Noise-induced shallow circuits and absence of barren plateaus

TL;DR

This work analyzes the impact of uncorrected non-unital noise on ensembles of random quantum circuits in the NISQ regime. It proves that observable expectation values effectively depend only on the last $\Theta(\log n)$ layers, enabling an efficient average-case classical simulation for local observables and revealing an absence of barren plateaus for local cost functions (though only the final layers are trainable). The authors develop a classical algorithm with runtime that is depth- and dimension-aware, and show contrasting concentration behaviors for local versus global costs and for fidelity vs projected kernels. These results suggest that, unless circuits are engineered to exploit the noise, noisy, non-unital devices may not outperform shallow quantum circuits for estimating local observable expectations. The work also outlines open questions on sampling complexity and potential algorithmic improvements, offering insight into fundamental noise-driven limits of quantum advantage.

Abstract

Motivated by realistic hardware considerations of the pre-fault-tolerant era, we comprehensively study the impact of uncorrected noise on quantum circuits. We first show that any noise `truncates' most quantum circuits to effectively logarithmic depth, in the task of estimating observable expectation values. We then prove that quantum circuits under any non-unital noise exhibit lack of barren plateaus for cost functions composed of local observables. But, by leveraging the effective shallowness, we also design an efficient classical algorithm to estimate observable expectation values within any constant additive accuracy, with high probability over the choice of the circuit, in any circuit architecture. The runtime of the algorithm is independent of circuit depth, and for any inverse-polynomial target accuracy, it operates in polynomial time in the number of qubits for one-dimensional architectures and quasi-polynomial time for higher-dimensional ones. Taken together, our results showcase that, unless we carefully engineer the circuits to take advantage of the noise, it is unlikely that noisy quantum circuits are preferable over shallow quantum circuits for algorithms that output observable expectation value estimates, like many variational quantum machine learning proposals. Moreover, we anticipate that our work could provide valuable insights into the fundamental open question about the complexity of sampling from (possibly non-unital) noisy random circuits.

Figures (8)

• Figure 1: For most of the quantum circuits with any possibly non-unital noise only the last $O(\log(n))$ influence significantly observables expectation values. Here, $n$ is the number of qubits.
• Figure 2: Example of the architecture that our model encompasses: A brickwork circuit composed of two-qubit gates followed by local noise (depicted with yellow circles).
• Figure 3: A graphical representation of $\Phi^*_{[L-2,L]}(P)$ with respect to the local Pauli observable $P$ represented by the blue shaded area. The (noisy) gates outside the blue shaded area are contracted trivially due to the fact that the adjoint of every channel is unital, and thus cannot influence the expectation value of the Pauli. Even if the qubits in the system are $n$, the computation of $\Phi^*_{[L-2,L]}(P)$ is restricted to only a constant number of qubits.
• Figure 4: The light-cone of a local Pauli observable with respect to $\Phi_{[1,L]}$ is the set of qubits within the shaded area. The (noisy) gates outside the blue shaded area are contracted trivially due to the fact that any adjoint channel is unital, and thus cannot influence the expectation value of the Pauli.
• Figure 5: Example of Clifford path choices. The shaded region indicates the fixed Clifford gates. Note that we choose some Clifford gates to be SWAP gates so that they connect one of the Paulis to $H_{\mu}$. We protect the other remaining Pauli from spreading across the circuit with identity Clifford gates.

Theorems & Definitions (99)

• Theorem 1: Effective logarithmic depth
• Theorem 2: General scaling
• proof : Proof sketch of Theorem \ref{['th:theor1']}
• Proposition 3: Worst-case effective depth with high noise
• Proposition 4: Average classical simulation of local expectation values
• Remark 5: Estimating classically any Pauli expectation value
• Theorem 6: Variance of expectation values of random circuits with non-unital noise
• Corollary 7: Lack of exponential concentration for local cost functions
• Corollary 8: Cost concentration for global expectation values
• Theorem 9: Only the last few layers are trainable for local cost functions