Nonstabilizerness and Error Resilience in Noisy Quantum Circuits

Abstract

We investigate how noise impacts nonstabilizerness - a key resource for quantum advantage - in many-body qubit systems. While noise typically degrades quantum resources, we show that amplitude damping, a nonunital channel, can generate or enhance magic, whereas depolarizing noise provably cannot. In an encoding-decoding protocol, we find that, unlike in the coherent case, a sharp decoding fidelity transition does not match a transition in nonstabilizerness. Our results point toward the possibility of leveraging, rather than merely mitigating, noise for quantum information processing.

Figures (11)

• Figure 1: (A) Schematic of the encoding-decoding protocol. A quantum state consisting of $k$ logical qubits is embedded into a larger system of $N$ physical qubits using a unitary operation. After the system undergoes noise, a decoding unitary is applied to attempt recovery of the logical information. The protocol is post-selected on outcomes where the ancillary qubits are found in the zero state. (B) Fidelity of the recovered state as a function of the damping parameter $p$, showing a transition from error-protecting to error-vulnerable behavior. Curves show analytical predictions, while data points correspond to numerical simulations. The inset highlights the boundary between these two phases. (C) Average robustness of magic under amplitude damping within the encoding-decoding framework. ROM values are averaged over 1000, 500, and 300 Clifford unitary realizations for increasing $N$.
• Figure 2: Magic under different noisy channels. (A) Stabilizer Rényi entropy of the state $|\Psi_p\rangle$ produced by the no-click amplitude damping protocol. The inset shows a schematic of the protocol. (B) ROM for various initial states under amplitude damping noise versus damping strength $p$. (C–D) Heat maps of ROM for the generalized amplitude damping channel applied to $|+\rangle$ and $|H\rangle$ across different channel parameters.
• Figure 3: Hilbert–Schmidt concentration of post-selected logical outputs. For amplitude damping (A), $\mathbb{E}_U[\|\tilde{\rho}_U-\bar{\rho}\|_2]$ decays with system size, indicating ensemble concentration onto the depolarizing channel. For coherent errors (B-C), defined as global $Z$-rotations of angle $\alpha$ with post-selection, the HS distance fails to decay above a critical $\alpha$, where the magic monotone $\widetilde{\mathcal{M}}_2$ shows a transition.
• Figure 4: Heatmaps of $\widetilde{\mathcal{M}}_2$ (left) and $\mathcal{R}$ (right) on the Bloch sphere, projected onto the $xz$-plane. The black diamond outlines the stabilizer polytope. This comparison highlights the limitation of using the SRE as a measure of mixed-state magic: $\widetilde{\mathcal{M}}_2$ assigns nonzero values within the stabilizer region, whereas the ROM correctly identifies the stabilizer states as having no magic.
• Figure 5: Bloch sphere trajectories of certain states under amplitude damping noise.