Robustness of quantum algorithms: Worst-case fidelity bounds and implications for design

TL;DR

The paper introduces an algorithm-centered robustness framework to assess quantum algorithms under general error models, including coherent and time-varying incoherent errors described by set-membership uncertainty. It derives explicit worst-case fidelity bounds $F_{\mathrm{wc}}$ that depend on error magnitude, circuit depth, and an averaged interaction Hamiltonian, and extends the analysis to general quantum operations with vectorized noisy channels. The framework enables design-and-compile-time optimization, showing how minimizing the robustness parameter $\gamma$ can yield intrinsically more fault-tolerant algorithms and how to apply these insights to composite pulses, circuit transpilation, and large-scale modular circuits. Practical demonstrations and scalable techniques, such as circuit partitioning, illustrate the bounds’ usefulness for guiding robust quantum algorithm design on noisy hardware with potentially time-varying errors.

Abstract

Errors occurring on noisy hardware pose a key challenge to reliable quantum computing. Existing techniques such as error correction, mitigation, or suppression typically separate the error handling from the algorithm analysis and design. In this paper, we develop an alternative, algorithm-centered framework for understanding and improving the robustness against errors. For a given quantum algorithm and error model, we derive worst-case fidelity bounds which can be efficiently computed to certify the robustness. We consider general error models including coherent and (Markovian) incoherent errors and allowing for set-based error descriptions to address uncertainty or time-dependence in the errors. Our results give rise to guidelines for robust algorithm design and compilation by optimizing our theoretical robustness measure. We demonstrate the practicality of the framework with numerical results on algorithm analysis and robust optimization, including the robustness analysis of a 50-qubit modular adder circuit.

Figures (11)

• Figure 1: The figure displays the infidelity bound \ref{['eq:scaling_infidelity_bound']} depending on the noise/depth-factor $\delta N$, i.e., the product of the noise level $\delta$ and the circuit depth $N$.
• Figure 2: The figures show the infidelity $1-F_{\mathrm{wc}}$ with systematic and independent errors depending on the error level $\delta$ for the Pauli-$X$ rotation $R_X(\frac{\pi}{4})$, the pulses \ref{['eq:application_composite_pulse_sequence']} by Jones jones2003robust, and our newly designed pulses \ref{['eq:application_composite_pulse_sequence_ours']}. The displayed worst-case fidelity bounds are computed based on Theorem \ref{['thm:avg_algo']}.
• Figure 3: Infidelities of the three-qubit QFT circuit as a function of the noise level $\delta$ for different gate sets. Square markers indicate worst-case fidelity bounds computed using Theorem \ref{['thm:avg_algo']}, while circular markers correspond to the worst simulated infidelity obtained from 10,000 randomly drawn noise samples. The number of gates for the transpiled circuits are: A: 59, B: 42, C: 20, D: 14.
• Figure 4: Infidelity bounds for coherent control errors on the transpiled QFT circuits. Theoretical bounds are computed using Theorem \ref{['thm:avg_algo']} with two different strategies for obtaining $\gamma$: via an optimization problem ($\gamma_\mathrm{opt}$, Appendix \ref{['app:robustness_bounds_2']}) and via a norm-based bound ($\gamma_\mathrm{norm}$, Appendix \ref{['app:robustness_bounds_4']}). For comparison, the bound from berberich2024robustness is also shown. Additionally, we simulate 10,000 randomly sampled noise realizations, reporting both the mean and worst-case infidelity. The noise level is set to $\delta=0.001$.
• Figure 5: Infidelities of different Pauli-errors and coherent control errors over the noise level $\delta$ for a 50 qubit modular adder circuit. The infidelity bounds are computed using Theorem \ref{['thm:avg_algo']} with $\gamma_\mathrm{opt}$ and partitioning into sub-circuits with at most three qubits.

Theorems & Definitions (2)

• proof
• proof