Does provable absence of barren plateaus imply classical simulability? Or, why we need to rethink variational quantum computing (2312.09121v2)

Abstract: A large amount of effort has recently been put into understanding the barren plateau phenomenon. In this perspective article, we face the increasingly loud elephant in the room and ask a question that has been hinted at by many but not explicitly addressed: Can the structure that allows one to avoid barren plateaus also be leveraged to efficiently simulate the loss classically? We present strong evidence that commonly used models with provable absence of barren plateaus are also classically simulable, provided that one can collect some classical data from quantum devices during an initial data acquisition phase. This follows from the observation that barren plateaus result from a curse of dimensionality, and that current approaches for solving them end up encoding the problem into some small, classically simulable, subspaces. Thus, while stressing quantum computers can be essential for collecting data, our analysis sheds serious doubt on the non-classicality of the information processing capabilities of parametrized quantum circuits for barren plateau-free landscapes. We end by discussing caveats in our arguments, the role of smart initializations and the possibility of provably superpolynomial, or simply practical, advantages from running parametrized quantum circuits.

• The paper demonstrates that eliminating barren plateaus often restricts variational quantum circuits to polynomial subspaces.
• It reveals that these restrictions can enable efficient classical simulation of problems originally thought to require quantum resources.
• The study challenges the conventional quantum advantage, prompting a reevaluation of hybrid quantum-classical strategies in variational computing.

Rethinking Variational Quantum Computing: Implications of Barren Plateau Elimination

The paper "Does provable absence of barren plateaus imply classical simulability? Or, why we need to rethink variational quantum computing" explores a nuanced exploration of the challenges associated with variational quantum computing, particularly focusing on the phenomenon known as "barren plateaus." The authors investigate whether the elimination of barren plateaus, a significant barrier in training quantum algorithms, inadvertently suggests that such quantum problems might also be classically simulable.

Background and Motivation

Barren plateaus refer to the concentration of the loss landscape in variational quantum algorithms, where gradients become exponentially small with increasing system size, rendering the optimization process intractable. Various strategies to circumvent this issue have emerged, such as restricting circuit depth and embedding symmetries that preserve useful information structures. These approaches have been pivotal in advancing the feasibility of variational quantum algorithms.

Key Contributions

The core inquiry of the paper is whether strategies that circumvent barren plateaus inherently suggest that these quantum models can be efficiently simulated by classical means. The investigation centers on the realization that avoiding barren plateaus often relies on confining operations within polynomially sized subspaces of the exponentially large operator space.

1. Subspace Analysis: The authors posit that many barren plateau-free architectures result in loss functions that inhabit classically identifiable polynomial subspaces. Through an analysis of these architectures, they argue that in all cases studied, a proof of non-exponential concentration coincides with a projection onto a smaller subspace, suggesting inherent classical tractability.
2. Implicative Simulability: By establishing that quantum problems within these subspaces are simulable via classical algorithms, sometimes with initial quantum data acquisition phases, the paper challenges the inherent quantum advantage claimed by such algorithms. The use of quantum resources could often be reduced to non-adaptive data acquisition, transforming the hybrid approach typically seen in variational algorithms.
3. Theoretical and Practical Implications: This line of reasoning entails significant implications for the perceived quantum advantage. By moving computations to a classical domain, it raises a flag on the necessity and practical relevance of deploying variational quantum algorithms on quantum devices, especially when classical surrogates might suffice.

Future Directions and Caveats

While the paper is thorough in its analysis, several caveats and future directions are identified:

• Heuristic Arguments: Not all non-concentrated loss functions are inherently classically simulable. Theoretical constructs leveraging conventional quantum algorithms could still hold potential for exponential speedups that the current variational techniques overlook.
• Hybrid Approaches: The authors propose a refined hybrid quantum-classical approach, where iterative data acquisition enhances classical simulations' fidelity, presenting opportunities for more efficient optimization strategies.
• Smart Initialization: The potential of initialization strategies that intelligently avoid barren plateaus without known subspaces poses an intriguing future research avenue. These may reveal unexplored territories where quantum processing retains superiority.

Conclusion

This paper provides a critical lens through which the relationship between barren plateaus and classical simulability is analyzed, urging a reevaluation of variational quantum computing paradigms. It encourages a shift toward understanding when quantum circuits genuinely provide computational advantages and how classical algorithms may serve as scalable surrogates in many proven barren plateau-free landscapes. This insight bridges a vital gap in identifying the true potential and limitations of current quantum computing approaches.