# Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms

## TL;DR

This paper introduces descriptors to evaluate the expressibility and entangling capability of parameterized quantum circuits, aiding in the design of efficient hybrid quantum-classical algorithms by analyzing various circuit structures and their performance.

## Contribution

It proposes statistical measures for circuit expressibility and entangling capability, and systematically compares different circuit structures and gate sequences to identify optimal configurations.

## Key findings

- Two-qubit gates in ring or all-to-all arrangements outperform line configurations.
- Sequences of controlled X-rotation gates are more expressive than controlled Z-rotation sequences.
- Expressibility saturates with circuit depth, serving as a distinguishing feature of circuit templates.

## Abstract

Parameterized quantum circuits play an essential role in the performance of many variational hybrid quantum-classical (HQC) algorithms. One challenge in implementing such algorithms is to choose an effective circuit that well represents the solution space while maintaining a low circuit depth and number of parameters. To characterize and identify expressible, yet compact, parameterized circuits, we propose several descriptors, including measures of expressibility and entangling capability, that can be statistically estimated from classical simulations of parameterized quantum circuits. We compute these descriptors for different circuit structures, varying the qubit connectivity and selection of gates. From our simulations, we identify circuit fragments that perform well with respect to the descriptors. In particular, we quantify the substantial improvement in performance of two-qubit gates in a ring or all-to-all connected arrangement compared to that of those on a line. Furthermore, we quantify the improvement in expressibility and entangling capability achieved by sequences of controlled X-rotation gates compared to sequences of controlled Z-rotation gates. In addition, we investigate how expressibility "saturates" with increased circuit depth, finding that the rate and saturated-value appear to be distinguishing features of a parameterized quantum circuit template. While the correlation between each descriptor and performance of an algorithm remains to be investigated, methods and results from this study can be useful for both algorithm development and design of experiments for general variational HQC algorithms.