Fast-forwarding of Hamiltonians and Exponentially Precise Measurements

Understand

In the early days of quantum mechanics, it was believed that the time energy uncertainty principle (TEUP) bounds the efficiency of energy measurements, relating the duration ($\Delta t$) of the measurement, and its accuracy error ($\Delta E$) by $\Delta t\Delta E \ge$ 1/2.

• In 1961 Y.
• Aharonov and Bohm gave a counterexample, whereas Aharonov, Massar and Popescu [2002] showed that under certain conditions the principle holds.
• Can we classify when and to what extent the TEUP is violated? Our main theorem asserts that such violations are in one to one correspondence with the ability to "fast forward" the associated Hamiltonian, namely, to simulate its evolution for time $t$ using much less than $t$ quantum gates.