Equation 32 · Part 1 · A Horizon Is a Toll Booth, Not a Loophole

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2\times10^{10}

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What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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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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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the article section

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