Variational Quantum Algorithms (VQAs) are one of the most prominent methods used during the Noisy Intermediate Scale Quantum (NISQ) era as they adapt to the constraints of NISQ devices. VQAs are used in a wide range of applications from dynamical quantum simulation to machine learning. NISQ devices do not have the resources to exploit the benefits arising from quantum error correction (QEC) techniques, therefore such algorithms suffer from the effects of noise, which decreases their performance. As a substitute of full QEC, many algorithms employ error mitigation techniques to counter the effects of noise. An example of a VQA is the Variational Quantum Eigensolver (VQE), a setting which does offer some strategies to mitigate errors, however additional strategies are required to complement the existing advantages over noise.

It has been proven that noise can significantly decrease the trainability of linear (or super-linear) depth VQAs. One effect of noise is the flattening of the cost landscape as a function of the variational parameters, thus requiring an exponential precision in system size to resolve its features. This phenomenon is known as Noise-Induced Barren Plateaus (NIBPs), because the algorithm can no longer resolve a finite cost gradient (known as Barren Plateau). It can be advantageous to pursue algorithms with sublinear circuit depth and utilizing strategies that limit the hardware noise.

In this work, the authors investigate the effects of error mitigation on the trainability of noisy cost function landscapes in two regimes: i) asymptotic regime (in terms of scaling with system size) and ii) non-asymptotic regime for various error mitigation strategies including Zero Noise Extrapolation (ZNE), Virtual Distillation (VD), Probabilistic Error Cancellation (PEC), and Clifford Data Regression (CDR). The error mitigation protocols are subjected to a class of local depolarizing noise, known to cause NIBP. The first case of an asymptotic regime was considered in terms of scaling with system size. The theoretical results show that, if VQA suffers from exponential cost concentration, the error mitigation strategies cannot remove this exponential scaling, implying that at least an exponential number of resources is required to extract accurate information from the cost landscape in order to find a cost-minimizing optimization direction.

In the second case, it is shown that VD decreases the resolvability of the noisy cost landscape, and impedes trainability. ZNE also shows similar results under more restrictive assumptions on the cost landscape. It is also observed that any improvement in the resolvability after applying PEC under local depolarizing noise exponentially degrades with increasing number of qubits. Finally, for Clifford Data Regression, there is no change to the resolvability of any pair of cost values if the same ansatz is used.

The work also numerically investigates the effects of error mitigation on VQA trainability for the case when the effects of cost concentration is minimal. This is done by simulating the Quantum Approximate Optimization Algorithm (QAOA) for 5-qubit MaxCut problems on a realistic noise model of an IBM quantum computer, obtained by gate set tomography of IBM’s Ourense quantum device. The results compare the quality of the solutions of noisy (unmitigated) and CDR-mitigated optimization and demonstrate that CDR-mitigated optimization outperforms noisy optimization for all considered implementations.

Unlike all the considered error mitigation strategies, it is shown that CDR reverses the concentration of cost values more than it increases the statistical uncertainty as it has a neutral impact on resolvability under a global depolarizing noise model. Also, it can remedy the effects of more complex noise models. This indicates that CDR could resolve trainability issues arising due to corruptions of the cost function outside of cost concentration, whilst having a neutral effect on cost concentration itself, and thus improving overall trainability. A potential direction for future work will be investigating other mechanisms which can allow mitigation strategies to improve the trainability of noisy cost landscapes. As is known from the theory of error correction, once sufficient resources exist, then NIBPs can indeed be avoided.