Quantum Complexity and Chaos in Many-Qudit Doped Clifford Circuits

• URL: http://arxiv.org/abs/2506.02127v3
• Date: Sat, 12 Jul 2025 10:33:33 GMT
• Title: Quantum Complexity and Chaos in Many-Qudit Doped Clifford Circuits
• Authors: Beatrice Magni, Xhek Turkeshi,
• Abstract summary: We investigate the emergence of quantum complexity and chaos in doped Clifford circuits acting on qudits of odd prime dimension $d$.<n>Using doped Clifford Weingarten calculus and a replica tensor network formalism, we derive exact results and perform large-scale simulations.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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