Quantum Computing · Part C — 11

VQE — Ground-State Energies

The Variational Quantum Eigensolver puts the hybrid loop to work on the killer chemistry question: what is a molecule’s lowest energy? One law makes it possible — the energy of any trial state can never dip below the true ground energy. So just push the cost as low as it’ll go, and you’ve found E₀.

↩ before you start · keep these handy
·From QC 10: the hybrid loop minimizes a cost C(θ) = ⟨H⟩ by tuning circuit angles.
·From QC 09: a Hamiltonian H is a sum of simple Pauli terms.
·From QI Ch. 3: you estimate ⟨P⟩ by averaging many measurements in the right basis.
🔑 symbol decoder · every new mark, in plain words
E₀ (ground energy)the lowest eigenvalue of H — the molecule’s most stable energy. eigenvaluean allowed energy level of the system; H’s smallest one is E₀. variational principlethe guarantee that ⟨ψ|H|ψ⟩ ≥ E₀ for every trial state — you can only approach from above. Pauli decompositionwriting H = Σ cᵢ Pᵢ (a weighted sum of Pauli strings) so each piece can be measured. bond length Rthe distance between two atoms; energy depends on it. Hartreethe natural energy unit of chemistry (about 27.2 eV).
feel

You can only push down to the floor

Here is the gift that makes VQE work: no matter what trial state you cook up, its energy is never lower than the true ground energy. The ground energy is a floor you can press against but never break through. So the recipe is almost embarrassingly simple — build a flexible trial state, and let the optimizer shove its energy down as far as the floor allows. Where it stops is the answer.

⚫ everyday picture

Drop a marble into a bowl. However you toss it, it can’t end up below the bowl’s lowest point — it rolls around and settles there. The molecule’s energy landscape is that bowl; the trial state is the marble. VQE just lets it roll. And the bowl’s shape as you stretch the atoms apart — the bond curve — tells you the equilibrium bond length and how much energy it takes to break the bond.

recapAny trial energy ≥ E₀, so minimizing the cost lands you on the ground-state energy.
play

Trace the H₂ bond curve

At each bond length R, VQE minimizes the energy and plots one point. Drag the two hydrogen atoms apart and read the curve: a steep wall when they’re crammed together, a well at the equilibrium distance, and a flattening as the bond breaks. The bottom of the well is the ground-state energy.

▸ H₂ potential curveE₀(R) from VQE
E R → equilibrium
H H
bond length R = {{ Rstr }} Å
energy E₀(R){{ energy }} Ha
equilibrium R₀0.74 Å
{{ status }}
recapVQE finds E₀ at each R; the curve’s minimum is the equilibrium bond length, its depth the bond strength.
math

Why minimizing ⟨H⟩ finds E₀

Expand any trial state in H’s energy eigenstates, |ψ⟩ = Σ cᵢ|Eᵢ⟩. Then its energy is a weighted average of the levels — and an average can’t beat the smallest one:

⟨ψ|H|ψ⟩ = Σᵢ |cᵢ|² Eᵢ ≥ E₀ Σᵢ |cᵢ|² = E₀
equality ⇔ |ψ⟩ = the ground state

To measure that energy, split H into Pauli pieces and add up their measured averages:

⟨H⟩ = Σᵢ cᵢ ⟨Pᵢ⟩  — measure each ⟨Pᵢ⟩, scale, sum
✎ worked example · the floor, on a 2-level system
1.say H has levels E₀ = −1 and E₁ = +1, trial |ψ⟩ = c₀|E₀⟩ + c₁|E₁⟩
2.⟨H⟩ = |c₀|²(−1) + |c₁|²(+1), with |c₀|² + |c₁|² = 1
3.this is most negative when |c₀|² = 1: ⟨H⟩ = −1 = E₀
4.any other mix gives ⟨H⟩ > −1 — you can never go below the floor. ✓
recap⟨H⟩ is a weighted average of energy levels, so it’s ≥ E₀; measure it via H = Σ cᵢ⟨Pᵢ⟩.
⚠ common misconceptions

“VQE could land below the true energy if it’s lucky.” Impossible — the variational principle forbids it. VQE always gives an upper bound on E₀; a lower number is strictly better, never an error in the other direction.

“If the ansatz is too simple it’ll just take longer.” Worse — if the true ground state isn’t reachable by any setting of the angles, VQE converges to a value stuck above E₀. Expressive enough ansätze are essential.

“VQE on today’s hardware already beats classical chemistry.” Not yet. Small molecules have been demonstrated, but noise and measurement costs keep VQE from clear advantage on problems classical methods can’t already handle.

✓ you can now
✓state the variational principle: ⟨ψ|H|ψ⟩ ≥ E₀ for every trial state
✓explain why minimizing the cost finds the ground-state energy
✓measure ⟨H⟩ by decomposing H into weighted Pauli terms
✓read a bond-dissociation curve and explain why VQE only ever overestimates E₀
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