[论文解读] The Inductive Bias of Quantum Kernels

Jonas M. Kübler, Simon Buchholz+1 more|arXiv (Cornell University)|Jun 7, 2021

Quantum Computing Algorithms and Architecture参考文献 44被引用 6

一句话总结

本文研究了量子核方法的归纳偏置，表明只有当量子核编码了经典计算难以模拟的问题特异性知识时，量子机器学习中的量子优势才会出现。作者证明，低维再生核希尔伯特空间（RKHS）且包含难以计算的函数时可实现量子优势，但估计此类核需要指数级多的测量次数，构成重大实际障碍。

ABSTRACT

It has been hypothesized that quantum computers may lend themselves well to applications in machine learning. In the present work, we analyze function classes defined via quantum kernels. Quantum computers offer the possibility to efficiently compute inner products of exponentially large density operators that are classically hard to compute. However, having an exponentially large feature space renders the problem of generalization hard. Furthermore, being able to evaluate inner products in high dimensional spaces efficiently by itself does not guarantee a quantum advantage, as already classically tractable kernels can correspond to high- or infinite-dimensional reproducing kernel Hilbert spaces (RKHS). We analyze the spectral properties of quantum kernels and find that we can expect an advantage if their RKHS is low dimensional and contains functions that are hard to compute classically. If the target function is known to lie in this class, this implies a quantum advantage, as the quantum computer can encode this inductive bias, whereas there is no classically efficient way to constrain the function class in the same way. However, we show that finding suitable quantum kernels is not easy because the kernel evaluation might require exponentially many measurements. In conclusion, our message is a somewhat sobering one: we conjecture that quantum machine learning models can offer speed-ups only if we manage to encode knowledge about the problem at hand into quantum circuits, while encoding the same bias into a classical model would be hard. These situations may plausibly occur when learning on data generated by a quantum process, however, they appear to be harder to come by for classical datasets.

研究动机与目标

• 理解量子核方法在何种条件下可为经典模型提供计算优势。
• 通过其再生核希尔伯特空间（RKHS）的谱分析，形式化量子核的归纳偏置。
• 在高维特征空间的挑战下，识别量子核在何种情况下能实现良好泛化。
• 研究量子核方法中表达能力与泛化能力之间的权衡。
• 澄清量子优势是否可在经典数据集上实现，或仅在量子过程生成的数据上可行。

提出的方法

• 通过分析量子核的谱特性来表征其归纳偏置，重点关注核矩阵的特征值谱。
• 证明过度表达的量子嵌入会导致泛化性能差，从而确立核设计的理论极限（定理1）。
• 提出一种将量子核投影到低维子空间的方法，以编码经典难以实现的归纳偏置（定理2）。
• 使用数值模拟评估合成数据集上核的性能，比较完整量子核、RBF核和有偏量子核。
• 采用带正则化的核岭回归评估泛化性能，通过选择最优正则化参数来界定性能上限。
• 模拟完整的量子态演化以计算精确的核值，避免测量噪声以确保理论清晰性。

实验结果

研究问题

• RQ1 在何种条件下，量子核方法可在机器学习中超越经典模型？
• RQ2 量子核的谱结构如何影响其泛化性能？
• RQ3 量子核能否编码经典难以模拟的归纳偏置？
• RQ4 估计有偏量子核的成本是多少，其随系统规模的增长是否有利？
• RQ5 量子优势在经典数据集上是否可行，还是仅在量子过程生成的数据上可行？

主要发现

• 当数据嵌入到量子希尔伯特空间时过于表达，量子核方法将无法泛化，如定理1所示。
• 只有当量子核的RKHS为低维且包含经典难以计算的函数时，才可能实现量子优势。
• 通过投影可工程化量子核的谱偏差，从而创建经典难以复制的归纳偏置。
• 估计此类有偏量子核需要指数级多的测量次数，这一限制与量子神经网络中的“ barren plateaus”现象类似。
• 实验结果证实，即使使用最优正则化，完整量子核和RBF核的性能仍表现不佳，而有偏核因具有低维RKHS而能实现良好泛化。
• 对于有偏核，核目标对齐度随量子比特数增加而提升，且与学习性能的改善呈正相关。