  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 of
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. 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 $8$ 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.

Quantum Physics