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

Beatrice Magni, Markus Heinrich, Lorenzo Leone, Xhek Turkeshi

TL;DR

The paper analyzes anticoncentration and state design in real (orthogonal) Clifford circuits and tensor networks doped with magic and imaginary resources. By developing Weingarten calculus for the real Clifford group, it derives the orthogonal Clifford Porter-Thomas (OCPT) distribution and shows that real Matrix Product States and shallow local circuits reach this universal statistic at logarithmic depth, with bond dimension scaling polynomial in system size. It reveals a sharp resource hierarchy: polylogarithmic real or complex magic suffices to approximate Haar statistics, while a single imaginary state is enough to reproduce unitary Clifford statistics. These results provide a precise, resource-aware framework for achieving desired randomness properties in real-clifford-based quantum architectures and have implications for benchmarking and complexity in near-term devices.

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.

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

TL;DR

The paper analyzes anticoncentration and state design in real (orthogonal) Clifford circuits and tensor networks doped with magic and imaginary resources. By developing Weingarten calculus for the real Clifford group, it derives the orthogonal Clifford Porter-Thomas (OCPT) distribution and shows that real Matrix Product States and shallow local circuits reach this universal statistic at logarithmic depth, with bond dimension scaling polynomial in system size. It reveals a sharp resource hierarchy: polylogarithmic real or complex magic suffices to approximate Haar statistics, while a single imaginary state is enough to reproduce unitary Clifford statistics. These results provide a precise, resource-aware framework for achieving desired randomness properties in real-clifford-based quantum architectures and have implications for benchmarking and complexity in near-term devices.

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.

Paper Structure

This paper contains 6 sections, 33 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Methods
  3. Results
  4. Discussion
  5. Review on anticoncentration of unitary, orthogonal and Clifford ensemble
  6. Doped real Clifford RMPS

Figures (1)

  • Figure 1: Numerically sampled overlap distribution for different settings. Panel (a) presents the sampling of real random stabilizer states which agree with the distribution in Eq. \ref{['eq:Orthocliff']} (orange dashed line), compared to the unitary Clifford Porter-Thomas distribution (red dashed line). Panel (b) samples from $\mathbb{R}$MPS comparing the data with the distribution in Eq. \ref{['eq:RMPSdistr']} (black dashed line) for different $x_0 = N/\chi$ and the limiting distribution for real random stabilizer states. Panels (c) and (d) show the data sampled from a random brickwork circuit at different depths $t$ for $N = 128$ and $N=256$. Notably, the black dashed lines are obtained from the distribution \ref{['eq:RMPSdistr']} with the scaling variable $x$ as a single fitting parameter, while the orange line represents the OCPT.