Interpretation of tomography and spectroscopy as dual forms of quantum computation

It is important to be able to determine the state of a quantum system and to measure properties of its evolution. State determination can be achieved using tomography1, in which the system is subjected to a series of experiments, whereas spectroscopy can be used to probe the energy spectrum associated with the system's evolution. Here we show that, for a quantum system whose state or evolution can be modelled on a quantum computer, tomography and spectroscopy can be interpreted as dual forms of quantum computation2. Specifically, we find that the phase estimation algorithm3 (which underlies a quantum computer's ability to perform efficient simulations4 and to factorize large numbers5) can be adapted for tomography or spectroscopy. This is analogous to the situation encountered in scattering experiments, in which it is possible to obtain information about both the state of the scatterer and its interactions. We provide an experimental demonstration of the tomographic application by performing a measurement of the Wigner function (a phase space distribution) of a quantum system. For this purpose, we use three qubits formed from spin-1/2 nuclei in a quantum computation involving liquid-state nuclear magnetic resonance.