SPARTA: $χ^2$-calibrated, risk-controlled exploration-exploitation for variational quantum algorithms

TL;DR

Variational quantum algorithms suffer from barren plateaus, where gradient information vanishes with system size, impeding scalable optimization. SPARTA provides a $χ^2$-calibrated, sequential regime detector that guarantees anytime-valid control while alternating between probabilistic trust-region exploration on plateaus and variance-proportional, gCANS-style exploitation in informative basins, guided by Lie-algebraic insights. Theoretical results connect dynamical Lie algebra structure to finite-sample decision rules, yielding geometric plateau exit bounds and linear convergence in informative regions, with Lie-informed shot allocation enhancing test power without compromising calibration. Empirically, SPARTA demonstrates deeper minima and faster convergence than gradient-only baselines on TFIM and synthetic barren-plateau landscapes, highlighting its practical potential for near-term quantum optimization under shot noise and budget constraints.

Abstract

Variational quantum algorithms face a fundamental trainability crisis: barren plateaus render optimization exponentially difficult as system size grows. While recent Lie algebraic theory precisely characterizes when and why these plateaus occur, no practical optimization method exists with finite-sample guarantees for navigating them. We present the sequential plateau-adaptive regime-testing algorithm (SPARTA), the first measurement-frugal scheduler that provides explicit, anytime-valid risk control for quantum optimization. Our approach integrates three components with rigorous statistical foundations: (i) a $χ^2$-calibrated sequential test that distinguishes barren plateaus from informative regions using likelihood-ratio supermartingales; (ii) a probabilistic trust-region exploration strategy with one-sided acceptance to prevent false improvements under shot noise; and (iii) a theoretically-optimal exploitation phase that achieves the best attainable convergence rate. We prove geometric bounds on plateau exit times, linear convergence in informative basins, and show how Lie-algebraic variance proxies enhance test power without compromising statistical calibration.

Figures (5)

• Figure 1: Validation of the chi‐squared model in plateau and informative regions. (a) Histogram of the whitened statistic $s$ in a near‐plateau region ($\|\nabla f(\theta)\|\approx0$) from $N=5000$ samples (blue bars), overlaid with the central $\chi^2_4$ density (red curve); Kolmogorov–Smirnov test: $p=0.147$. (b) Histogram of $s$ in an informative region ($\|\nabla f(\theta)\|\approx0.87$) from $N=5000$ samples (blue bars), overlaid with the noncentral $\chi^2_4(\lambda)$ density (red curve); Kolmogorov–Smirnov test: $p=0.431$.
• Figure 2: Multi-run robustness comparison on 6-qubit TFIM with varied starting positions. SPARTA (solid blue lines, $n=10$ runs) versus standard gCANS (dashed red lines, $n=10$ runs) on a transverse-field Ising model with chain topology. Each trial uses different random seed (42, 123, 456, 789, 1011, 2022, 3033, 4044, 5055, 6066) and varied initial parameters ($\gamma = 0.5 \pm 0.1$, $\beta = 0.4 \pm 0.1$) with identical shot budget (25,000). SPARTA adaptively switches between plateau-region trust exploration (PTR) and gradient-based exploitation (gCANS), achieving significantly lower mean cost ($-3.455 \pm 0.829$) versus gCANS ($-2.667 \pm 0.446$) with robust 90% win rate (9/10 trials).
• Figure 3: Convergence on Lie-inspired barren plateau. Comparison on a 12D synthetic landscape with exponentially suppressed gradients (75% variance suppression, gorge width $w \approx 0.51$). SPARTA (blue) discovers the deep gorge via PTR exploration, converging to near-optimal cost ($-29.12$), while gCANS (red) remains trapped in the plateau region (final cost: $0.00$). Both use 250,000 shots starting from distance $\approx 6.1\times$ the gorge width. The orange dashed line marks the global minimum ($-30.0$).
• Figure 4: Barren plateau landscape structure. Gradient magnitude heatmap (log scale) reveals exponentially suppressed signal outside the narrow gorge (cyan dashed circle, radius $w \approx 0.51$). SPARTA's trajectory (blue) successfully locates the gorge via random PTR exploration from the plateau region, while gCANS (red) wanders aimlessly following weak, noisy gradients. The yellow star marks the global minimum. This 2D projection (first two coordinates of 12D space) visualizes why gradient-based methods fail under barren plateau conditions: directional information vanishes outside the gorge.
• Figure 5: Scaling comparison of SPARTA vs. gCANS on TFIM chains. Energy trajectories are shown as a function of cumulative shots for systems of 2, 4, 6, and 8 qubits. SPARTA (blue solid) employs sequential hypothesis testing to detect plateau regimes and switches between trust-region exploration and variance-adaptive exploitation. gCANS (red dashed) uses pure exploitation with variance-proportional shot allocation. SPARTA achieves superior convergence on all scales through regime-aware optimization, reaching lower final costs with fewer iterations.

Theorems & Definitions (16)

• Remark 1
• Definition 1: Whitened gradient statistic
• Proposition 1: Distribution under competing hypotheses
• proof
• Definition 2: Dynamical Lie algebra
• Theorem 1: Variance formula, Ragone et al.
• Proposition 2: Expected non-centrality
• Corollary 2: Plateau detection difficulty
• Proposition 3: Optimal exploration allocation
• proof