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Equation 30 · Part 2 · A Horizon Is a Toll Booth, Not a Loophole

Symbol E_0

⟨H⟩−E0=hf≈3.3×10−24 J\langle H\rangle-E_0=hf\approx3.3\times10^{-24}\,\mathrm J
E0E_0

What this part means

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

Its job in the formula

E0E_0 is part of the quantity the equation computes from the expression on the right.

Where the article explains it

For a system with a fixed, time-independent Hamiltonian H evolving from a state ∣\lvertψ\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: τ⊥\tau_\perp ≥\geq πℏ2(⟨H⟩−E0)\frac{\pi\hbar}{2\left(\langle H\rangle - E_0\right)}, where ⟨\langle H⟩\rangle is the mean energy in the evolving state and E0E_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\,GHz\mathrm{GHz} carries a local energy gap ⟨\langle H⟩\rangle-E0E_0=hf≈\approx3.3×\times10^{-24}\,J\mathrm J . Substituted into the instantaneous rate 2(⟨\langle H⟩\rangle-E0E_0)/π\piℏ\hbar , this gives an upper ceiling of roughly 2×\times10^{10} orthogonal state changes per proper second; over one proper microsecond of continuous operation, N⊥N_\perp≲\lesssim2×\times10^4 . 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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