$$N \leq \frac{2}{\pi\hbar}\left(\langle H\rangle-E_0\right) t$$

Equation 15 · Evolutionary Physics

A Horizon Is a Toll Booth, Not a Loophole

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

where $\langle$ H$\rangle$ is the mean energy in the evolving state and $E_0$ is the ground-state energy of H [ 2 , 1 ] . Inverting this gives a maximum rate of orthogonal state changes per unit proper time, 2($\langle$ H$\rangle$-$E_0$)/$\pi$$\hbar$ , and Giovannetti, Lloyd, and Maccone showed that for a periodic evolution this rate integrates cleanly: a system with mean energy $\langle$ H$\rangle$-$E_0$ above its ground state can pass through at most $N \leq \frac{2}{\pi\hbar}\left(\langle H\rangle-E_0\right) t$. mutually orthogonal states in elapsed time t [ 3 ] . Lloyd used the same bound to price computation itself: no amount of clever engineering lets a system with energy E execute logical operations faster than about 2E/$\pi$$\hbar$ per second, a…

Meanings in this article

• $E_0$: the ground-state energy of H [ 2 , 1 ].

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