Robustness of quantum algorithms against coherent control errors

TL;DR

The paper addresses the robustness of quantum algorithms to coherent control errors by developing a Lipschitz-based framework that bounds worst-case fidelity. It derives both norm-based and pairwise Lipschitz bounds, connecting state sensitivity to gate-generating Hamiltonians and their sequential coupling, and links these bounds to diamond-distance-based fault-tolerance criteria. A practical design guideline is proposed: reduce Hamiltonian norms during circuit design or transpilation to enhance robustness, demonstrated on the 3-qubit QFT across multiple gate sets and validated on IBM hardware. The framework is extended to variational quantum algorithms via parameter regularization, showing that larger regularization can substantially improve robustness and potentially aid convergence. Overall, the work provides scalable, computable robustness guarantees and a concrete methodology to design more resilient quantum circuits for near-term devices.

Abstract

Coherent control errors, for which ideal Hamiltonians are perturbed by unknown multiplicative noise terms, are a major obstacle for reliable quantum computing. In this paper, we present a framework for analyzing the robustness of quantum algorithms against coherent control errors using Lipschitz bounds. We derive worst-case fidelity bounds which show that the resilience against coherent control errors is mainly influenced by the norms of the Hamiltonians generating the individual gates. These bounds are explicitly computable even for large circuits, and they can be used to guarantee fault-tolerance via threshold theorems. Moreover, we apply our theoretical framework to derive a novel guideline for robust quantum algorithm design and transpilation, which amounts to reducing the norms of the Hamiltonians. Using the $3$-qubit Quantum Fourier Transform as an example application, we demonstrate that this guideline targets robustness more effectively than existing ones based on circuit depth or gate count. Furthermore, we apply our framework to study the effect of parameter regularization in variational quantum algorithms. The practicality of the theoretical results is demonstrated via implementations in simulation and on a quantum computer.

Figures (12)

• Figure 1: The figure shows the evolution of the quantum state resulting from applying $U_{\mathrm{ideal}}=R_{\mathrm{z}}(\frac{\pi}{4})R_{\mathrm{y}}(\frac{\pi}{2})$ (blue, dotted) and $U_{\mathrm{ideal}}'=R_{\mathrm{z}}(-\frac{3\pi}{4})R_{\mathrm{y}}(-\frac{\pi}{2})$ (red, dashed) to $\ket{0}$. Additionally, the final states for random realizations of coherent control errors are shown for both circuits (in blue and red, respectively).
• Figure 2: Ideal and noisy quantum circuits.
• Figure 3: Fidelities, Lipschitz bounds, gate counts, and circuit depths for five elementary gate set implementations of the $3$-qubit QFT, which are affected by coherent control errors. For each gate set, the figure shows the average fidelity including the standard deviation (left bar, orange), the worst-case fidelity (middle bar, blue), the Lipschitz bound (right bar, gray), and gate count (left number in parentheses) as well as the circuit depth (right number in parentheses).
• Figure 4: Ideal circuits $U_A$ and $U_B$, and their noisy versions $U_A(\varepsilon_A)$ and $U_B(\varepsilon_B)$.)
• Figure 5: Fidelities for circuits $U_A$ (orange) and $U_B$ (blue) depending on the noise level: average (solid), standard deviation (shaded area), and worst case (dashed) on ibm_nairobi over 80 noise samples for each noise level $\bar{\varepsilon}$, and worst-case bound from Theorem \ref{['thm:lipschitz_unitary']} (dotted).

Theorems & Definitions (4)

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