Patient Safety Improvement Act of 2020 This bill directs the Department of Health and Human Services (HHS) to undertake actions related to (1) health care-associated infections (HAI), (2) antimicrobial stewardship, and (3) patient safety. With respect to HAI, HHS must award grants for state and regional infection control programs and must establish a pilot program related to HAI data reporting by long-term care facilities and others. HHS must also award grants to eligible entities for infection control during patient care transitions, and the Centers for Medicare & Medicaid Services (CMS) must develop guidance regarding transfers or discharges of patients with HAI diagnoses. The Government Accountability Office must evaluate the adequacy of oversight and validation of specified HAI data reporting. With respect to antimicrobial stewardship, the Centers for Disease Control and Prevention must award grants to states to implement antimicrobial stewardship plans and must otherwise take steps to address antibiotic prescribing. With respect to patient safety, the HHS must incentivize state medical boards to require education on patient safety topics as a condition of licensure, and the CMS must harmonize quality measure reporting requirements through a collaborative that includes representatives from, among other groups, insurers and patient groups. In addition, as a condition of a hospital's participation in Medicare, members of the hospital's board must periodically receive training on relevant patient safety topics.