Quantum conditional operations

TL;DR

This paper characterizes when sets of quantum gates can be conditionally controlled, revealing a fundamental link between controllability and information extraction in quantum operations.

Contribution

It provides a necessary and sufficient condition for controllability of quantum gate sets and characterizes controllable sets in two dimensions.

Findings

Identifies conditions for quantum gate controllability.

Characterizes controllable sets in 2D quantum systems.

Links controllability to information extraction from quantum gates.

Abstract

An essential element of classical computation is the "if-then" construct, that accepts a control bit and an arbitrary gate, and provides conditional execution of the gate depending on the value of the controlling bit. On the other hand, quantum theory prevents the existence of an analogous universal construct accepting a control qubit and an arbitrary quantum gate as its input. Nevertheless, there are controllable sets of quantum gates for which such a construct exists. Here we provide a necessary and sufficient condition for a set of unitary transformations to be controllable, and we give a complete characterization of controllable sets in the two dimensional case. This result reveals an interesting connection between the problem of controllability and the problem of extracting information from an unknown quantum gate while using it.