Circuit Design and Efficient Simulation of Quantum Inner Product and Empirical Studies of Its Effect on Near-Term Hybrid Quantum-Classic Machine Learning
Authors: Hao Xiong, Yehui Tang, Xinyu Ye, Junchi Yan

For the essential operation namely inner product (IP) as widely adopted in classic computing e.g. matrix multiplication its quantum counterpart: quantum inner product (QIP) has also been recently theoretically explored with a verifiable lower complexity on quantum computers. However it remains unclear for the embodiment of the quantum circuits (QC) for QIP let alone a (thorough) evaluation of the QIP circuits especially in a practical context in the NISQ era by applying QIP to ML via hybrid quantum-classic pipelines. In this paper we carefully design the QIP circuits from scratch whose complexity is in accordance with the theoretical complexity. To make the simulation tractable on classic computers especially when it is integrated in the gradient-based hybrid ML pipelines we further devise a highly-efficient simulation scheme by directly simulates the output state. Experiments show that the scheme accelerates the simulation for more than 68k times compared with the previous circuit simulator. This allows our empirical evaluation on typical machine learning tasks ranging from supervised and self-supervised learning via neural nets to K-Means clustering. The results show that the calculation error brought by typical quantum mechanisms would incur in general little influence on the final numerical results given sufficient qubits. However certain tasks e.g. ranking in K-Means could be more sensitive to quantum noise.