The Fed is not state-run.
The Federal Reserve is a private institution authorized by Congress and an independent institution jointly established by member banks. Among them, member banks are controlled by many commercial banks and are subject to the supervision of the US Congress. They must also work under the economic policy framework established by the United States. The full name of the Federal Reserve is the Federal Reserve System, which consists of the Federal Reserve Board in Washington, D.C. and 12 regional Federal Reserve Banks located in major cities across the country. The main monetary policy of the Federal Reserve is jointly formulated by the members of the Federal Reserve and the chairman of the Federal Reserve Bank.