Near-Heisenberg-limited parallel amplitude estimation with logarithmic depth circuit

Quantum amplitude estimation is one of the core subroutines in quantum algorithms. This paper gives a parallelized amplitude estimation (PAE) algorithm, that simultaneously achieves near-Heisenberg scaling in the total number of queries and sub-linear scaling in the circuit depth, with respect to the estimation precision. The algorithm is composed of a global GHZ state followed by separated low-depth Grover circuits; the number of qubits in the GHZ state and the depth of each circuit is tunable as a trade-off way, which particularly enables even near-Heisenberg-limited and logarithmic-depth algorithm for amplitude estimation. The quantum signal processing technique is effectively used to build the algorithm. The proposed algorithm has a form of distributed quantum computing, which may be suitable for device implementation.