Diagnosing Quantum Circuits: Noise Robustness, Trainability, and Expressibility

TL;DR

This work introduces 2MC-OBPPP, a polynomial-time classical framework for diagnosing parameterized quantum circuits under PCS1 noise by jointly estimating noise robustness, trainability, and expressibility via unbiased estimators. The two-layer approach samples discrete rotation configurations and back-propagates Pauli paths to compute bounded diagnostics, enabling efficient pre-deployment benchmarking and targeted noise mitigation through a noise-bottleneck map. The study reveals a nuanced role for amplitude damping, where moderate noise can alleviate barren plateaus while degrading expressibility, and demonstrates substantial gains from bottleneck-focused interventions. Collectively, the method provides a scalable, actionable toolkit for hardware-aware circuit design and optimization toward practical quantum advantage.

Abstract

Achieving practical quantum advantage on near-term noisy hardware is a central goal of quantum computation. However, without efficient pre-execution diagnostics, circuit design and scheme selection often rely on costly hardware-in-the-loop trial-and-error, inflating experimental overhead and impeding progress. To address this challenge, we introduce 2MC-OBPPP, a polynomial-time classical estimator that, for parameterized quantum circuits, jointly estimates trainability, expressibility, and robustness to noise. For example, our approach visually demonstrates that moderate amplitude damping alleviates barren plateaus (improving trainability) while decreasing expressibility. Moreover, the method produces a spatiotemporal ``noise-hotspot" map that pinpoints the most noise-sensitive qubits/gates, enabling targeted noise suppression. In a representative circuit, interventions on fewer than $2\%$ qubits reduce the error up to $90\%$. Together, before execution, our approach provides an efficient diagnostic benchmark for circuit/scheme design, and in deployment, guides for targeted interventions that substantially reduce the cost of error suppression.

Figures (12)

• Figure 1: Workflow overview of 2MC-OBPPP. For a given PQC, each rotation angle is sampled from a Clifford-compatible discrete set (blue arrow). For each sampled configuration of rotation angles, the algorithm starts from Pauli terms in final observable and recursively back-propagates the operators through the circuit. Upon encountering a PCS1 channel (red points), a Pauli operator (solid circles) is sampled from the set of candidate operators (dashed circles), the resulting sequence of Pauli operators defines a Pauli path (solid line) and produces a sample value. Finally, post-processing and averaging over samples yield estimators of the diagnostic quantities.
• Figure 2: (a) Sensitivity of qubits in a 75-layer, 435-qubit circuit with 1-weight measurement. Dots represent qubits, and lines indicate two-qubit gates. The color of each qubit reflects the logarithm of the gradient norm of the MSE \ref{['eq:noise_gradient']} with respect to the qubit's noise strength. (b) Reduction of MSE \ref{['eq:noise_mse']} under the bottleneck-first error-suppression strategy. The x-axis denotes the number of mitigated qubits, and the y-axis shows the corresponding MSE. The legend of $\lambda$ presents the noise strength of the selected qubits after error mitigation. Shaded regions represent the standard deviation over $100$ independent trials. The $\star$ marks the case where $7$ qubits are mitigated to $\lambda=0.001$. The inset highlights the $7$ qubits with the largest gradient norms on the chip layout.
• Figure 3: The sum of the variance of gradient over all parameters $\sum_k \mathrm{Var}\left(\frac{\partial \langle \widetilde{O} \rangle_{\bm{\theta}}}{\partial \theta_k}\right)$ in circuits with 5-weight measurement under amplitude damping noise. The x-axis represents the damping intensities $\gamma$. Different colors indicate different circuit depths. The shaded area denotes the standard deviation over $100$ independent trials. The inset shows the spatial distribution of the variance of gradient at $\gamma=0.25$ and circuit depth $15$.
• Figure 4: Deviation from $2$-design for circuits with 8 qubits under amplitude damping noise. The circuits comprise $R_X$, $R_Z$ and nearest-neighbor $CZ$ gates on a ring, which can be viewed as a reduced unit used in Fig. \ref{['fig:variance_estimation']}. The x-axis represents the damping intensities $\gamma$; different colors indicate different circuit depths. Markers $\bullet$ and $\blacktriangle$ denote the noiseless metric $\mathcal{M}_2^2$ and the noisy lower bound $\widetilde{\mathcal{M}}_{2\leq}^2$, respectively. The shaded area denotes the standard deviation over $100$ independent trials.
• Figure 5: The noisy gate $\widetilde{U}$: ideal gate $U$ followed by noise channel $\mathcal{N}$ acting on the output.

Theorems & Definitions (16)

• Definition 1: PCS1
• Definition 2
• lemma 1
• proof
• Definition 3
• Theorem 1
• proof
• Corollary 1
• proof
• lemma 2