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

TL;DR

This work analyzes anticoncentration in Clifford circuits by comparing overlap statistics to those of random stabilizer states. Using Clifford Weingarten calculus and Clifford replica tensor networks, it derives the exact overlap distribution for stabilizer states (the Clifford-Porter-Thomas form) and shows that random Clifford circuits reach this distribution in depth $O(\log N)$, effectively mimicking universal randomness for the overlaps. It further demonstrates that injecting a polylogarithmic amount of magic resources (T-states) into Clifford dynamics drives the system toward Porter-Thomas statistics, yielding pseudo-magic states and providing a controlled route to Haar-like randomness in sampling tasks. The results illuminate the interplay between Clifford dynamics, magic-state injection, and complexity, with practical implications for quantum circuit sampling, NISQ benchmarking, and many-body physics.

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.

Figures (3)

• Figure 1: (a) Numerical sampling of random stabilizer states over $N$ qubits ($d=2$), compared to the Haar prediction (red line in the plot) in Eq. \ref{['eq:IPRHaarCl']}. The data is in quantitative agreement with our analytical predictions. (b) Numerical sampling of the overlap distribution of Clifford random matrix product states for $N=256$ varying $x=N/\chi$. The dashed black line is the analytical prediction Eq. \ref{['eq:IPRrandomMPS']} which is in robust agreement with our data. For reference, we plot the Clifford Haar distribution Eq. \ref{['eq:IPRHaarCl']} in red, highlighting the substantial difference in the scaling regime $N,\chi\to\infty$ with $x_0=N/\chi$ fixed. In both cases, we consider $\mathcal{N}=2\times 10^7$ circuit realizations.
• Figure 2: Dynamics of Eq. \ref{['eq:DeltaS3']} for qutrits $d=3$ (a) and qupents $d=5$ (b) for different system sizes. In both cases, the difference between the Haar value and the time-evolving IPR decreases exponentially fast with circuit depth, saturating for a fixed $\varepsilon$ in a timescale that is logarithmic in system size. Distribution of the overlaps for shallow circuits with $t$ layers for $N=128$ (c) and $N=256$ (d), compared to Eq. \ref{['eq:ziopera']} where we fitted $x$ to match the most probable value of the distribution. In all cases, we consider $\mathcal{N}=2\times 10^7$ data to sample the distribution, which is in quantitative agreement with our prediction.
• Figure 3: (a) Fluctuations between the doped random matrix product state $I_3$ and the Haar unitary value for different $N\le 512$ and $4\le r\le 12$ compared to the log-normal fluctuations expected for unitary circuits, and the Poisson fluctuations of Clifford circuits. The data displays strong agreement with the latter, corroborating Eq. \ref{['eq:rmpsIPRdoped']}. (b) Evolution of Eq. \ref{['eq:DeltaS32']} under random circuits of qutrits ($d=3$) for different system sizes starting from a system with a number $N_T=\lfloor \log_2(N)/2\rfloor$ of $|T\rangle$ states. This quantity decreases exponentially fast in system size up to a convergence in $\log(N)$ depth toward a value that $\Delta\tilde{S}^\mathcal{U}_3(\infty)$ that decreases polynomially in system size. (c) Evolution of Eq. \ref{['eq:maria']} with respect to the doped global Clifford value Eq. \ref{['eq:maria2']}. This quantity decreases exponentially fast in system size, saturating in a timescale that is logarithmic in system size $\log(N)$.