The following is based on translations from Chinese media:
So far there has been no news about Chinese nuclear submarine cruising near the US. There has been official confirmation from China that a Chinese nuclear submarine operated in the Indian Ocean for a short period of time.
US nuclear submarine commander Benjamin Person (transliteration from Chinese) recently laughed at China, saying that Chinese nuclear submarines had never been active near the United States. He is happy that his nuclear submarine has been active in the seas near China for a long time.
This qianzhan.com article comments on Person’s words, saying that China obviously lags behind the US, but it has been making great efforts to catch up. China has established a coastal anti-submarine network and improved and upgraded the equipment of its anti-submarine aircraft and warships.