Anticoncentration and State Design of Doped Real Clifford Circuits and Tensor Networks

TL;DR

This paper studies the statistical behavior of doped real Clifford circuits, revealing a new universality class and resource hierarchy for approximating Haar and Clifford distributions.

Contribution

It develops Weingarten calculus for real Clifford groups and identifies a new orthogonal Porter-Thomas distribution, advancing understanding of quantum circuit statistics.

Findings

Orthogonal Clifford Porter-Thomas distribution identified

Logarithmic depth local architectures recover global statistics

Resource hierarchy established for Haar and Clifford statistics

Abstract

We investigate the statistical properties of orthogonal, or real, Clifford circuits doped with magic and imaginary resources. By developing the Weingarten calculus for the real Clifford group, we derive the exact overlap distribution of real stabilizer states, identifying a new universality class: the orthogonal Clifford Porter-Thomas distribution. We prove that local real architectures recover this global statistic in logarithmic depth. Furthermore, we uncover a sharp hierarchy in resource requirements: while retrieving Haar statistics necessitates a polylogarithmic amount of magic states, recovering the full unitary Clifford statistics requires only a single phase gate.