Quantum Phase Estimation Using Multivalued Logic

Abstract

Quantum phase estimation (QPE) is one of the most important quantum algorithms which is used as a subroutine for other important quantum algorithms like Shor's factoring algorithm, simulation of quantum systems, quantum counting and QFT on arbitrary Zp. In this paper we develop the theoretical framework for the multivalued quantum logic version of the QPE algorithm using d valued qudits and show a quantum circuit to implement QPE with a complexity of O(nlogn) single qudit operations. The multivalued QPE algorithm, when compared to the binary quantum logic version, turns out to be more robust and leads to a significant decrease in the number of qudits required along with drastic improvement in the precision and success probability. We derive the requirements to amplify the probability of success to a value very close to 1 (for a given precision), thereby generalizing the previously obtained result in the binary case. Also, we note that the failure probability of QPE algorithm decreases exponentially as d increases.