Quantum Complexity and Chaos in Many-Qudit Doped Clifford Circuits
Institut für Theoretische Physik, Universität zu Köln, Zülpicher Strasse 77, 50937 Köln, Germany
| Published: | 2025-12-24, volume 9, page 1956 |
| Editor: | Alioscia Hamma |
| Eprint: | arXiv:2506.02127v5 |
| Doi: | https://doi.org/10.22331/q-2025-12-24-1956 |
| Citation: | Quantum 9, 1956 (2025). |
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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.
Featured image: $\mathrm{Onset\ of\ quantum\ complexity\ in\ doped\ Clifford\ circuits.}$
As the density $q$ of non-Clifford gates increases, the magic density grows linearly and undergoes a sharp transition at a critical threshold $q_c(d)$, after which it saturates to a universal value $\log d$. The transition sharpens with system size and occurs at smaller $q_c$ for higher-dimensional qudits. Numerical results for local brickwork and staircase circuits agree with exact predictions, showing that locality plays a marginal role in the spreading of state magic.
Popular summary
We find that local dimension matters: higher-dimensional qudits reach this complexity threshold faster than standard qubits. Furthermore, we distinguish between two complexity regimes, showing that the resource cost to maximize “magic” is exactly half the cost required to induce full quantum chaos (scrambling). These scaling laws hold for both global random circuits and local brickwork architectures, providing a unified framework for diagnosing complexity in multi-level quantum processors.
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