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