QuickQudits: A Framework for Efficient Simulation of Noisy Qudit Clifford Circuits via an Extended Stabilizer Tableau Formalism

Abstract

We present a comprehensive and self-contained framework for the efficient classical simulation of Clifford circuits acting on $d$-dimensional qudits, including realistic Pauli/Weyl noise via stochastic simulation. Our approach uses the stabilizer tableau formalism for qudits of arbitrary dimension and tracks both stabilizer and destabilizer generators under Clifford updates. The classical simulation remains efficient with simple algebraic Clifford update rules over $\mathbb{Z}_d$. Computational basis measurements in prime dimensions are handled by a generalized Aaronson-Gottesman (CHP) procedure. In composite dimensions, $\mathbb{Z}_d$ is not a field and the standard tableau reduction fails, so we employ an exact Smith normal form decomposition to enable efficient sampling. Noise is modelled as probabilistic mixtures of Weyl operators that act only on the tableau's phase column. For fast simulation of noisy circuits, we support Pauli frames, respectively generalized Weyl frames, and introduce a noise-pushing technique that allows all noise processes to be consolidated into a single phase update at the end of the circuit. Using this representation, circuit fidelity can be determined entirely by the single accumulated phase-shift parameter $Δτ$, reducing fidelity estimation to a simple phase check per shot. Our codebase supports tableau simulation and conventional state-vector and density-matrix backends for qudits, and also includes circuit and tableau visualisations. Additionally, we provide tests and Jupyter notebooks for validation and illustration. This framework forms the basis for a scalable, open-source strong+weak stabilizer simulator including noise and can be found publicly available at https://github.com/QUICK-JKU/QuickQudits.

Figures (8)

• Figure 1: Overview of the qudit Clifford simulator and its main components. a) The simulator implements algebraic update rules for Clifford gates within the stabilizer tableau formalism, enabling efficient simulation of $n$-qudit circuits of local dimension $d$, including measurement, exact sampling, and visualization utilities. b) We benchmark the runtime against state-of-the-art stabilizer simulators such as Sdim and Stim, demonstrating competitive performance (see \ref{['sub:compare']}). c) Noisy Clifford circuits are supported via Pauli frames and a noise-pushing technique that converts stochastic Weyl errors into explicit tableau phase updates for circuit-fidelity estimation.
• Figure 2: Stabilizer tableau evolution for the preparation of the three-qutrit ($d=3$) GHZ state $|\mathrm{GHZ}_3^{(3)}\rangle = \frac{1}{\sqrt{3}}\, (|000\rangle + |111\rangle + |222\rangle )$. Here, $H$ is a generalized Hadamard gate and the entangling gate is an extension of CNOT to qudits.
• Figure 3: The noise pushing approach takes a given noise realization and can be viewed as implicitly pushing it to the end of the circuit. We compute a corresponding phase update $\Delta\tau$ from the stored tableau columns and add it to the tableau. Importantly, note that we only compute the phase update and not the explicit final noise layer.
• Figure 4: Runtimes are averaged over $3$ runs of $10$ randomly generated Clifford circuits with $1000$ Monte Carlo shots each. After every Clifford gate, a single-qudit depolarizing noise channel is applied with error probability $p=0.5$. No measurement is performed, only the final phase-update vector $\Delta\tau$ for noise pushing and the final Pauli-frame exponent vectors $(\mathbf a,\mathbf b)$ for Pauli-frame simulation are computed.
• Figure 5: The plot shows the execution time (in seconds) as a function of the number of qudits for QuickQudits (solid green), sdim (dashed orange), and stim (dotted blue). Markers denote the local dimension: circles ($d=2$), squares ($d=3$), and triangles ($d=6$). We introduce a cutoff point at $n=32$ for composite dimensions as the runtime rapidly grows for the Sdim and QuickQudits simulators. While stim provides a baseline for the specialized $d=2$ case, QuickQudits demonstrates a consistent order-of-magnitude speedup over Sdim across all tested dimensions.

Theorems & Definitions (10)

• Definition 1: Stabilizer
• Definition 2: Stabilizer Group
• Example 1
• Definition 3: Set of destabilizers
• Definition 4: Clifford Group
• Example 2
• Example 3
• Proposition 1
• proof
• Corollary 1