Does provable absence of barren plateaus imply classical simulability?

TL;DR

The paper investigates whether the absence of barren plateaus in variational quantum algorithms implies efficient classical simulability. By formalizing BP-free losses as confined to polynomial subspaces via adjoint action, it shows that many BP-free architectures admit polynomial-time classical or quantum-enhanced classical simulation after an initial data-acquisition phase. The authors outline a general three-step framework to perform such simulations and illustrate case-by-case where simulability holds, while explicitly noting caveats and potential exceptions. They discuss new opportunities, including hybrid data-driven schemes and architecture-aware resource considerations, and argue that BP absence does not automatically guarantee quantum advantage, prompting a principled re-evaluation of VQA approaches as hardware scales. Overall, the work provides a nuanced perspective on the dequantization of VQAs and highlights directions for balanced, resource-aware quantum-classical workflows.

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 collect evidence-on a case-by-case basis-that many commonly used models whose loss landscapes avoid barren plateaus can also admit classical simulation, 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 that quantum computers can be essential for collecting data, our analysis sheds doubt on the information processing capabilities of many parametrized quantum circuits with provably barren plateau-free landscapes. We end by discussing the (many) caveats in our arguments including the limitations of average case arguments, the role of smart initializations, models that fall outside our assumptions, the potential for provably superpolynomial advantages and the possibility that, once larger devices become available, parametrized quantum circuits could heuristically outperform our analytic expectations.

Figures (6)

• Figure 1: Schematic description of the simulation classes. a) Problems in ${\rm CSIM}$ are those for which there exists a fully classical algorithm that takes as input the problem description and estimates the loss function in polynomial time with a classical computer. Here, access to a quantum computer is not needed. b) Problems in ${\rm QESIM}$ are those where one is allowed access to a quantum computer for an initial data acquisition phase that takes no more than polynomial time. At the end of this phase, access to the quantum computer is no longer allowed. A classical algorithm then takes the problem's description, and the data obtained from the quantum device, to estimate the loss function in polynomial time. Note that one can compute problems in ${\rm QESIM}$ without needing to run a parametrized quantum circuit on the quantum hardware. c) Problems in ${\rm QSIM}$ are those where one allows 'on demand' access to a quantum computer. Here, one usually implements the parametrized quantum circuit on the device.
• Figure 2: Class inclusions. We show the inclusions between the classes ${\rm CSIM}$, ${\rm QESIM}$ and ${\rm QSIM}$. The region outside of ${\rm CSIM}$ (highlighted with stripes) corresponds to problems where a quantum advantage could potentially be achieved.
• Figure 3: Adjoint action subspaces. a) Given an operator $P_\lambda$ and some parameters $\boldsymbol{\theta}$, we say $\mathcal{B}_\lambda\subset\mathcal{B}$ is a proper subspace if $\left\langle U(\boldsymbol{\theta})P_\lambda U^{\dagger}(\boldsymbol{\theta}),P_j \right\rangle^2$ is non-zero only for operators $P_j$ in $\mathcal{B}_\lambda$. b) We will instead say that $\mathcal{B}_\lambda$ is an effective subspace if $\left\langle U(\boldsymbol{\theta})P_\lambda U^{\dagger}(\boldsymbol{\theta}),P_j \right\rangle^2$ is non-zero for many operators outside of $\mathcal{B}_\lambda$, but is only large for operators in $\mathcal{B}_\lambda$. Proper and effective subspaces can arise either for all $\boldsymbol{\theta}$, or with high probability when sampling $\boldsymbol{\theta}$ from $\mathcal{P}$.
• Figure 4: Subspaces for shallow hardware efficient ansätze. We consider classes of problems where the unitary $U(\boldsymbol{\theta})$ is an $L$-layered hardware efficient ansatz with two-qubit gates acting on alternating pairs of neighboring qubits in a brick-like fashion. We further assume that $L\in\mathcal{O}(\log(n))$. a) For a global operator such as $O=Z^{\otimes n}$, the subspace obtained by adjoint action of $U(\boldsymbol{\theta})$ is exponentially large $\forall L$. b) Given a local operator, such as $O=Z_\mu$, one can see that for all $\boldsymbol{\theta}$ the ensuing subspace is proper and only contains Pauli operators acting on at most $\mathcal{O}(\log(n))$ neighboring qubits. Hence, this subspace is only polynomially large. Colored regions depict the backwards light cone of the measurement operator when Heisenberg evolved leone2022practical.
• Figure 5: Absence of barren plateaus, simulability, and faithful training. a) We consider a problem in $\overline{{\rm BP}}$ such that, with high probability for some $\boldsymbol{\theta}\sim\mathcal{P}$, the circuit's adjoint action leads to a polynomial effective subspace as in Fig. \ref{['fig:modules']}(b). Training on the classically estimated loss can be faithful for the first few optimization steps. However, as the optimization continues, a vital contribution to the loss could arise from operators not in $\mathcal{B}_\lambda$. If this occurs, training on the classically estimated loss can be unfaithful and one can converge towards a parameter region that does not correspond to the minima of the true 'full' loss function. b) To mitigate the aforementioned issues, one could use an alternative form of hybrid variational quantum computing where multiple, iterative, data acquisition steps are used. This information is then used by a classical computer to update a classical simulation of the loss and make it more faithful as the optimization progresses. As schematically shown, this scheme could make training the loss on a classical computer more faithful.

Theorems & Definitions (4)

• Claim 1
• Claim 2
• Theorem 1: Classical simulability of QCNNs
• proof