Quantum Complexity and Chaos in Many-Qudit Doped Clifford Circuits

TL;DR

This paper explores how quantum complexity and chaos emerge in doped Clifford circuits acting on qudits of prime dimension, revealing phase transitions, resource thresholds, and the role of local dimension in magic and entanglement dynamics.

Contribution

It introduces a unified framework for analyzing complexity in doped Clifford circuits on qudits, including exact results, phase transition characterization, and insights into resource requirements for chaos.

Findings

Identification of a dynamical phase transition in doping rate

Faster convergence to Haar randomness in higher-dimensional qudits

Finite non-Clifford gate density induces chaos with a sharp transition

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.