量子生成模型中谱滤波的经典极限
Classical Limits of Spectral Filtering in Quantum Generative Models
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中文总结 AI 辅助
该研究探讨量子生成模型的谱滤波是否能产生经典后处理无法匹配的结果，推导了幅度滤波器的二分条件，发现其未产生量子-经典差距，纯相位滤波器是唯一不受约束的对角谱操作。
中文摘要 AI 辅助
谱滤波已被提出作为量子生成模型的一种正则化途径：量子傅里叶变换可揭示量子电路Born机的振幅谱，而对角滤波器能抑制与有限样本噪声相关的高频分量，其经典对应操作看似需要处理指数长度的振幅向量。我们研究这种相干操作是否能产生未滤波模型样本的经典后处理无法匹配的结果。在匹配采样成本（考虑衰减的后选择开销）下，将滤波器与对称概率核的卷积进行对比，我们推导了两者差距消失的必要充分条件。幅度（衰减）滤波器遵循二分法：在固定可承受阈值下，滤波输出要么是具有高效经典采样器的恒定大小傅里叶对象，要么必须拓宽通带直至不再衰减任何固定频率且滤波器不再平滑。两种情况下均未产生量子-经典差距，剩余差距由输入态的谱相位继承。对训练后的电路Born机的数值实验证实了该分类，且决定性相位对Born规则训练损失不可见，由初始化设定。在对角族中，纯相位滤波器仍是唯一不受这些约束的谱操作。
英文摘要
Spectral filtering has been proposed as a route to regularization in quantum generative models: the quantum Fourier transform exposes the amplitude spectrum of a quantum circuit Born machine, and a diagonal filter suppresses the high frequencies associated with finite-sample noise, an operation whose classical counterpart seemingly requires manipulating an exponentially long amplitude vector. We examine whether this coherent operation produces anything that classical post-processing of samples from the unfiltered model cannot match. Measuring the filter against convolution with a symmetric probability kernel at matched sampling cost, which accounts for the post-selection overhead of attenuation, we derive necessary and sufficient conditions for the gap between the two to vanish. Magnitude (attenuating) filters obey a dichotomy: at a fixed affordability threshold, the filtered output is either a constant-size Fourier object with an efficient classical sampler, or the passband must widen until no fixed frequency is attenuated and the filter no longer smooths. In neither case does the filter create a quantum-classical separation. Whatever separation survives is inherited from the spectral phase of the input state. Numerical experiments on trained circuit Born machines confirm the classification and show that the deciding phases are invisible to the Born-rule training loss and set by the initialization. Within the diagonal family, pure phase filters remain the only spectral operations exempt from these constraints.
发表机构
- Fraunhofer Institute for Industrial Engineering IAO（弗劳恩霍夫工业工程研究所IAO）
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