Case Study 36.2: The Catalyst Proposal

The situation

A pharmaceutical company's computational chemistry group is asked to evaluate a vendor proposal. The vendor offers early access to a 156-qubit device and a joint project on a nitrogenase-inspired iron–sulfur cluster — a transition-metal system where the group's existing methods genuinely struggle.

The proposal's technical case is three claims:

1. The active space of interest is 50 spatial orbitals, which is past the point where classical methods stop being exact.
2. 50 orbitals is 100 qubits, which fits on a 156-qubit device.
3. Therefore the calculation is in reach, and the project would be the first quantum computation of real chemical value.

The group's quantum lead is asked whether to sign. Each claim is individually true. The conclusion does not follow from them, and this case study is the analysis that shows why.

Step 1: price the shot budget

Claims 1 and 2 are about qubits. The bill is denominated in terms.

>>> from vqelab.chemistry import shot_budget
>>> b = shot_budget(n_orbitals=50, n_parameters=781_250)
>>> b["n_terms"], b["total_shots"], b["qpu_years"]
(6250000, 190734497070000000000, 605506339.9047619)

Working through it, as §36.7 did:

   orbitals  qubits         terms      params   shots/gradient        total    QPU years
         50     100     6,250,000     781,250         9.54e+17     1.91e+20     6.06e+08

$1.91\times10^{20}$ shots. At 10,000 shots per second, 6.06 × 10⁸ QPU-years.

The device fits the qubits. It does not fit the calculation, and the gap is eight orders of magnitude — not a margin that better engineering closes.

Step 2: apply every mitigation in the book, honestly

The group's quantum lead does not stop at the headline number, because the honest question is what the techniques from Parts IV–VI are worth here:

Against a deficit of 6.06 × 10⁸ QPU-years, a 600,000× saving leaves about 10³ QPU-years.

Roughly four orders of magnitude of mitigation against a deficit of eight. Every technique in the table is real, and several are the best available. Stacked perfectly and optimistically, they leave the project a thousand QPU-years short.

⚠️ Note what this table is not. It is not a claim that the techniques do not work — they do, and Chapters 24 and 35 measured them working. It is Chapter 24 §24.7's lesson in its sharpest form: every remedy is denominated in the currency of the disease, and constant factors do not beat exponents. A 600,000× improvement is spectacular engineering and is still not enough.

Step 3: identify what the vendor's claim 1 actually says

This is the step the group's lead considers most important, and it is not about arithmetic.

Claim 1 says 50 orbitals is past where classical methods stop being exact. That is true. It is also not the relevant threshold, and §36.7 named the confusion:

The crossover is not where VQE becomes useful. It is where classical methods stop being exact.

Past that point the classical answer is approximate — but CCSD(T) is approximate the way a well-characterized instrument is approximate. It reaches chemical accuracy on most main-group chemistry at 100+ orbitals, at $\mathcal{O}(n^7)$, on hardware the group already owns. That is the number the project must beat, not full CI, which nobody has run at this size and which Chapter 21 §21.7 already identified as the wrong comparison.

Step 4: find the part of the proposal that is right

The group's lead does not recommend rejecting the proposal, and the reason is §36.6's narrowing.

An iron–sulfur cluster is strongly correlated. It is precisely the regime where CCSD(T) degrades — where the single-reference assumption underneath it breaks down — and where DMRG's structure assumptions are also strained:

>>> classical_baseline(50, strongly_correlated=True)["method"]
'dmrg'
>>> classical_baseline(50, strongly_correlated=True)["note"]
'CCSD(T) degrades on strong correlation; DMRG is the baseline and its own
limits are where the quantum case lives'

So the vendor picked a good system. The honest target is narrower than "chemistry" — transition-metal catalysis, some excited states, bond-breaking — and this is squarely inside it. The problem is not the science. The problem is that the method proposed for it is priced eight orders of magnitude out of reach, and the proposal reached its conclusion by checking the qubit count.

The recommendation

The lead's memo, in full:

Do not sign for a production calculation. Sign for a benchmark, with a different deliverable.

The system is well chosen — strongly correlated, genuinely hard for CCSD(T), the right kind of target. The method is not viable: at 50 orbitals VQE needs $1.9\times10^{20}$ shots, and every mitigation we know stacked together leaves us ~10³ QPU-years short. That gap is not an engineering schedule.

What is worth buying is calibration: run the largest active space the device can do, measure the active-space error against DMRG, and find out how the truncation error behaves as the space grows. That number is what we would need for any future fault-tolerant proposal, it is not obtainable from a paper, and it is not gated on the device being good — §36.4's error is a physics quantity, not a hardware one.

The route that survives the arithmetic is phase estimation on error-corrected hardware, where precision costs $\mathcal{O}(1/\epsilon)$ instead of $\mathcal{O}(1/\epsilon^2)$. If we want to be ready for that, the useful work now is the active-space characterization, not the VQE run.

What this case study is really about

The group's lead did not need to know more chemistry than the vendor. They needed to notice that a capability quoted in qubits was being matched against a cost denominated in terms — and §36.2 already showed those come apart badly, with BeH₂ and H₂O differing by 420 terms at the same 14 qubits.

Every claim in the proposal was true. The proposal was still wrong, because the claims were about different quantities than the conclusion required. That is the same shape as Case Study 36.1's slide, one organizational level up.

Questions

1. The mitigation table stacks four techniques multiplicatively. Is that justified? Name a pair that might not compose, and say what you would measure to find out.
2. The lead recommends measuring the active-space error against DMRG rather than exact diagonalization. Why can't they use exact diagonalization, and what does the substitution cost the argument?
3. The proposal is for a 156-qubit device and the calculation needs 100. Under what circumstance would the surplus 56 qubits matter? (Chapters 25 and 29 both have answers.)
4. Rewrite the vendor's three-claim technical case so it supports a conclusion that is actually true. You may not remove the iron–sulfur system.
5. The hard one. The lead's memo says the active-space characterization "is not gated on the device being good." Is that entirely true? Identify the step in that work where device quality does enter, and say whether it changes the recommendation.