Quantum Phase Estimation with an Arbitrary Number of Qubits

TL;DR

This paper introduces a quantum phase estimation method optimized for limited qubits, reducing complexity from quadratic to near-linear, and proposes a scheme for factorization using available qubits.

Contribution

It presents a novel quantum phase estimation algorithm that efficiently operates with an arbitrary number of qubits, improving complexity and enabling factorization.

Findings

Complexity reduced to O(n log k) from O(n^2)

Efficient phase estimation with limited qubits

Scheme for quantum factorization using classical approximation methods

Abstract

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) and adopts classical bits to implement such a task. Our algorithm has the complexity of O(n \log k), instead of O(n^2) 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.