A leading approach to algorithm design aims to minimize the number of operations in an algorithm's compilation. One intuitively expects that reducing the number of operations may decrease the chance of errors. This paradigm is particularly prevalent in quantum computing, where gates are hard to implement and noise rapidly decreases a quantum computer's potential to outperform classical computers. Here, we find that minimizing the number of operations in a quantum algorithm can be counterproductive, leading to a noise sensitivity that induces errors when running the algorithm in non-ideal conditions. To show this, we develop a framework to characterize the resilience of an algorithm to perturbative noises (including coherent errors, dephasing, and depolarizing noise). Some compilations of an algorithm can be resilient against certain noise sources while being unstable against other noises. We condense these results into a tradeoff relation between an algorithm's number of operations and its noise resilience. We also show how this framework can be leveraged to identify compilations of an algorithm that are better suited to withstand certain noises.
A leading approach to algorithm design aims to minimize the number of operations in an algorithm's compilation. One intuitively expects that reducing the number of operations may decrease the chance of errors. This paradigm is particularly prevalent in quantum computing, where gates are hard to implement and noise rapidly decreases a quantum computer's potential to outperform classical computers. Here, we find that minimizing the number of operations in a quantum algorithm can be counterproductive, leading to a noise sensitivity that induces errors when running the algorithm in non-ideal conditions. To show this, we develop a framework to characterize the resilience of an algorithm to perturbative noises (including coherent errors, dephasing, and depolarizing noise). Some compilations of an algorithm can be resilient against certain noise sources while being unstable against other noises. We condense these results into a tradeoff relation between an algorithm's number of operations and its noise resilience. We also show how this framework can be leveraged to identify compilations of an algorithm that are better suited to withstand certain noises.
As quantum processors move into the noisy intermediate-scale quantum (NISQ) era, the central question is no longer whether a quantum algorithm is asymptotically faster than its classical counterpart, but whether it can retain a meaningful speedup under realistic noise, compilation, and measurement constraints. This work develops a quantitative framework for characterizing the noise resilience of quantum advantage. We introduce the advantage robustness function RA(ε), which measures how the asymptotic and concrete performance of an algorithm A degrades with physical error rate ε, and we compute it analytically or numerically for twelve representative algorithm families, including Grover search, amplitude estimation, phase estimation, HHL, VQE/QAOA, Hamiltonian simulation, and sampling algorithms. For each class, we derive critical noise thresholds and depth limits beyond which classical algorithms regain supremacy, explicitly incorporating realistic connectivity and compilation overheads. We show that algorithms based on coherent amplitude amplification are more fragile to coherent errors than depolarizing noise, while variational algorithms display a tradeoff between expressivity and robustness. By combining these algorithm-specific robustness curves with current and projected hardware error rates for superconducting, trapped-ion, neutralatom, and photonic platforms, we map out a phase diagram of practical quantum advantage that links problem size, noise rate, and algorithm type. Our analysis identifies promising near-term application windows in risk analytics, small-molecule chemistry, and structured optimization, and also delineates regions where no advantage is realistic, providing valuable guidance for both algorithm designers and hardware architects.
The rapid evolution of Noisy Intermediate-Scale Quantum (NISQ) devices has intensified the need for robust compilation strategies capable of mitigating hardware-induced errors while maximizing computational efficiency. Quantum compilers play a pivotal role in translating high-level quantum algorithms into hardware-executable instructions, yet they face significant challenges related to qubit connectivity constraints, limited coherence times, and stochastic noise. This paper examines error-resilient quantum compiler design for efficient qubit mapping, gate optimization, and noise mitigation in NISQ-era devices, proposing a framework that integrates topology-aware mapping, adaptive gate synthesis, and probabilistic error suppression techniques. By synthesizing advances in quantum circuit optimization, hardware-aware scheduling, and hybrid classical–quantum error correction strategies, the study evaluates how compiler-level innovations can substantially enhance execution fidelity and resource efficiency. The findings suggest that intelligent compiler architectures are critical enablers of practical quantum advantage during the transitional NISQ phase, bridging theoretical algorithm design with real-world hardware constraints.
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 guideline for robust quantum algorithm design and transpilation, which amounts to reducing the norms of the Hamiltonians. Using the three-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.
Abstract Noisy Intermediate-Scale Quantum (NISQ) devices are near-term quantum computing devices that inherently suffer from imperfect gate implementation, decoherence, and measurement noise. The main type of variational quantum algorithms(VQAs) is based on Parameterized Quantum Circuits (PQCs), which, however, face practical challenges due to optimization instability of optimization caused by noise and reduced performance. In this paper, we suggest a comprehensive noise-resilient optimization framework for PQCs, which explicitly incorporates noise-resilient circuit architecture design, incorporating noise-aware training strategies into the variational optimization loop. The framework uses circuit structures that are efficient in depth and are hardware compatible in addition to adaptive optimizer choice, stochastic optimization, and averaging as a part of repeated measurement to reduce the impacts of realistic noise. Large-scale experiments are performed with different amounts of qubits, depths of the circuit and noise strengths with realistic noise models. The findings indicate that the convergence rate is faster, the stability is increased, the variability of run-to-run variability is minimized, and other robustness is better than the noise-unaware optimization techniques. Moreover, systematic studies determine optimal operating regions that optimally trade-off expressibility and noise tolerance to offer useful design advice on trusted implementation of VQAs with near-term quantum hardware.
Optimizing quantum circuits is a critical step in the design and deployment of efficient, scalable, and fault tolerant quantum algorithms. As quantum circuits grow in depth and complexity, they often accumulate redundant or unnecessary gate operations. These redundancies not only increase the resource requirements of the circuit but also exacerbate one of the most pressing challenges in quantum computing—quantum noise. Unlike classical noise, quantum noise can irreversibly disturb the fragile quantum state, leading to decoherence and erroneous computational outcomes.
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.