Quantum Complexity and Chaos in Many-Qudit Doped Clifford Circuits

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 $\mathcal{O}(\log N)$ 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.