[Paper Review] Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?

Samson Wang, Piotr Czarnik+4 more|arXiv (Cornell University)|Sep 2, 2021

Quantum Computing Algorithms and Architecture86 references38 citations

TL;DR

The paper analyzes whether error mitigation can improve the trainability of noisy VQAs and shows that many EM strategies cannot overcome exponential cost concentration due to local depolarizing noise; some methods may hinder trainability, while Clifford Data Regression can help in certain settings.

ABSTRACT

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.

Motivation & Objective

• Motivate understanding of how noise affects trainability and the role EM could play in mitigating NISBPs (Noise-Induced Barren Plateaus).
• Characterize a broad class of EM protocols and determine whether they can reverse exponential cost concentration without exponential resources.
• Evaluate specific EM techniques (ZER, Virtual Distillation, Probabilistic Error Cancellation, Clifford Data Regression) for their impact on cost resolvability and trainability.

Proposed method

• Model the VQA cost as Tr[U(θ) ρ_in U†(θ) O] and analyze how noise degrades the landscape via local depolarizing channels.
• Prove that EM protocols that rely on linear combinations of measured quantities cannot remove exponential estimator concentration without exponential resource overhead (Theorem 1 and Corollary 1).
• Analyze four EM protocols (Zero Noise Extrapolation, Virtual Distillation, Probabilistic Error Cancellation, Clifford Data Regression) to assess their effect on cost resolvability and training.
• Introduce the concept of error mitigation cost γ(θ, ε) as Var[C_m(θ, ε)] / Var[Ĉ(θ, ε)].
• Provide non-asymptotic assessment of EMs using relative resolvability metrics to gauge improvement in cost landscape resolvability.

Experimental results

Research questions

• RQ1 Can error mitigation erase exponential cost concentration caused by local depolarizing noise without paying exponential resource costs?
• RQ2 Do EM techniques improve the resolvability of the noisy cost landscape enough to enhance VQA trainability under NIBPs?
• RQ3 Which EM protocols can either worsen, fail to improve, or improve trainability under specific landscape conditions?
• RQ4 Is Clifford Data Regression capable of improving trainability in settings with moderate cost concentration?
• RQ5 How does the error mitigation cost compare across EM methods and how does this impact practical training budgets?

Key findings

• Exponential cost concentration from local depolarizing noise cannot be undone by a broad class of EM strategies without exponential resource expenditure.
• Some EM protocols, such as Virtual Distillation, can decrease the resolvability of the noisy cost landscape and impede trainability.
• Probabilistic Error Cancellation under local depolarizing noise can exponentially degrade resolvability with increasing qubits.
• Zero Noise Extrapolation shows similar potential limitations under restrictive assumptions on the cost landscape.
• Clifford Data Regression does not change the resolvability of cost pairs when using the same linear ansatz, but numerically can improve trainability in certain settings.
• Overall, EM can worsen or not improve trainability in many scenarios; careful design of EM methods is needed to enhance VQA trainability.