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  • Open Access

Anticoncentration in Clifford Circuits and Beyond: From Random Tensor Networks to Pseudomagic States

Beatrice Magni1,*, Alexios Christopoulos2, Andrea De Luca3, and Xhek Turkeshi1,†

  • *Contact author: bmagni@uni-koeln.de
  • †Contact author: xturkesh@uni-koeln.de
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Phys. Rev. X 15, 031071 – Published 15 September, 2025

DOI: https://doi.org/10.1103/p8dn-glcw

Abstract

Anticoncentration describes how an ensemble of quantum states spreads over the allowed Hilbert space, leading to statistically uniform output probability distributions. In this work, we investigate the anticoncentration of random Clifford circuits toward the overlap distribution of random stabilizer states. Using exact analytical techniques and extensive numerical simulations based on Clifford replica tensor networks, we demonstrate that random Clifford circuits fully anticoncentrate in logarithmic circuit depth; namely, higher-order moments of the overlap distribution converge to those of random stabilizer states. Moreover, we investigate the effect of introducing a controlled number of non-Clifford (magic) resources into Clifford circuits. We show that inserting a polylogarithmic in qudit number of T states is sufficient to drive the overlap distribution toward the Porter-Thomas statistics, effectively recovering full quantum randomness. In short, this fact presents doped tensor networks and shallow Clifford circuits as pseudomagic quantum states. Our results clarify the interplay between Clifford dynamics, magic-state injection, and quantum complexity, with implications for quantum circuit sampling, many-body quantum physics, and the benchmarking of quantum computational advantage.

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Popular Summary

A central question in quantum computing is how quantum circuits generate randomness, since this property underlies complexity, chaos, and the potential for computational advantage. A key concept is anticoncentration, which measures how evenly a quantum state spreads across all possible outcomes. In this work, we study anticoncentration in Clifford circuits and tensor networks, which play a central role in error correction and fault-tolerant quantum computation.

We calculate the overlap distribution for random stabilizer states, giving a precise description of how Clifford circuits anticoncentrate. Our analysis shows that Clifford tensor networks whose bond dimension grows like some power of the system fully anticoncentrate, as do Clifford circuits with only logarithmic depth. We then go beyond Clifford dynamics by introducing non-Clifford, or “magic,” resources. By adding only a logarithmic number of special quantum states known as T states, we find that circuits transition to fully chaotic behavior.

These results reveal the fine balance between Clifford operations, which are structured and efficiently simulable, and non-Clifford resources, which push circuits into the regime of quantum complexity. Understanding this transition informs strategies for quantum sampling, provides new tools for benchmarking devices, and points to resource requirements for true quantum advantage.

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Phys. Rev. X 15, 031071– Published 15 September, 2025
  • Received 7 March 2025
  • Revised 16 July 2025
  • Accepted 15 August 2025
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