A Unified Quantum NOT Gate

TL;DR

This paper develops an analytical framework for implementing optimized approximate quantum NOT gates for states between two latitudes on the Bloch sphere, generalizing previous models and enabling sequential generation via matrix product states.

Contribution

It introduces a unified analytical formula for 1-to-M quantum NOT gates applicable to states between two latitudes, extending prior work on uniform and phase covariant states.

Findings

Derived an analytical formula for optimized 1-to-M quantum NOT gates.

Generalized previous models to states between two latitudes on the Bloch sphere.

Demonstrated implementation via sequential matrix product states.

Abstract

We study the feasibility of implementing a quantum NOT gate (approximate) when the quantum state lies between two latitudes on the Bloch's sphere and present an analytical formula for the optimized 1-to-$M$ quantum NOT gate. Our result generalizes previous results concerning quantum NOT gate for a quantum state distributed uniformly on the whole Bloch sphere as well as the phase covariant quantum state. We have also shown that such 1-to-$M$ optimized NOT gate can be implemented using a sequential generation scheme via matrix product states (MPS).