$$N_\perp\lesssim2\times10^4$$

Equation 33 · Evolutionary Physics

A Horizon Is a Toll Booth, Not a Loophole

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

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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