Recursive quantum eigenvalue and singular-value transformation: Analytic construction of matrix sign function by Newton iteration

Quantum eigenvalue transformation (QET) and its generalization, quantum singular value transformation (QSVT), are versatile quantum algorithms that allow us to apply broad matrix functions to quantum states, which cover many significant quantum algorithms such as Hamiltonian simulation. However, finding a parameter set which realizes preferable matrix functions in these techniques is difficult for large-scale quantum systems: there is no analytical result other than trivial cases as far as we know and we often suffer also from numerical instability. In this Letter, 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, by exploiting the recursive relation of Newton iteration, we construct the matrix sign function, which can be applied for eigenstate filtering for example, in a tractable way. We show that an analytically obtained parameter set composed of only eight different values is sufficient for executing QET of the matrix sign function with an arbitrarily small error $\varepsilon$. Our protocol will serve as an alternative protocol for constructing QET or QSVT for some useful matrix functions without numerical instability.