Polynomial-Time Classical Simulation of Noisy Quantum Circuits with Naturally Fault-Tolerant Gates

TL;DR

This work shows that for geometrically local, non-resettable quantum circuits built from Clifford or IQP+CNOT gates, introducing depolarizing noise at a constant rate forces a percolation‑driven decay of long‑range entanglement once depth passes a threshold $d^* = O(\gamma^{-1}\log n)$. By moving noise to the input and analyzing surviving Pauli operators via a Pauli‑path/percolation framework, the circuit decomposes into small, independent islands, enabling a polynomial‑time classical sampler for the output distribution without relying on anticoncentration. The results extend to Clifford‑Magic, Conjugated Clifford, and IQP+CNOT circuit classes, and yield both exact (Las Vegas) and Monte Carlo sampling guarantees with explicit runtimes; in particular, they imply that noisy, non‑adaptive versions of these models do not realize scalable quantum advantage under the stated locality constraints. The findings illuminate a fundamental role for noise‑driven percolation and locality in delimiting near‑term quantum advantage, with implications for experimental designs and the interpretation of noisy quantum devices.

Abstract

We construct a polynomial-time classical algorithm that samples from the output distribution of noisy geometrically local Clifford circuits with any product-state input and single-qubit measurements in any basis. Our results apply to circuits with nearest-neighbor gates on an $O(1)$-D architecture with depolarizing noise after each gate. Importantly, we assume that the circuit does not contain qubit resets or mid-circuit measurements. This class of circuits includes Clifford-magic circuits and Conjugated-Clifford circuits, which are important candidates for demonstrating quantum advantage using non-universal gates. Additionally, our results can be extended to the case of IQP circuits augmented with CNOT gates, which is another class of non-universal circuits that are relevant to current experiments. Importantly, these results do not require randomness assumptions over the circuit families considered (such as anticoncentration properties) and instead hold for every circuit in each class as long as the depth is above a constant threshold. This allows us to rule out the possibility of fault-tolerance in these circuit models. As a key technical step, we prove that interspersed noise causes a decay of long-range entanglement at depths beyond a critical threshold. To prove our results, we merge techniques from percolation theory and Pauli path analysis.

Figures (1)

• Figure 1: Overview of the simulation algorithm. (a) Depolarizing errors are sampled and propagated to the beginning of the circuit. $\ket{A}$ represents a single-qubit magic-state but can be any single-qubit state. (b) The circuit can now equivalently be represented as one error channel followed by the original noiseless circuit. (c) We prove that this input error channel has the effect of depolarizing many of the input qubits and so it remains to simulate the lightcones of qubits that are not depolarized. When these lightcones intersect they must be simulated together but can otherwise be simulated independently. Any measurements that are not in the lightcone of a depolarized qubit can be simulated by a random coin flip.

Theorems & Definitions (39)

• Theorem 1: Informal
• Theorem 2
• Corollary 3
• Definition 4
• Definition 5: Pauli projection channel
• Definition 6: Error Propagation
• proof
• Definition 8
• Lemma 9
• proof