Impact of Clifford operations on non-stabilizing power and quantum chaos

TL;DR

This paper investigates how Clifford and non-Clifford operations influence non-stabilizerness, thermalization, and quantum chaos in quantum circuits, revealing relationships and behaviors crucial for quantum computation and understanding quantum chaos.

Contribution

It establishes a direct link between non-stabilizing power and non-Clifford gates, demonstrating thermalization and its role in quantum chaos within mixed Clifford and non-Clifford circuits.

Findings

Non-stabilizing power relates directly to non-Clifford gate powers.

Non-stabilizing power thermalizes to Haar-averaged values in generic circuits.

Non-stabilizing power influences the emergence of quantum chaos.

Abstract

Non-stabilizerness, alongside entanglement, is a crucial ingredient for fault-tolerant quantum computation and achieving a genuine quantum advantage. Despite recent progress, a complete understanding of the generation and thermalization of non-stabilizerness in circuits that mix Clifford and non-Clifford operations remains elusive. While Clifford operations do not generate non-stabilizerness, their interplay with non-Clifford gates can strongly impact the overall non-stabilizing dynamics of generic quantum circuits. In this work, we establish a direct relationship between the final non-stabilizing power and the individual powers of the non-Clifford gates, in circuits where these gates are interspersed with random Clifford operations. By leveraging this result, we unveil the thermalization of non-stabilizing power to its Haar-averaged value in generic circuits. As a precursor, we analyze two-qubit gates and illustrate this thermalization in analytically tractable systems. Extending this, we explore the operator-space non-stabilizing power and demonstrate its behavior in physical models. Finally, we examine the role of non-stabilizing power in the emergence of quantum chaos in brick-wall quantum circuits. Our work elucidates how non-stabilizing dynamics evolve and thermalize in quantum circuits and thus contributes to a better understanding of quantum computational resources and of their role in quantum chaos.