpolation with quantum singular value transformations

The problem of phase estimation (or amplitude estimation) admits a quadratic quantum speedup. Wang, Higgott, and Brierley [Wang, Brierley, Phys. Rev. Lett. 122, 140504 (2019)] have shown that there is continuous tradeoff between speedup circuit depth [by defining family algorithms known as $\ensuremath{\alpha}$-quantum $(\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{QPE})$]. In this paper, we show the scaling $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{QPE}$ can be naturally succinctly derived within framework singular value transformation (QSVT). From QSVT perspective, greater number coherent oracle calls translates into better polynomial approximation to sign function, which key routine for solving estimation. fewer samples one needs determine accurately. With idea, simplify proof $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{QPE}$, while providing an interpretation interpolation parameters, promising reasoning about classical-quantum interpolations.