Quantum advantage requires overcoming noise-induced degradation of quantum systems. Conventional methods for reducing noise such as error mitigation face scalability issues in deep circuits. Specifically, noise hampers the extraction of amplitude and observable information from quantum systems. In this work, we present a novel algorithm that enhances the estimation of amplitude and observable under noise. Remarkably, our algorithm exhibits robustness against noise that varies across different depths of the quantum circuits. We assess the accuracy of amplitude and observable using numerical analysis and theoretically analyze the impact of gate-dependent noise on the results. This algorithm is a potential candidate for noise-resilient approaches that have high computational accuracy.
Quantum advantage requires overcoming noise-induced degradation of quantum systems. Conventional methods for reducing noise such as error mitigation face scalability issues in deep circuits. Specifically, noise hampers the extraction of amplitude and observable information from quantum systems. In this work, we present a novel algorithm that enhances the estimation of amplitude and observable under noise. Remarkably, our algorithm exhibits robustness against noise that varies across different depths of the quantum circuits. We assess the accuracy of amplitude and observable using numerical analysis and theoretically analyze the impact of gate-dependent noise on the results. This algorithm is a potential candidate for noise-resilient approaches that have high computational accuracy.
We show how phase and amplitude estimation algorithms can be parallelized. This can reduce the gate depth of the quantum circuits to that of a single Grover operator with a small overhead. Further, we show that for quantum amplitude estimation, the parallelization can lead to vast improvements in resilience against quantum errors. The resilience is not caused by the lower gate depth, but by the structure of the algorithm. Even in cases with errors that make it impossible to read out the exact or approximate solutions from conventional amplitude estimation, our parallel approach provided the correct solution with high probability. The results on error resilience hold for the standard version and for low depth versions of quantum amplitude estimation. Methods presented are subject of a patent application [Quantum computing device: Patent application EP 21207022.1].
The recent quantum age has allowed quantum algorithms to perform in the real world. However, they have significant performance errors in noisy environments. To investigate this issue, we conduct an analysis and simulation to evaluate maximum likelihood quantum amplitude estimation (MLQAE) circuit depths under various conditions: (1) ideal, (2) theoretical noise model, and (3) real noise model. These show variant performance errors, especially in a high-qubit circuit with high error rates, even when using a low-noise model.
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.
Despite rapid advances in quantum hardware, noise remains a central obstacle to deploying quantum algorithms on near-term devices. In particular, random coherent errors that accumulate during circuit execution constitute a dominant and fundamentally challenging noise source. We introduce a noise-resilient framework for Quantum Signal Processing (QSP) that mitigates such coherent errors without increasing circuit depth or ancillary qubit requirements. Our approach uses ensembles of noisy QSP circuits combined with measurement-level averaging to suppress random phase errors in Z rotations. Building on this framework, we develop robust QSP algorithms for implementing polynomial functions of Hermitian matrices and for estimating observables, with applications to Hamiltonian simulation, quantum linear systems, and ground-state preparation. We analyze the trade-off between approximation error and hardware noise, which is essential for practical implementation under the stringent depth and coherence constraints of current quantum hardware. Our results establish a practical pathway for integrating error mitigation seamlessly into algorithmic design, advancing the development of robust quantum computing, and enabling the discovery of scientific applications with near- and mid-term quantum devices.
Amplitude estimation is a fundamental quantum algorithmic primitive that enables quantum computers to achieve quadratic speedups for a large class of statistical estimation problems, including Monte Carlo methods. The main drawback from the perspective of near term hardware implementations is that the amplitude estimation algorithm requires very deep quantum circuits. Recent works have succeeded in somewhat reducing the necessary resources for such algorithms, by trading off some of the speedup for lower depth circuits, but high quality qubits are still needed for demonstrating such algorithms. Here, we report the results of an experimental demonstration of amplitude estimation on a state-of-the-art trapped ion quantum computer. The amplitude estimation algorithms were used to estimate the inner product of randomly chosen four-dimensional unit vectors, and were based on the maximum likelihood estimation (MLE) and the Chinese remainder theorem (CRT) techniques. Significant improvements in accuracy were observed for the MLE based approach when deeper quantum circuits were taken into account, including circuits with more than ninety two-qubit gates and depth sixty, achieving a mean additive estimation error on the order of 10−2. The CRT based approach was found to provide accurate estimates for many of the data points but was less robust against noise on average. Last, we analyze two more amplitude estimation algorithms that take into account the specifics of the hardware noise to further improve the results.
Noise in quantum hardware is the primary obstacle to realizing the transformative potential of quantum computing. Quantum error mitigation (QEM) offers a promising pathway to enhance computational accuracy on near-term devices, yet existing methods face a difficult trade-off between performance, resource overhead, and theoretical guarantees. In this work, we introduce neighbor-informed learning (NIL), a versatile and scalable QEM framework that unifies and strengthens existing methods such as zero-noise extrapolation (ZNE) and probabilistic error cancellation (PEC), while offering improved flexibility, accuracy, efficiency, and robustness.
NIL learns to predict the ideal output of a target quantum circuit from the noisy outputs of its structurally related ``neighbor'' circuits. A key innovation is our 2-design training method, which generates training data for our machine learning model. In contrast to conventional learning-based QEM protocols that create training circuits by replacing non-Clifford gates with uniformly random Clifford gates, our approach achieves higher accuracy and efficiency, as demonstrated by both theoretical analysis and numerical simulation. Furthermore, we prove that the required size of the training set scales only \emph{logarithmically} with the total number of neighbor circuits, enabling NIL to be applied to problems involving large-scale quantum circuits. Our work establishes a theoretically grounded and practically efficient framework for QEM, paving a viable path toward achieving quantum advantage on noisy hardware.