This paper presents a quantum cardinality estimation (QCardEst) technique using quantum machine learning and hybrid quantum-classical networks. It encodes SQL queries into a compact quantum state that requires only the number of qubits equal to the number of tables contained in the query, enabling the entire query to be processed by a single variational quantum circuit (VQC) on current hardware. Furthermore, we compare several classical postprocessing layers to convert the probability vector output of VQC into a cardinality value, and introduce quantum cardinality correction (QCardCorr), which multiplies the coefficients generated by VQC to improve the classical cardinality estimator. Using QCardCorr, we achieve a 6.37x performance improvement over the standard PostgreSQL optimizer on JOB-light, an 8.66x improvement on STATS, and a 3.47x improvement over MSCN on JOB-light.