Quantum measurements and the Abelian Stabilizer Problem

TL;DR

This paper presents a polynomial-time quantum algorithm for the Abelian Stabilizer Problem, which encompasses factoring and discrete logarithm as special cases. The key idea is to leverage quantum measurements of eigenvalues and a generalized quantum Fourier transform on finite Abelian groups to extract stabilizer information, enabling recovery of a polynomial-size basis for the stabilizer. The work also develops a reversible, garbage-free framework for quantum computation, including a reversible construction of the QFT and controlled operations, along with a detailed treatment of precision and measurement. Together, these components provide an alternative, information-theoretic route to Shor-type quantum speedups and deepen the foundational understanding of quantum computation and measurements.

Abstract

We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor's results. Our method is based on a procedure for measuring an eigenvalue of a unitary operator. Another application of this procedure is a polynomial quantum Fourier transform algorithm for an arbitrary finite Abelian group. The paper also contains a rather detailed introduction to the theory of quantum computation.