Beyond MNIST: Limitations of Amplitude Encoding on Quantum Classification

It remains unclear whether quantum machine learning (QML) truly holds an advantage when tackling practical and meaningful tasks. In exploring this question, the importance of classical data encoding, often overlooked, becomes increasingly evident. Amplitude encoding, which can embed $2^n$ classical data into $n$ qubits, is widely used due to its apparent efficiency. However, its potential limitations for QML have yet to be fully explored. In this paper, we establish a theoretical result for the limitations caused by quantum encoding and point out the existence of a concentration phenomenon in amplitude encoding. Compared to prior work, our theoretical result operates under more general conditions and leads to a stronger conclusion. This concentration phenomenon causes the classification predictions to remain close to random guessing, regardless of the training process. Our findings shed new light on a long-standing puzzle in the field of QML: why some QML models perform well on simple datasets like MNIST but fail to generalize to more complex practical tasks. By highlighting the pivotal role of encoding design in QML, our work clearly indicates that future research needs to focus more on the design of classical data encoding to advance the effectiveness of QML.