Equation 31 · Part 2 · A Horizon Is a Toll Booth, Not a Loophole

Symbol E_0

2(\langle H\rangle-E_0)/\pi\hbar

E_0

What this part means

the ground-state energy of H [ 2 , 1 ].

Its job in the formula

$E_0$ is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

Full expressionSymbol E_0Article meaning

Where the article explains it

For a system with a fixed, time-independent Hamiltonian H evolving from a state $\lvert$ $\psi$ $\rangle$ toward any state orthogonal to it, the minimum time required is bounded below by the tight, unified Margolus-Levitin/Mandelstam-Tamm result of Levitin and Toffoli: $\tau_\perp$ $\geq$ $\frac{\pi\hbar}{2\left(\langle H\rangle - E_0\right)}$ , where $\langle$ H $\rangle$ is the mean energy in the evolving state and $E_0$ is the ground-state energy of H [ 2 , 1 ] .

The passage around this formula

A concrete number anchors the scale before the subtler chaining question is addressed. A superconducting transmon-style qubit with a transition frequency near 5\, $\mathrm{GHz}$ carries a local energy gap $\langle$ H $\rangle$ - $E_0$ =hf $\approx$ 3.3 $\times$ 10^{-24}\, $\mathrm J$ . Substituted into the instantaneous rate 2( $\langle$ H $\rangle$ - $E_0$ )/ $\pi$ $\hbar$ , this gives an upper ceiling of roughly 2 $\times$ 10^{10} orthogonal state changes per proper second; over one proper microsecond of continuous operation, $N_\perp$ $\lesssim$ 2 $\times$ 10^4 . Real superconducting processors run single-qubit gates in tens of nanoseconds, a realized rate several orders of magnitude below this ceiling. The Levitin-Toffoli bound is, in…

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Sources cited in the article section

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