Entanglement Distance of Two- and Multi-Qubit Variational States and Its Quantification with Quantum Computing

TL;DR

This paper analyzes the entanglement distance in variational quantum states for two- and multi-qubit systems, deriving analytical expressions and exploring how quantum correlations spread with circuit depth.

Contribution

It provides analytical formulas for entanglement distance in variational circuits and extends the analysis to multi-qubit chains, supported by numerical validation.

Findings

Entanglement distance increases with circuit depth.

Quantum correlations spread across more qubits as depth increases.

Analytical expressions match numerical simulations.

Abstract

We study the entanglement distance of variational quantum states for two-qubit and multi-qubit systems. These states are constructed using variational quantum circuits with $R_Y$ rotations and entangling $CZ$ gates.For the two-qubit case, we analytically derive recurrence relations for expectation values of Pauli observables using. This approach allows us to analytically calculate quantum correlators and evaluate the entanglement distance depending on the circuit parameters and depth. The analysis were extended to a closed one-dimensional chain of $N$ qubits. It is shown that with increasing circuit depth, more qubits influence a given qubit, which reflects the spreading of quantum correlations in the system. For a closed one-dimensional chain of $N$ qubits, explicit analytical expressions are derived for the case of two layers. The results are compared with numerical simulations performed using quantum programming tools. The results agree with the theoretical predictions.