Optimal quantum measurements of expectation values of observables

of A or on their tail distribution. These algorithms are particu larly useful for simulating quantum systems on quantum computers because they enable efficient measurement of observables and correlation functions. Our algorithms are based on a method for efficiently measuring the complex overlap of |ψi and U |ψi , where U is an implementable unitary operator. We explicitly consider the issue of confidence level s in measuring observables and overlaps and show that, as expected, confidence levels can be improved exponentially with linear overhead. We further show that the algorithms given here can typically be parallelized with minimal increase in resource usage.