The unitary dependence theory for understanding quantum circuits and states

TL;DR

This paper introduces a unitary dependence theory that characterizes quantum circuit behaviors by how gates influence qubit measurement probabilities, offering a practical and scalable alternative to entanglement-based descriptions.

Contribution

It develops a new dependence framework for quantum states and circuits, enhancing understanding and robustness over traditional entanglement-based methods.

Findings

Dependence on 1-qubit unitaries determines measurement probabilities.

CNOT gates copy dependence from control to target qubits.

Dependence picture offers practical insights and better scalability.

Abstract

We develop a unitary dependence theory to characterize the behaviors of quantum circuits and states in terms of how quantum gates manipulate qubits and determine their measurement probabilities. A qubit has dependence on a 1-qubit unitary gate if its measurement probabilities depend on the parameters of the gate. A 1-qubit unitary creates such a dependence onto the target qubit, and a CNOT gate copies all the dependences from the control qubit to the target qubit. The complete dependence picture of the output state details the connections qubits may have when being manipulated or measured. Compared to the conventional entanglement description of quantum circuits and states, the dependence picture offers more practical information, easier generalization to many-qubit systems, and better robustness upon partitioning of the system. The unitary dependence theory is a useful tool for understanding quantum circuits and states that is based on the practical ideas of how manipulations and measurements are affected by elementary gates.