Anticoncentration in Clifford Circuits and Beyond: From Random Tensor Networks to Pseudo-Magic States

TL;DR

This paper studies how random Clifford circuits and their variants spread quantum states over the Hilbert space, showing that shallow circuits and limited magic resources can produce near-random output distributions relevant for quantum advantage benchmarking.

Contribution

It analytically and numerically demonstrates anticoncentration in Clifford circuits at logarithmic depth and shows that adding polylogarithmic magic states induces full quantum randomness.

Findings

Clifford circuits fully anticoncentrate at logarithmic depth.

Injecting polylogarithmic T-states achieves Porter-Thomas distribution.

Doped tensor networks act as pseudo-magic quantum states.

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 pseudo-magic 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.