Optimal Quantum Measurements of Expectation Values of Observables
  Experimental characterizations of a quantum system involve the measurement of expectation values of observables for a preparable state |psi> of the quantum system. Such expectation values can be measured by repeatedly preparing |psi> and coupling the system to an apparatus. For this method, the precision of the measured value scales as 1/sqrt(N) for N repetitions of the experiment. For the problem of estimating the parameter phi in an evolution exp(-i phi H), it is possible to achieve precision 1/N (the quantum metrology limit) provided that sufficient information about H and its spectrum is available. We consider the more general problem of estimating expectations of operators A with minimal prior knowledge of A. We give explicit algorithms that approach precision 1/N given a bound on the eigenvalues of A or on their tail distribution. These algorithms are particularly 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 |psi> and U|psi>, where U is an implementable unitary operator. We explicitly consider the issue of confidence levels 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. 
Emanuel Knill, Gerardo Ortiz, Rolando D. Somma