量子电路拟合洛伦兹混沌系统失败,根源是架构容量上限,非梯度消失。
An architectural capacity ceiling, not a barren plateau: why a fixed-encoding variational quantum circuit cannot fit the Lorenz-63 attractor
- 固定时间编码导致电路表达能力受限,深度增加无法突破
- 梯度衰减仅九倍且饱和,远高于训练门槛,非指数崩溃
- 输出雅可比秩上限33,与傅里叶分析预测完全一致,适合架构研究者
变分量子电路在混沌预报中表现不佳,常归因于梯度消失(巴伦平原)。通过一个可精确模拟的四量子比特变分量子物理信息电路拟合洛伦兹-63系统,我们发现巴伦平原解释不成立:失败根源在于电路时间编码固定的架构容量上限,而非可训练深度。四项证据支持:(i) McClean可比梯度方差估计器在n=4时为3.9×10⁻³,接近局部代价哈亚/2设计尺度,结构活跃参数下深度增加使梯度方差下降约九倍后饱和,数值足够训练,非指数衰减;(ii) 在200次优化器迭代(分三阶段共600次)预算下,梯度下降、分层优化和SPSA达到相同量级损失,无优化器能解锁更优基域;(iii) 输出雅可比秩从五层起饱和于33,深度无法引入新输出方向;(iv) 傅里叶分析表明:量子比特1相位编码作用于初始|0>态,呈惰性,最大可达频率为2.5/t_max=0.83 Hz,各深度相同,约低于洛伦兹最窄分量带宽4.4倍。修正后频带维度每可观测量为1+2×5=11,3×11=33与实测秩上限一致,统一两项诊断。训练深度扫描验证:平均损失随深度提升后趋于平坦,恰在秩饱和处。修正此前未归一化梯度范数对比McClean阈值的误判,将固定储水库与经典回声状态网络的优势归于架构,而非量子力学。
原文摘要 · Abstract (English)
Variational quantum circuits train poorly on chaotic forecasting, usually blamed on barren plateaus (exponentially vanishing gradients). Using an exactly simulable four-qubit variational quantum physics-informed circuit fit to Lorenz-63, we show the barren-plateau explanation fails: the failure is an architectural capacity ceiling fixed by the circuit time-encoding, not its trainable depth. Four measurements support this. (i) A McClean-comparable gradient-variance estimator sits at the local-cost Haar/2-design scale 2^(-2n)=3.9e-3 at n=4; on structurally live parameters it decays about ninefold with depth then saturates there, large enough to train, not an exponential collapse. (ii) At a common budget of 200 optimiser iterations (600, in three stages, for layer-wise), gradient descent, layer-wise, and SPSA reach the same order of magnitude of loss, so no optimiser unlocks a better basin. (iii) The output-Jacobian rank saturates at 33 from five layers on, so depth buys no new output directions. (iv) A Fourier analysis explains why: the qubit-1 phase encoding acts on the initial |0> and is inert, so the maximum accessible frequency is 2.5/t_max=0.83 Hz, identical at every depth and about 4.4x below the narrowest Lorenz component bandwidth. The corrected band has dimension 1+2x5=11 per observable, and 3x11=33 equals the measured rank ceiling exactly, unifying the two diagnostics. A trained depth sweep agrees: mean loss improves with depth then flattens once the rank saturates. We correct our earlier preprint diagnosis, which compared unnormalised gradient norms to the McClean threshold, and place the advantage of fixed reservoirs and classical echo-state networks in architecture, not quantum mechanics.
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