Optimal Manipulations with Qubits: Universal NOT Gate

TL;DR

This paper introduces the Universal-NOT (U-NOT) gate for qubits, which approximates the complement of an unknown quantum state with optimal fidelity based on estimation, outperforming estimation-based methods when prior information is available.

Contribution

It defines and analyzes the U-NOT gate, showing its fidelity is tied to optimal estimation and can be improved with prior knowledge about the input state.

Findings

U-NOT gate fidelity is (N+1)/(N+2) for N input qubits

The U-NOT operation can be implemented via classical estimation and re-preparation

Prior information about the input state enhances the fidelity of the NOT operation

Abstract

It is not a problem to complement a classical bit, i.e. to change the value of a bit, a 0 to a 1 and vice versa. This is accomplished by a NOT gate. Complementing a qubit in an unknown state, however, is another matter. We show that this operation cannot be done perfectly. We define the Universal-NOT (U-NOT) gate which out of N identically prepared pure input qubits generates M output qubits in a state which is as close as possible to the perfect complement. This gate can be realized by classical estimation and subsequent re-preparation of complements of the estimated state. Its fidelity is therefore equal to the fidelity F= (N+1)/(N+2) of optimal estimation, and does not depend on the required number of outputs. We also show that when some additional a priori information about the state of input qubit is available, than the fidelity of the quantum NOT gate can be much better than the fidelity of estimation.