Recursive Quantum Eigenvalue/Singular-value Transformation*

Quantum eigenvalue transformation (QET) and quantum singular value transformation (QSVT), are versatile quantum algorithms that allow us to apply broad matrix functions to quantum states, covering quantum algorithms such as Hamiltonian simulation. However, finding a parameter set for preferable matrix functions is generally difficult: there is no analytical result other than trivial cases and we often suffer also from numerical instability. We propose recursive QET or QSVT (r-QET or r-QSVT), in which we can execute complicated matrix functions by recursively organizing block-encoding by low-degree QET or QSVT. Owing to the simplicity of recursive relations, it works only with a few parameters with exactly determining the parameters, while its iteration results in complicated matrix functions. In particular, Newton iteration allows us to analytically construct a parameter set of the matrix sign function, which can be applied for eigenstate filtering for example. The analytically-obtained parameter set composed of only different values is sufficient for executing QET of the matrix sign function with an arbitrarily small error . Our protocol will serve as an alternative protocol for constructing QET or QSVT for some useful matrix functions without numerical instability.