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Equation 2 · Part 5 · Variational Evolution: How Quantum Computers Learn Their Answers

=

E(θ)=⟨ψ(θ)∣H∣ψ(θ)⟩.E(\theta) = \langle \psi(\theta) \mid H \mid \psi(\theta) \rangle .
=

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

This is the variational algorithm, and by 2026 it is not one technique among several — it is close to the entire working repertoire of near-term quantum hardware. The archetype is the variational quantum eigensolver (VQE), first demonstrated in 2014 on a photonic quantum processor, where Alberto Peruzzo and coauthors used the loop to compute the ground-state molecular energy of the helium hydride cation, HeH+, to within chemical accuracy [ 1 ] . The demonstration used only a small photonic chip, but its structural claim was the important part: instead of demanding the long coherent evolution that quantum phase estimation requires, VQE splits the work. The quantum processor’s only job is to…

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the surrounding passage

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