Stacking the Deck: Tunable Trainability in Stacked LCUs

2026-07-27

AI Summary

This work addresses the tension between expressivity and trainability in variational quantum circuits, where highly expressive ansätze often suffer from barren plateaus, while structures avoiding this issue are frequently classically simulable. The authors propose a Stacked Linear Combination of Unitaries (S-LCU) ansatz that systematically balances trainability and computational complexity through a single tunable parameter—the number of layers $l$. Constructed from fermionic Gaussian unitaries, the S-LCU enables analytical derivation of a lower bound on the variance of the loss landscape, proven to be $\Omega(1/(n k^{3l}))$. Crucially, its classical simulation cost scales as $O(k^{2l} n^3)$, whereas the quantum gate complexity grows only as $O(l k n^2)$, thereby achieving a tunable trade-off between quantum advantage and trainability.

Abstract

Variational quantum circuits have been central to many proposed near-term applications of quantum computing, but a growing body of evidence suggests that trainability and quantum advantage are fundamentally at odds: ansätze expressive enough to resist efficient classical simulation tend to exhibit barren plateaus, while structures that provably rule out barren plateaus typically render them classically simulable. We propose a stacked linear combination of unitaries (S-LCU) as a variational ansatz which provides a tunable trade-off between barren plateaus and classical simulability. Using a diagrammatic analysis, we bound the loss-landscape variance of the Free Fermion S-LCU, whose elements are fermionic Gaussian unitaries. We prove a variance lower bound of $Ω(1/(n k^{3l}))$, with a simulation cost of $O(k^{2l} n^3)$ using the best known classical algorithm, compared to a quantum gate complexity of only $O(lkn^2)$. The number of layers $l$ serves as a single dial that trades computational complexity against the rate of cost concentration. This offers practitioners a systematic method for constructing ansätze with a complexity-trainability trade-off that best suits their application and hardware.

Problem

• barren plateaus
• classical simulability
• variational quantum circuits
• quantum advantage
• trainability

Innovation

• stacked LCU
• barren plateaus
• classical simulability
• variational quantum circuits
• trainability