  Quantum Architecture Search (QAS) is a promising approach to designing
quantum circuits for variational quantum algorithms (VQAs). However, existing
QAS algorithms require to evaluate a large number of quantum circuits during
the search process, which makes them computationally demanding and limits their
applications to large-scale quantum circuits. Recently, predictor-based QAS has
been proposed to alleviate this problem by directly estimating the performances
of circuits according to their structures with a predictor trained on a set of
labeled quantum circuits. However, the predictor is trained by purely
supervised learning, which suffers from poor generalization ability when
labeled training circuits are scarce. It is very time-consuming to obtain a
large number of labeled quantum circuits because the gate parameters of quantum
circuits need to be optimized until convergence to obtain their ground-truth
performances. To overcome these limitations, we propose GSQAS, a graph
self-supervised QAS, which trains a predictor based on self-supervised
learning. Specifically, we first pre-train a graph encoder on a large number of
unlabeled quantum circuits using a well-designed pretext task in order to
generate meaningful representations of circuits. Then the downstream predictor
is trained on a small number of quantum circuits' representations and their
labels. Once the encoder is trained, it can apply to different downstream
tasks. In order to better encode the spatial topology information and avoid the
huge dimension of feature vectors for large-scale quantum circuits, we design a
scheme to encode quantum circuits as graphs. Simulation results on searching
circuit structures for variational quantum eigensolver and quantum state
classification show that GSQAS outperforms the state-of-the-art predictor-based
QAS, achieving better performance with fewer labeled circuits.
