Fast-forwarding of Hamiltonians and exponentially precise measurements

Abstract

The time-energy uncertainty relation (TEUR) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {\it{t}}\Delta {\it{E}}\,{\bf{ \ge }}\,{\textstyle{{\bf{1}} \over {\bf{2}}}}$$\end{document} 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 Δ t 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.

Abstract

A complete theory of when and to what extent time-energy uncertainty relation violations can occur is lacking. Here, the authors set the ground for a rigorous theory to investigate this possibility.