Due to the great difficulty in scalability, quantum computers are limited in the number of qubits during the early stages of the quantum computing regime. In addition to the required qubits for storing the corresponding eigenvector, suppose we have additional k qubits available. Given such a constraint k, we propose an approach for the phase estimation for an eigenphase of exactly n-bit precision. This approach adopts the standard recursive circuit for quantum Fourier transform (QFT) in [R. Cleve and J. Watrous, Fast parallel circuits for quantum fourier transform, Proc. 41st Annual Symp. on Foundations of Computer Science (2000), pp. 526–536.] and adopts classical bits to implement such a task. Our algorithm has the complexity of O(n log k), instead of O(n2) in the conventional QFT, in terms of the total invocation of rotation gates. We also design a scheme to implement the factorization algorithm by using k available qubits via either the continued fractions approach or the simultaneous Diophantine approximation.
Simultaneous diophantine approximation
