The Republic of Ireland International goalkeeper, who has recently returned to action following a nine-month drugs ban, rejected a new contract offer with the Blades to join up with former boss Neil Warnock at Loftus Road, bringing to an end his eight-year stay at Bramall Lane.
Birch told United's official website: "We are very disappointed and frustrated by what has happened. Paddy was offered a new, three-year enhanced contract, which he has chosen to reject.
"Paddy has now had discussions with QPR and agreed terms. As a result we have been left with no alternative but to let him go.
"We now have to move on and swiftly find a replacement. That process is well under way to bring in the right player in good time for pre-season. I am very confident of finding that player."