Variational quantum algorithms (VQAs) are expected to become a practical application of near-term noisy quantum computers. Although the effect of the noise crucially determines whether a VQA works, the heuristic nature of VQAs makes it difficult to establish analytic theories. Analytic estimations of the impact of the noise are urgent for searching for quantum advantages, as numerical simulations of noisy quantum computers on classical computers are heavy and quite limited to small-scale problems. In this paper, we establish analytic estimations of the error in the cost function of VQAs due to the noise. The estimations are applicable to any typical VQAs under Gaussian noise, which is equivalent to a class of stochastic noise models. Notably, depolarizing noise is included in this model. As a result, we obtain estimations of the noise level to guarantee a required precision. Our formulas show how the Hessian of the cost function, the spectrum of the target operator, and the geometry of the ansatz affect the sensitivity to the noise. This insight implies trade-off relations between the trainability and the noise resilience of the cost function. We also obtain rough estimations which can be easily calculated without detailed information of the cost function. As a highlight of the applications of the formula, we propose a quantum error mitigation method which is different from the extrapolation and the probabilistic error cancellation.
Variational quantum algorithms (VQAs) are expected to become a practical application of near-term noisy quantum computers. Although the effect of the noise crucially determines whether a VQA works, the heuristic nature of VQAs makes it difficult to establish analytic theories. Analytic estimations of the impact of the noise are urgent for searching for quantum advantages, as numerical simulations of noisy quantum computers on classical computers are heavy and quite limited to small-scale problems. In this paper, we establish analytic estimations of the error in the cost function of VQAs due to the noise. The estimations are applicable to any typical VQAs under Gaussian noise, which is equivalent to a class of stochastic noise models. Notably, depolarizing noise is included in this model. As a result, we obtain estimations of the noise level to guarantee a required precision. Our formulas show how the Hessian of the cost function, the spectrum of the target operator, and the geometry of the ansatz affect the sensitivity to the noise. This insight implies trade-off relations between the trainability and the noise resilience of the cost function. We also obtain rough estimations which can be easily calculated without detailed information of the cost function. As a highlight of the applications of the formula, we propose a quantum error mitigation method which is different from the extrapolation and the probabilistic error cancellation.
Variational Quantum Algorithms (VQAs) are a promising application for near-term quantum processors, however the quality of their results is greatly limited by noise. For this reason, various error mitigation techniques have emerged to deal with noise that can be applied to these algorithms. Recent work introduced a technique for mitigating expectation values against correlated measurement errors that can be applied to measurements of 10s of qubits. We apply these techniques to VQAs and demonstrate its effectiveness in improving estimates to the cost function. Moreover, we use the data resulting from this technique to experimentally characterize measurement errors in terms of the device connectivity on devices of up to 20 qubits. These results should be useful for better understanding the near-term potential of VQAs as well as understanding the correlations in measurement errors on large, near-term devices.
Variational quantum algorithms (VQAs) offer the most promising path to obtaining quantum advantages via noisy intermediate-scale quantum (NISQ) processors. Such systems leverage classical optimization to tune the parameters of a parameterized quantum circuit (PQC). The goal is minimizing a cost function that depends on measurement outputs obtained from the PQC. Optimization is typically implemented via stochastic gradient descent (SGD). On NISQ computers, gate noise due to imperfections and decoherence affects the stochastic gradient estimates by introducing a bias. Quantum error mitigation (QEM) techniques can reduce the estimation bias without requiring any increase in the number of qubits, but they in turn cause an increase in the variance of the gradient estimates. This work studies the impact of quantum gate noise on the convergence of SGD for the variational eigensolver (VQE), a fundamental instance of VQAs. The main goal is ascertaining conditions under which QEM can enhance the performance of SGD for VQEs. It is shown that quantum gate noise induces a nonzero error-floor on the convergence error of SGD (evaluated with respect to a reference noiseless PQC), which depends on the number of noisy gates, the strength of the noise, as well as the eigenspectrum of the observable being measured and minimized. In contrast, with QEM, any arbitrarily small error can be obtained. Furthermore, for error levels attainable with or without QEM, QEM can reduce the number of required iterations, but only as long as the quantum noise level is sufficiently large, and a sufficiently large number of measurements is allowed at each SGD iteration. Numerical examples for a max-cut problem corroborate the main theoretical findings.
As we are entering the era of constructing practical quantum computers, suppressing the inevitable noise to accomplish reliable computational tasks will be the primary goal. Leakage noise, as the amplitude population leaking outside the qubit subspace, is a particularly damaging source of error that error-correction approaches cannot handle. However, the impact of this noise on the performance of variational quantum algorithms (VQAs), a type of near-term quantum algorithm that is naturally resistant to a variety of noises, is yet unknown. Here, we consider a typical scenario with the widely used hardware-efficient ansatz and the emergence of leakage in two-qubit gates, observing that leakage noise generally reduces the expressive power of VQAs. Furthermore, we benchmark the influence of leakage noise on VQAs in real-world learning tasks. Results show that, for both data fitting and data classification, leakage noise generally has a negative impact on the training process and final outcomes. Our findings give strong evidence that VQAs are vulnerable to leakage noise in most cases, implying that leakage noise must be effectively suppressed in order to achieve practical quantum computing applications, whether for near-term quantum algorithms or for long-term error-correcting quantum computing.
Variational Quantum Algorithms (VQAs) are often viewed as the best hope for near-term quantum advantage. However, recent studies have shown that noise can severely limit the trainability of VQAs, e.g., by exponentially flattening the cost landscape and suppressing the magnitudes of cost gradients. Error Mitigation (EM) shows promise in reducing the impact of noise on near-term devices. Thus, it is natural to ask whether EM can improve the trainability of VQAs. In this work, we first show that, for a broad class of EM strategies, exponential cost concentration cannot be resolved without committing exponential resources elsewhere. This class of strategies includes as special cases Zero Noise Extrapolation, Virtual Distillation, Probabilistic Error Cancellation, and Clifford Data Regression. Second, we perform analytical and numerical analysis of these EM protocols, and we find that some of them (e.g., Virtual Distillation) can make it harder to resolve cost function values compared to running no EM at all. As a positive result, we do find numerical evidence that Clifford Data Regression (CDR) can aid the training process in certain settings where cost concentration is not too severe. Our results show that care should be taken in applying EM protocols as they can either worsen or not improve trainability. On the other hand, our positive results for CDR highlight the possibility of engineering error mitigation methods to improve trainability.
Variational Quantum Algorithms (VQA) are one of the most promising candidates for near-term quantum advantage. Traditionally, these algorithms are parameterized by rotational gate angles whose values are tuned over iterative execution on quantum machines. The iterative tuning of these gate angle parameters make VQAs more robust to a quantum machine's noise profile. However, the effect of noise is still a significant detriment to VQA's target estimations on real quantum machines - they are far from ideal. Thus, it is imperative to employ effective error mitigation strategies to improve the fidelity of these quantum algorithms on near-term machines.While existing error mitigation techniques built from theory do provide substantial gains, the disconnect between theory and real machine execution characteristics limit the scope of these improvements. Thus, it is critical to optimize mitigation techniques to explicitly suit the target application as well as the noise characteristics of the target machine.We propose VAQEM, which dynamically tailors existing error mitigation techniques to the actual, dynamic noisy execution characteristics of VQAs on a target quantum machine. We do so by tuning specific features of these mitigation techniques similar to the traditional rotation angle parameters -by targeting improvements towards a specific objective function which represents the VQA problem at hand. In this paper, we target two types of error mitigation techniques which are suited to idle times in quantum circuits: single qubit gate scheduling and the insertion of dynamical decoupling sequences. We gain substantial improvements to VQA objective measurements - a mean of over 3x across a variety of VQA applications, run on IBM Quantum machines.More importantly, while we study two specific error mitigation techniques, the proposed variational approach is general and can be extended to many other error mitigation techniques whose specific configurations are hard to select a priori. Integrating more mitigation techniques into the VAQEM framework in the future can lead to further formidable gains, potentially realizing practically useful VQA benefits on today's noisy quantum machines.
Variational Quantum Algorithms (VQAs) are a class of hybrid quantum-classical algorithms that leverage on classical optimization tools to find the optimal parameters for a parameterized quantum circuit. One relevant application of VQAs is the Variational Quantum Eigensolver (VQE), which aims at steering the output of the quantum circuit to the ground state of a certain Hamiltonian. Recent works have provided global convergence guarantees for VQEs under suitable local surjectivity and smoothness hypotheses, but little has been done in characterizing convergence of these algorithms when the underlying quantum circuit is affected by noise. In this work, we derive an upper bound on the error on the optimal parameters of a VQE under the effect of different coherent and incoherent noise processes. We then procced to show robust convergence guarantees of the algorithm to the perturbed optimal parameters. Our work provides novel theoretical insight into the behavior of VQAs subject to noise. Furthermore, we accompany our results with numerical simulations implemented via Pennylane.