The time-energy uncertainty relation (TEUR) \Delta t\Delta t over Δ t Δ E ≥ 1 2 holds if the Hamiltonian is completely unknown, but can be violated otherwise; here we initiate a rigorous study describing when and to what extent such violations can occur. To this end, we propose a computational version of the TEUR (cTEUR), in which Δt is replaced by the computational complexity of simulating the measurement. cTEUR violations are proved to occur if and only if the Hamiltonian can be fast forwarded (FF), namely, simulated for time t with complexity significantly smaller than t. Shor's algorithm provides an example of exponential cTEUR violations; we show that so do commuting local Hamiltonians and quadratic fermion Hamiltonians. A general FF method is ruled out, but finding further examples, as well as experimental demonstrations, are left for future work. We discuss possible connections to sensing and quantum gravity. This work initiates a rigorous theory of efficiency versus accuracy in energy measurements using computational complexity language.
Bibliographical noteFunding Information:
We are grateful to S. Aaronson, F. Brandão, J. Dowling, N. Katz, A. Kempf, L. Palter-ovitch, D. Roberts, O. Sattath, and N.Y. Yao, for very useful comments. In particular, we acknowledge a discussion with N.Y. Yao that eventually led to Theorem 4. This research was supported by ERC grant number 280157 and Simons foundation grant number 385590.
© 2017 The Author(s).