Quantum for Universal Advancement in Nationwide Technology Use and Modernization for National Security Act of 2021 or the QUANTUM for National Security Act of 2021 This bill addresses quantum science research and its role in national security through the Department of Defense (DOD). For purposes of the public-private talent exchange program, DOD must ensure there are exchanges of personnel that are working on the modernization priorities of DOD, including quantum science and technology research. Among other requirements, DOD must seek to establish public-private exchange programs in the Army, Navy, and Marine Corps focused on private sector entities working on quantum information sciences and technology research applications; implement the National Security Pipeline Pilot Program to seek to enter into partnerships with minority-serving institutions to diversify participants in the Science, Mathematics, and Research Transformation (SMART) Defense Education Program; and implement a fellowship program in quantum information science and technology research and development for individuals who have graduate or post-graduate degrees. The Government Accountability Office (GAO) must assess the National Quantum Initiative Program and report on its findings. Additionally, the GAO must brief Congress on its assessment of the quantum information fellowship program established by this bill. The President must establish, through the National Science and Technology Council, a temporary Subcommittee on the Economic and Security Implications of Quantum Science. The National Quantum Initiative Advisory Committee must include defense researchers and ensure the integration of DOD in committee actions and programs.