Information-Theoretic Constraints on Variational Quantum Optimization: Efficiency Transitions and the Dynamical Lie Algebra

TL;DR

This work reframes variational quantum optimization as a thermodynamic problem governed by information flow through a single ancilla. It derives a constitutive law linking extracted work to mutual information and demonstrates a quantum advantage from entanglement over classical Landauer bounds. By comparing polynomial (Ordered) and exponential (Chaotic) Dynamical Lie Algebra growth, it reveals a complexity-dependent efficiency transition with an efficiency collapse near six qubits, tying trainability to information-channel capacity limits. The results offer a thermodynamic lens on quantum learning, suggesting design principles to maintain information superconductivity and guiding future strategies to extend the tractable regime of VQAs.

Abstract

Variational quantum algorithms are the leading candidates for near-term quantum advantage, yet their scalability is limited by the ``Barren Plateau'' phenomenon. While traditionally attributed to geometric vanishing gradients, we propose an information-theoretic perspective. Using ancilla-mediated coherent feedback, we demonstrate an empirical constitutive relation $ΔE \leq ηI(S:A)$ linking work extraction to mutual information, with quantum entanglement providing a factor-of-2 advantage over classical Landauer bounds. By scaling the system size, we identify a distinct efficiency transition governed by the dimension of the Dynamical Lie Algebra. Systems with polynomial algebraic complexity exhibit sustained positive efficiency, whereas systems with exponential complexity undergo an ``efficiency collapse'' ($η\to 0$) at $N \approx 6$ qubits. These results suggest that the trainability boundary in variational algorithms correlates with information-theoretic limits of quantum feedback control.

Figures (4)

• Figure 1: The Thermodynamic Constitutive Law. Extracted Work ($-\Delta \langle H \rangle$) vs. Mutual Information $I(S:A)$ for a 4-qubit transverse Ising system. The robust linear relationship ($R^2 \approx 0.90$, slope $\eta = 0.247$ energy/bit) confirms the constitutive law $\Delta E \leq \eta I$. The sensing time $\tau$ was varied from 0 to 1.5 while the feedback strength was held constant at $\theta_{gain} = 0.5$ rad, isolating information as the sole variable driving work extraction.
• Figure 2: Quantum Advantage via Landauer Analysis. Extracted work (blue) vs. Landauer erasure cost (red dashed) as a function of sensing time $\tau$. The green shaded region indicates net positive work after accounting for information erasure. The constant ratio $I(S:A)/S(A) = 2.00$ confirms that quantum entanglement (not classical correlation) fuels the engine.
• Figure 3: The Complexity-Dependent Efficiency Transition. Normalized algorithmic efficiency ($\eta/N^2$) vs. System Size $N$ for Ordered (Ferromagnet, blue) and Chaotic (Spin Glass, red) Hamiltonians. The Ordered phase maintains positive efficiency, while the Chaotic phase undergoes efficiency collapse, with efficiency approaching zero at $N \ge 6$. Error bars represent $\pm 1\sigma$ over 5 random seeds. The transition is consistent with the ancilla channel capacity (1 bit) becoming insufficient to resolve the scrambled gradient information.
• Figure 4: DLA Efficiency Scaling. Comparison of thermodynamic efficiency for Ordered (Complete Graph, polynomial DLA $\sim O(N^3)$) vs. Chaotic (Spin Glass, exponential DLA $\sim O(4^N)$) systems. The polynomial scaling of the algebra ensures sustained positive efficiency, while the exponential algebra exhibits the efficiency crash observed in Fig. \ref{['fig:bifurcation']}.