With a finite amount of measurement data acquired in variational quantum algorithms, the statistical benefits of several optimized numerical estimation schemes, including the scaled parameter-shift (SPS) rule and finite-difference (FD) method, for estimating gradient and Hessian functions over analytical schemes [unscaled parameter-shift (PS) rule] were reported by the present author [Phys. Rev. A 107, 042421 (2023)]. We continue the saga by exploring the extent to which these numerical schemes remain statistically more accurate for a given number of sampling copies in the presence of noise. For noise-channel error terms that are independent of the circuit parameters, we demonstrate that without any knowledge about the noise channel, using the SPS and FD estimators optimized specifically for noiseless circuits can still give lower mean-squared errors than PS estimators for substantially wide sampling-copy number ranges---specifically for SPS, closed-form mean-squared error expressions reveal that these ranges grow exponentially in the qubit number and reciprocally with a decreasing error rate. Simulations also demonstrate similar characteristics for the FD scheme. Lastly, if the error rate is known, we propose a noise-model-agnostic error-mitigation procedure to optimize the SPS estimators under the assumptions of two-design circuits and circuit-parameter-independent noise-channel error terms. We show that these heuristically optimized SPS estimators can significantly reduce mean-squared-error biases that naive SPS estimators possess even with realistic circuits and noise channels, thereby improving their estimation qualities even further. The heuristically optimized FD estimators possess as much mean-squared-error biases as the naively optimized counterparts and are thus not beneficial with noisy circuits.
With a finite amount of measurement data acquired in variational quantum algorithms, the statistical benefits of several optimized numerical estimation schemes, including the scaled parameter-shift (SPS) rule and finite-difference (FD) method, for estimating gradient and Hessian functions over analytical schemes [unscaled parameter-shift (PS) rule] were reported by the present author [Phys. Rev. A 107, 042421 (2023)]. We continue the saga by exploring the extent to which these numerical schemes remain statistically more accurate for a given number of sampling copies in the presence of noise. For noise-channel error terms that are independent of the circuit parameters, we demonstrate that without any knowledge about the noise channel, using the SPS and FD estimators optimized specifically for noiseless circuits can still give lower mean-squared errors than PS estimators for substantially wide sampling-copy number ranges---specifically for SPS, closed-form mean-squared error expressions reveal that these ranges grow exponentially in the qubit number and reciprocally with a decreasing error rate. Simulations also demonstrate similar characteristics for the FD scheme. Lastly, if the error rate is known, we propose a noise-model-agnostic error-mitigation procedure to optimize the SPS estimators under the assumptions of two-design circuits and circuit-parameter-independent noise-channel error terms. We show that these heuristically optimized SPS estimators can significantly reduce mean-squared-error biases that naive SPS estimators possess even with realistic circuits and noise channels, thereby improving their estimation qualities even further. The heuristically optimized FD estimators possess as much mean-squared-error biases as the naively optimized counterparts and are thus not beneficial with noisy circuits.
Sampling noisy intermediate-scale quantum devices is a fundamental step that converts coherent quantum-circuit outputs to measurement data for running variational quantum algorithms that utilize gradient and Hessian methods in cost-function optimization tasks. This step, however, introduces estimation errors in the resulting gradient or Hessian computations. To minimize these errors, we discuss tunable numerical estimators, which are the finite difference (including their generalized versions) and scaled parameter-shift estimators [introduced in Phys. Rev. A 103, 012405 (2021)], and propose operational circuit-averaged methods to optimize them. We show that these optimized numerical estimators offer estimation errors that drop exponentially with the number of circuit qubits for a given sampling-copy number, revealing a direct compatibility with the barren-plateau phenomenon. In particular, there exists a critical sampling-copy number below which an optimized difference estimator gives a smaller average estimation error in contrast to the standard (analytical) parameter-shift estimator, which exactly computes gradient and Hessian components. Moreover, this critical number grows exponentially with the circuit-qubit number. Finally, by forsaking analyticity, we demonstrate that the scaled parameter-shift estimators beat the standard unscaled ones in estimation accuracy under any situation, with comparable performances to those of the difference estimators within significant copy-number ranges, and are the best ones if larger copy numbers are affordable.
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.
Many optimization methods for training variational quantum algorithms are based on estimating gradients of the cost function. Due to the statistical nature of quantum measurements, this estimation requires many circuit evaluations, which is a crucial bottleneck of the whole approach. We propose a new gradient estimation method to mitigate this measurement challenge and reduce the required measurement rounds. Within a Bayesian framework and based on the generalized parameter shift rule, we use prior information about the circuit to find an estimation strategy that minimizes expected statistical and systematic errors simultaneously. We demonstrate that this approach can significantly outperform traditional gradient estimation methods, reducing the required measurement rounds by up to an order of magnitude for a common QAOA setup. Our analysis also shows that an estimation via finite differences can outperform the parameter shift rule in terms of gradient accuracy for small and moderate measurement budgets.
Variational quantum algorithms (VQAs) have attracted a lot of attention from the quantum computing community for the last few years. Their hybrid quantum-classical nature with relatively shallow quantum circuits makes them a promising platform for demonstrating the capabilities of noisy intermediate scale quantum (NISQ) devices. Although the classical machine learning community focuses on gradient-based parameter optimization, finding near-exact gradients for variational quantum circuits (VQCs) with the parameter-shift rule introduces a large sampling overhead. Therefore, gradient-free optimizers have gained popularity in quantum machine learning circles. Among the most promising candidates is the simultaneous perturbation stochastic approximation (SPSA) algorithm, due to its low computational cost and inherent noise resilience. We introduce a novel approach that uses the approximated gradient from SPSA in combination with state-of-the-art gradient-based classical optimizers. We demonstrate numerically that this out-performs both standard SPSA and the parameter-shift rule in terms of convergence rate and absolute error in simple regression tasks. The improvement of our novel approach over SPSA with stochastic gradient descent (SGD) is even amplified when shot- and hardware-noise are taken into account. We also demonstrate that error mitigation does not significantly affect our results.
We initiate the study of weak Schur sampling (WSS) under noisy inputs. Many optimal quantum learning algorithms rely on this measurement, for instance in quantum spectrum estimation (QSE) and quantum state tomography (QST). The standard analysis of WSS, QSE and QST assumes that the n input copies of the target state ρ are identical, i.e. ρ⊗n. We study what happens when they are not, a more realistic noisy scenario that breaks the very permutation symmetry on which WSS is built. In the simplest model the input is a noisy product state: the copies are independent but not necessarily identical and each is ε-close in trace distance to the d-dimensional target state ρ. A naive argument via the data-processing inequality allows the per-copy errors to accumulate to nε on the measurement outcome. We show that for QSE via WSS this does not happen: the expected estimation error in total variation is at most ε+η(n,d), where η(n,d)=O(d/n) is the noiseless error, and the additive noise term ε is optimal. The output error does not grow with n even though the input error does; in this sense WSS is robust. Beyond noisy product states, for arbitrary input ω we show that the error is at most η(n,d)+n1∥ω−ρ⊗n∥W1, where W1 is the quantum Wasserstein distance of De Palma et al. In particular, our results imply, for the first time, noise-robustness of Keyl and Werner's seminal algorithm for QSE, more than two decades after its introduction. Finally, we show that the noise cannot be too bad: within our model of robustness the promise on the input cannot be weakened to closeness of single-copy marginals or to a global trace-distance budget alone, as in both cases some inputs defeat every measurement and estimator.
Noisy Intermediate-Scale Quantum (NISQ) devices show great promise for quantum advantage via Parametrized Quantum Circuits (PQCs) and Variational Quantum Algorithms (VQAs). Due to hardware imperfections, however, especially barren plateaus and environmental Pauli noise, the cost landscape is highly distorted, resulting in inaccurate classical optimization during training. To overcome these challenges, a robust quantum gradient descent framework is presented with the integration of error mitigation directly within the optimization loop. A noise-resilient parameter-shift rule is defined and modeled analytically, as is a variant of stochastic gradient descent with real-time Pauli noise mitigation. The framework is based on a 6-qubit system, which is tested on the Max-Cut problem in the presence of different amounts of depolarizing noise. It is shown experimentally that the proposed algorithm is able to achieve the final cost error of 0.045 under significant noise, whereas the standard quantum gradient descent fails to achieve better than 0.320 under this noise. Moreover, the methodology is consistent, converging without a noise-induced flat landscape and without local minima. The method offers a scalable route towards future applications of quantum machine learning on non-fault-tolerant hardware that does not suffer from the large sampling overhead often associated with conventional error mitigation techniques.