Quantum Complexity and Chaos in Many-Qudit Doped Clifford Circuits

TL;DR

This work develops a unified framework to diagnose quantum complexity, magic spreading, anticoncentration, entanglement, and chaos in doped Clifford circuits for qudits of odd prime dimension. By combining replica tensor networks with Clifford Weingarten calculus, it derives exact scaling laws for generalized stabilizer entropies, reveals a dynamical phase transition at a dimension-dependent critical doping, and shows that higher-dimensional qudits reach Haar-like behavior faster. It demonstrates that a logarithmic number of non-Clifford gates suffices for Haar-like anticoncentration and entanglement features, while a linear number is necessary for Haar-like OTOCs, with the chaos threshold twice the magic threshold. Numerical results on both global and local circuit architectures corroborate the analytical predictions, suggesting a universal, architecture-agnostic growth of complexity governed by the doping periodicity. The findings advance our understanding of nonstabilizerness in multiqudit systems and offer a concrete, replication-friendly framework for studying quantum complexity and universality in complex quantum circuits.

Abstract

We investigate the emergence of quantum complexity and chaos in doped Clifford circuits acting on qudits of odd prime dimension $d$. Using doped Clifford Weingarten calculus and a replica tensor network formalism, we derive exact results and perform large-scale simulations in regimes challenging for tensor network and Pauli-based methods. We begin by analyzing generalized stabilizer entropies, computable magic monotones in many-qudit systems, and identify a dynamical phase transition in the doping rate, marking the breakdown of classical simulability and the onset of Haar-random behavior. The critical behavior is governed by the qudit dimension and the magic content of the non-Clifford gate. Using the qudit $T$-gate as a benchmark, we show that higher-dimensional qudits converge faster to Haar-typical stabilizer entropies. For qutrits ($d=3$), analytical predictions match numerics on brickwork circuits, showing that locality plays a limited role in magic spreading. We also examine anticoncentration and entanglement growth, showing that $O(\log N)$ non-Clifford gates suffice for approximating Haar expectation values to precision $\varepsilon$, and relate antiflatness measures to stabilizer entropies in qutrit systems. Finally, we analyze out-of-time-order correlators and show that a finite density of non-Clifford gates is needed to induce chaos, with a sharp transition fixed by the local dimension, twice that of the magic transition. Altogether, these results establish a unified framework for diagnosing complexity in doped Clifford circuits and deepen our understanding of resource theories in multiqudit systems.

Figures (2)

• Figure 1: $(a)$ GSE densities for different local dimensions (d=3,5,11) and system sizes computed with the reference $T_d$-gate. The initial linear growth culminates in a transition, at the value $q_c$ in Eq. \ref{['eq:critical_q']}, that becomes sharper as the system size increases. After this point, the universal value $\log(d)$ is reached for $N\rightarrow \infty$. $(b)$ We observe the exact superposition of the numerical value of the GSE at $d=3$ and $k=3$ obtained from the local doped circuit (blue dots) and local doped staircase circuit (purple triangles) with the exact values given by Eq. \ref{['eq:gen_pur']} with $\xi_d(\theta)=0$. Due to the different construction and layer definition, the density $q$ of the staircase circuit has been doubled to match, per layer, the one of the brickwork. This result highlights the marginal role of locality for the spreading of magic in doped Clifford circuits.
• Figure 2: We illustrate the scaling of Eq. \ref{['eq:DeltaOTOC']} with the number of injected non-Clifford gates $N_T$ for local dimensions (a) $d = 3$ and (b) $d = 5$. Across different system sizes, we observe an exponential decay of $\Delta \mathrm{OTOC}_6$, which approaches the Haar-random value $\mathrm{OTOC}_6^{\mathrm{Haar}}$ at a characteristic doping rate $q=N_T/N\sim O(1)$. Evaluating the critical density marking the transition from the non-chaotic to the chaotic regime (dashed lines in orange), we find that it coincides with $q_c^\mathrm{OTOC}=2q_c$, the critical doping density for the magic spreading, cf. Eq. \ref{['eq:otocs']}. After this point, the system reaches the Haar value ($\Delta \mathrm{OTOC}_6\mapsto 1$). This observation supports our previous findings on the accelerated growth of complexity at higher local dimensions.