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Quantum eigenvalue transformation (QET) and its generalization, quantumsingular value transformation (QSVT), are versatile quantum algorithms thatallow us to apply broad matrix functions to quantum states, which cover many ofsignificant quantum algorithms such as Hamiltonian simulation. However, findinga parameter set which realizes preferable matrix functions in these techniquesis difficult for large-scale quantum systems: there is no analytical resultother than trivial cases as far as we know and we often suffer also fromnumerical instability. We propose recursive QET or QSVT (r-QET or r-QSVT), inwhich we can execute complicated matrix functions by recursively organizingblock-encoding by low-degree QET or QSVT. Owing to the simplicity of recursiverelations, it works only with a few parameters with exactly determining theparameters, while its iteration results in complicated matrix functions. Inparticular, by exploiting the recursive relation of Newton iteration, weconstruct the matrix sign function, which can be applied for eigenstatefiltering for example, in a tractable way. We show that ananalytically-obtained parameter set composed of only $8$ different values issufficient for executing QET of the matrix sign function with an arbitrarilysmall error $\varepsilon$. Our protocol will serve as an alternative protocolfor constructing QET or QSVT for some useful matrix functions without numericalinstability.

Recursive Quantum Eigenvalue/Singular-Value Transformation: Analytic Construction of Matrix Sign Function by Newton Iteration