It depends on the definition of efficiency, and what specific part of government we're talking about.
In terms of cost to produce, the government would always be less efficient. In terms of end cost to the user, the government can be more efficient, especially in industries where there are massive barriers to entry and a very steep demand curve. An example here is health care.
From societies' perspective, if we are defining efficiency in terms of the total benefit to society less the total cost of capital, there are also instances where the government can be more efficient. One big example here is education.
The biggest defining factor though is what your goals are. Certain goals lend themselves to the private industry, while others necessarily lend themselves to the public one. For example: how do you spur innovation by protecting intellectual property without the government?