Noise analysis of Grover and phase estimation algorithms implemented as quantum singular value transformations for a small number of noisy qubits

The quantum singular value transformation (QSVT) algorithm is a general framework to implement most of the known algorithms and provides a way forward for designing new algorithms. In the present work, the impact of noise on the QSVT algorithm is examined for bit flip, phase flip, bit-phase flip, and depolarizing noise models for a small number of qubits. The small number of noisy qubits approximates the currently available noisy quantum computers. For simulation results, the QSVT implementation of the Grover search and quantum phase estimation (QPE) algorithms is considered. These algorithms are among the basic quantum algorithms and form the building blocks of various applications of quantum algorithms. The results showed that the QSVT implementation of the Grover search and QPE algorithms has a consistently worse dependence upon noise than the original implementation for all four noise models. The probability of success of the Grover algorithm and phase measured by the QPE algorithm were found to exponentially depend upon the error probability in the noisy channels but only linearly dependent on the number of qubits.