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2(⟨H⟩−E0)/πℏ2(\langle H\rangle-E_0)/\pi\hbar

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where ⟨\langle H⟩\rangle is the mean energy in the evolving state and E0E_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-E0E_0)/π\piℏ\hbar , and Giovannetti, Lloyd, and Maccone showed that for a periodic evolution this rate integrates cleanly: a system with mean energy ⟨\langle H⟩\rangle-E0E_0 above its ground state can pass through at most

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HH

Symbol H

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π\pi

Symbol pi

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2(⟨H⟩−E0)/πℏ2(\langle H\rangle-E_0)/\pi\hbar

Equation 13 · 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.

where ⟨\langle H⟩\rangle is the mean energy in the evolving state and E0E_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-E0E_0)/π\piℏ\hbar , and Giovannetti, Lloyd, and Maccone showed that for a periodic evolution this rate integrates cleanly: a system with mean energy ⟨\langle H⟩\rangle-E0E_0 above its ground state can pass through at most

Meanings in this article

  • E0E_0: the ground-state energy of H [ 2 , 1 ].
Equation guide → · Article →
2(⟨H⟩−E0)/πℏ2(\langle H\rangle-E_0)/\pi\hbar

Equation 31 · 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\,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…

Meanings in this article

  • E0E_0: the ground-state energy of H [ 2 , 1 ].
Equation guide → · Article →