Key Takeaways: Quantum Chemistry with VQE
The measurement this chapter exists for
VQE reached the active space's exact answer to $2.04\times10^{-9}$ Hartree, from three parameters. That active space was wrong by $2.01\times10^{-2}$ Hartree — a ratio of 9,870,104.
Both halves matter. The variational method worked beautifully: three parameters, two parts in a billion, on an ansatz built from the allowed excitations. And the answer was wrong in the third decimal place before the circuit ran, because of a decision made in the classical setup to make the problem fit a device.
The approximation made to fit the device was ten million times larger than the error of the method being demonstrated. Reporting the VQE energy to nine decimal places is arithmetically correct and physically meaningless.
What the chapter measured
| Claim | Measured |
|---|---|
| Qubits per molecule | 2 × spatial orbitals — H₂ 4, LiH 12, BeH₂ 14, H₂O 14 |
| Hamiltonian terms | $\mathcal{O}(n^4)$ — 15, 631, 666, 1,086 |
| Same qubits, different terms | BeH₂ 666 vs H₂O 1,086 — 420 apart at 14 qubits each |
| Bravyi–Kitaev vs Jordan–Wigner | Same 631 terms. Weight mean 6.16 → 5.62, max 12 → 10 |
| LiH (2e,2o) active space | 12 qubits → 4, 631 terms → 27 |
| …and its error | 0.0201 Ha = 12.62 kcal/mol = 12.6× chemical accuracy |
| VQE error inside that space | $2.04\times10^{-9}$ Ha from 3 parameters |
| Ratio | 9,870,104× |
| Exact diagonalization | 22.996 s at 12 qubits; 0.001–0.044 s below |
| Shots at the 50-orbital crossover | $1.91\times10^{20}$ = 6.06 × 10⁸ QPU-years |
The seven things to carry forward
1. A calculation is only as accurate as its worst approximation, and the worst one is usually not reported. Every optimization in Part V — layout, transpilation, circuit shape, error mitigation — operates on a Hamiltonian whose accuracy was fixed by an active-space choice made before any quantum code ran. §36.4's decision dominates every engineering decision downstream of it, and it is not gated on hardware at all.
2. Bravyi–Kitaev does not reduce the number of terms. The common claim is wrong as stated: both mappings give 631 terms for LiH. What changes is Pauli weight, which sets the measurement circuit depth per term. At 12 qubits the improvement is modest — mean 6.16 → 5.62, max 12 → 10 — because the $\mathcal{O}(n)$ versus $\mathcal{O}(\log n)$ separation needs more qubits to show. §36.3.
3. A device sized in qubits is not a device sized for a molecule. BeH₂ and H₂O are both 14 qubits and differ by 420 terms, because the electron count decides how many four-index integrals survive. Capability is quoted in qubits; the bill is denominated in terms.
4. The crossover is not where VQE becomes useful — it is where classical methods stop being exact. Those are different points and they are conflated constantly. At 50 orbitals the shot budget is $1.91\times10^{20}$, and roughly four orders of magnitude of available mitigation face a deficit of eight. §36.7.
5. The real baseline is CCSD(T), not full CI. Nobody runs full CI past ~20 orbitals — it is exponential. CCSD(T) is $\mathcal{O}(n^7)$ and reaches chemical accuracy on most main-group chemistry at 100+ orbitals. Benchmarking against full CI is Chapter 21 §21.7's error with a molecule attached: comparing against the method nobody uses. §36.6.
6. The honest target is narrower than "chemistry". Strongly-correlated systems where CCSD(T) breaks down and DMRG's structure assumptions fail — transition-metal catalysis, some excited states, bond-breaking. A real and important set of problems. Not all of chemistry, and not most.
7. The route that survives the arithmetic needs fault tolerance. Phase estimation reaches precision $\epsilon$ in $\mathcal{O}(1/\epsilon)$ rather than $\mathcal{O}(1/\epsilon^2)$, because the precision comes from circuit depth rather than sampling. Chapter 22 §22.4 built that route; Chapter 25 §25.10 explained why it is gated on error correction. §36.8.
What went into vqelab
vqelab.chemistry, 25 tests passing:
total_error()RAISES on an unvalidated active space rather than silently treating its error as zero. The most important refusal in the module — assuming zero is exactly how a $2.04\times10^{-9}$ result gets reported for a calculation whose real error is $2.01\times10^{-2}$.ActiveSpace.claim_supportedis False until the approximation error has been measured against a named larger space. An unvalidated active space does not have a small error; it has an unknown one, and unknown is not small.chemical_accuracy_claim()returnsREACHED,REACHED_WITHIN_ACTIVE_SPACE,MISSED, orUNVALIDATED. The second is usually what is true and the first is usually what is said.shot_budget()prices the $\mathcal{O}(n^4)$ terms, not the qubits, and takes agrouping_factorso Chapter 24's commuting groups and Chapter 35's shadows enter as what they are: large constant factors against an exponent.classical_baseline()returns CCSD(T) at scale and never full CI.
The refrain, in its chemistry form
Part V's theme — a measurement that cannot detect the thing being asked about — returns here as a reported number that cannot detect the thing being claimed. The VQE error is real, correct, and answers a question nobody asked: how close the optimizer got to the exact answer of a problem that was already the wrong problem.
The number that is easy to get is not the number that answers the question. This chapter's version is that the easy number is off by seven orders of magnitude, and it is the one on the slide.
Next: Chapter 37 takes the same accounting to combinatorial optimization, where the classical baseline is even stronger and better documented — and where the comparison has been run enough times to have a literature of its own.