  Quantum Convolutional Layer (QCL) is considered as one of the core of Quantum
Convolutional Neural Networks (QCNNs) due to its efficient data feature
extraction capability. However, the current principle of QCL is not as
mathematically understandable as Classical Convolutional Layer (CCL) due to its
black-box structure. Moreover, classical data mapping in many QCLs is
inefficient. To this end, firstly, the Quantum Adjoint Convolution Operation
(QACO) consisting of a quantum amplitude encoding and its inverse is
theoretically shown to be equivalent to the quantum normalization of the
convolution operation based on the Frobenius inner product while achieving an
efficient characterization of the data. Subsequently, QACO is extended into a
Quantum Adjoint Convolutional Layer (QACL) by Quantum Phase Estimation (QPE) to
compute all Frobenius inner products in parallel. At last, comparative
simulation experiments are carried out on PennyLane and TensorFlow platforms,
mainly for the two cases of kernel fixed and unfixed in QACL. The results
demonstrate that QACL with the insight of special quantum properties for the
same images, provides higher training accuracy in MNIST and Fashion MNIST
classification experiments, but sacrifices the learning performance to some
extent. Predictably, our research lays the foundation for the development of
efficient and interpretable quantum convolutional networks and also advances
the field of quantum machine vision.
