Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Optimal Quantum Measurements of Expectation Values of Observables



Emanuel Knill, Gerardo Ortiz, Rolando Somma


Experimental characterizations of a quantum system involve the measurement of expectation values of observables for a preparable state |f> of the quantum system. Such expectation values can be measured by repeatedly preparing |f> 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 a in an evolution exp(-ia H), it is possible to achieve precision 1/ N (the quantum metrology limit) provided sufficient information about H and its spectrum is available. We consider the more general problem of estimating expectations of operators A with minimial 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 measurements of observables and correlation functions. Our algorithms are based on a method for efficiently measuring the complex overlap of |f> and U|f>, where U is an implementable unitary operator. We explicitly consider the issue of confidence in measuring observables and overlaps and show that as expected, confidence 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.
Physical Review A (Atomic, Molecular and Optical Physics)


expectations of observables, quantum computing, quantum measurements, quantum physics simulation, uncertainty principle.


Knill, E. , Ortiz, G. and Somma, R. (2007), Optimal Quantum Measurements of Expectation Values of Observables, Physical Review A (Atomic, Molecular and Optical Physics), [online], (Accessed May 27, 2024)


If you have any questions about this publication or are having problems accessing it, please contact

Created January 23, 2007, Updated October 12, 2021