"Classical" phase estimation versus iterative phase estimation

Question

In the article Arbitrary accuracy iterative phase estimation algorithm as a two qubit benchmark, the authors introduced implementation of phase estimation with two qubits only. The trick that bits representing a phase are found piece by piece. Although, this is at the cost of repeated calculations, in the end the number of qubits saved is material. Moreover, the circuit is much less complex. Both these features seems to me crucial in NISQ era.

I realized that such approach to the phase estimation was exploited in hybrid HHL algorithm (Large-scale quantum hybrid solution for linear systems of equations) and something similar in implementation of QFT in Shor's algorithm (Experimental Study of Shor's Factoring Algorithm on IBM Q).

Although the iterative phase estimation was introduced more than 14 years ago, I did not notice their significant exploitation. Therefore, my question is why this approach is not widely used? Is there any factor that hinders its application?

Answer 1

The iterative Quantum Phase Estimation(IQPE) procedure tries to measure the phase associated with a unitary matrix one bit at a time. While the idea of phase kickback used in the original QPE algorithm is used in IQPE too, one crucial aspect of the circuit is the classically controlled rotations.

Since we use the measurement values of the previously determined bits, implementing classically controlled operations is the main difference in executing an IQPE circuit vs executing a normal QPE circuit.

Although IBMs qiskit has this functionality, I personally think that due to a lack of awareness of this idea or the algorithm, this is not widely used despite a reduction in the number of qubits.

PS. I wrote this blog about how we can actually implement IQPE without using classically controlled gates. Maybe this would be a better way to familiarize people with IQPE!