  At present, in the face of the huge and complex data in cloud computing, the
parallel computing ability of quantum computing is particularly important.
Quantum principal component analysis algorithm is used as a method of quantum
state tomography. We perform feature extraction on the eigenvalue matrix of the
density matrix after feature decomposition to achieve dimensionality reduction,
proposed quantum principal component extraction algorithm (QPCE). Compared with
the classic algorithm, this algorithm achieves an exponential speedup under
certain conditions. The specific realization of the quantum circuit is given.
And considering the limited computing power of the client, we propose a quantum
homomorphic ciphertext dimension reduction scheme (QHEDR), the client can
encrypt the quantum data and upload it to the cloud for computing. And through
the quantum homomorphic encryption scheme to ensure security. After the
calculation is completed, the client updates the key locally and decrypts the
ciphertext result. We have implemented a quantum ciphertext dimensionality
reduction scheme implemented in the quantum cloud, which does not require
interaction and ensures safety. In addition, we have carried out experimental
verification on the QPCE algorithm on IBM's real computing platform, and given
a simple example of executing hybrid quantum circuits in the cloud to verify
the correctness of our scheme. Experimental results show that the algorithm can
perform ciphertext dimension reduction safely and effectively.
