Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?

Abstract

Variational Quantum Algorithms (VQAs) are widely 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.

Fundamental limits of quantum error mitigation

“…Our framework -suitably extended -could thus identify new performance bounds in each of these settings. Note added.-During the completion of our manuscript, we became aware of an independent work by Wang et al [62], which showed a result related to our Theorem 5 on the exponential scaling of required samples in variational algorithms.…”

Tight bounds on the convergence of noisy random circuits to the uniform distribution

“…Specifically, we study spatially local circuits of depth d on n qubits with Haar-random gates and local Pauli noise. The precise rate of convergence to the identity is a question of much significance in the complexity of random circuit sampling [GD18; BFLL21], the theory of benchmarking noisy circuits [ABIN96; Aha00; EAZ05; Boi+18; BSN17; Liu+21], and the investigation of near-term algorithms [SG21;Wan+21b;Wan+21a]. We prove upper and lower bounds on the expected total variation distance δ of the output distribution (when measuring in the computational basis) to the uniform distribution, which take the form δ ∼ exp − Θ(d) * .…”

Variational Quantum-Based Simulation of Waveguide Modes

“…An important question we leave open for future work is the extent to which noise affects the performance of our algorithm, and the degree to which error mitigation techniques [14], [43]- [47] can reduce the effects of the noise. Such an endeavor should take into consideration some recent theoretical and numerical work that have highlighted some limitations of quantum error mitigation on expectation estimation and training quantum circuits [48], [49].…”