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A projective measurement of energy (PME) on a quantum system is a quantummeasurement, determined by the Hamiltonian of the system. PME protocols existwhen the Hamiltonian is given in advance. Unknown Hamiltonians can beidentified by quantum tomography, but the time cost to achieve a given accuracyincreases exponentially with the size of the quantum system. In this letter, weimprove the time cost by adapting quantum phase estimation, an algorithmdesigned for computational problems, to measurements on physical systems. Wepresent a PME protocol without quantum tomography for Hamiltonians whosedimension and energy scale are given but otherwise unknown. Our protocolimplements a PME to arbitrary accuracy without any dimension dependence on itstime cost. We also show that another computational quantum algorithm may beused for efficient estimation of the energy scale. These algorithms show thatcomputational quantum algorithms have applications beyond their originalcontext with suitable modifications.

Quantum algorithm for universal implementation of projective measurement of energy