  A projective measurement of energy (PME) on a quantum system is a quantum
measurement, determined by the Hamiltonian of the system. PME protocols exist
when the Hamiltonian is given in advance. Unknown Hamiltonians can be
identified by quantum tomography, but the time cost to achieve a given accuracy
increases exponentially with the size of the quantum system. In this letter, we
improve the time cost by adapting quantum phase estimation, an algorithm
designed for computational problems, to measurements on physical systems. We
present a PME protocol without quantum tomography for Hamiltonians whose
dimension and energy scale are given but otherwise unknown. Our protocol
implements a PME to arbitrary accuracy without any dimension dependence on its
time cost. We also show that another computational quantum algorithm may be
used for efficient estimation of the energy scale. These algorithms show that
computational quantum algorithms have applications beyond their original
context with suitable modifications.
