Chapter 22: Scaling — From H₂ to FeMo-co
H₂ was our teacher. H₂O was our first real test. Now we look at where the pipeline goes — and where it meets its limits.
In This Chapter
- What you’ll learn: How the pipeline’s cost scales from small test systems through medium molecules to the grand challenge of quantum chemistry, and where encoding choice makes — or breaks — the simulation.
- Why this matters: The promise of quantum simulation is that it handles molecules classical computers cannot. This chapter asks: at what point does that promise become real, and what does the hardware need to deliver?
- Prerequisites: Chapters 17–19 (cost analysis, pipeline, bond angle scan).
The Scaling Landscape
Every quantity in our pipeline grows with the number of spin-orbitals $n$:
| Quantity | Growth | H₂ ($n{=}4$) | H₂O ($n{=}14$) | N₂ ($n{=}20$) | FeMo-co ($n{\approx}108$) |
|---|---|---|---|---|---|
| Qubits | $n$ | 4 | 14 | 20 | ~108 |
| Hamiltonian terms | $O(n^4)$ | 15 | ~600 | ~2,000 | ~$10^7$ |
| JW max weight | $n$ | 4 | 14 | 20 | 108 |
| TT max weight | $O(\log_3 n)$ | 3 | 4 | 5 | ~5 |
| Configurations (FCI) | $\binom{n}{n_e}$ | 6 | 1,001 | 38,760 | ~$10^{30}$ |
The configuration count is why classical methods fail. Full CI (exact diagonalisation) scales as $\binom{n}{n_e}$ — the number of ways to place $n_e$ electrons in $n$ spin-orbitals. For FeMo-co, that’s roughly $10^{30}$ determinants. No classical computer will ever enumerate them.
A quantum computer doesn’t enumerate configurations — it represents the quantum state directly in $n$ qubits. The cost is in the circuit, not the state space. And the circuit cost depends on the encoding.
The Encoding Crossover
At 4 qubits, all encodings cost the same (Chapter 18). At 14 qubits, the differences appear (Chapter 19). Let’s trace the crossover:
| Molecule | $n$ | JW CNOTs/step | TT CNOTs/step | Ratio |
|---|---|---|---|---|
| H₂ | 4 | 12 | 12 | 1.0× |
| LiH | 12 | ~400 | ~200 | 2.0× |
| H₂O | 14 | ~1,800 | ~600 | 3.0× |
| N₂ | 20 | ~8,000 | ~2,000 | 4.0× |
| FeMo-co | ~108 | ~$10^7$ | ~$10^5$ | ~100× |
The ratio grows because JW’s Pauli weights grow linearly while TT’s grow logarithmically. At FeMo-co scale, the difference is roughly two orders of magnitude for the logical (pre-transpilation) circuit. After hardware-specific transpilation (qubit routing, native gate decomposition), the absolute gate counts increase for all encodings, but the relative advantage of lighter Pauli weights is preserved.
Caveat on the estimates: The CNOT counts in this table are for unoptimised logical circuits — the direct output of FockMap’s CNOT staircase decomposition. Hardware transpilers typically reduce these by 20–40% through gate cancellation, commutation, and template matching. The estimates for N₂ and FeMo-co are order-of-magnitude projections based on scaling trends, not computed values.
Why the Ratio Matters
On near-term hardware, each CNOT gate has a finite error rate — typically 0.1–1%. The probability that a circuit executes correctly drops exponentially with the CNOT count:
\[P_\text{success} \approx (1 - \varepsilon)^{C_\text{CNOT}}\]At $\varepsilon = 0.5\%$ and 1,800 CNOTs (JW for H₂O), $P_\text{success} \approx 0.01\%$. At 600 CNOTs (TT for H₂O), $P_\text{success} \approx 5\%$. Under these typical assumptions, that’s roughly a 500× improvement in success probability from encoding choice alone — before error mitigation, before hardware improvements, before anything else. (In practice, error mitigation techniques like zero-noise extrapolation can partially compensate for low success rates, but they impose their own sampling overhead.)
This is why the chapters on encoding (5–9), tapering (10–13), and cost analysis (17) matter. They’re not academic exercises. They directly determine whether a simulation succeeds or fails on real hardware.
The Tapering Dividend
Tapering compounds with encoding choice. At each system size, tapering removes $k$ qubits and often reduces the term count:
| Molecule | $n$ | Tapered $n{-}k$ | Terms (before) | Terms (after) |
|---|---|---|---|---|
| H₂ | 4 | 2 | 15 | 5 |
| H₂O | 14 | ~11 | ~600 | ~300 |
| N₂ | 20 | ~16 | ~2,000 | ~1,200 |
Fewer terms means fewer Pauli rotations per Trotter step. Combined with lower Pauli weights from a good encoding, the circuit shrinks multiplicatively. The optimisation stack from Chapter 17 — encoding + tapering + Trotter order — compounds at every scale.
Application: Encoding Choice at H₂O Scale
With 14 spin-orbitals, H₂O is the smallest molecule where encoding choice makes a practical difference. Here’s the full comparison after tapering:
| Encoding | Tapered qubits | Max weight | CNOTs/step | Depth estimate |
|---|---|---|---|---|
| Jordan–Wigner | ~11 | 11 | ~1,800 | ~3,600 |
| Bravyi–Kitaev | ~11 | 5 | ~750 | ~1,500 |
| Parity | ~11 | 11 | ~1,800 | ~3,600 |
| Binary Tree | ~11 | 5 | ~700 | ~1,400 |
| Ternary Tree | ~11 | 4 | ~600 | ~1,200 |
| Vlasov Tree | ~11 | 4 | ~600 | ~1,200 |
The 3× reduction from JW to TT is the difference between a circuit that fries on near-term hardware and one that might just survive. On a device with ~99.5% two-qubit gate fidelity, the TT circuit has roughly a 10× higher success probability per shot. Over millions of VQE shots, that translates directly to better energy estimates.
This is the practical answer to “which encoding should I use?” — at H₂O scale and beyond, ternary tree with tapering is the best option in the FockMap toolkit.
The Grand Challenge: FeMo-co
The iron-molybdenum cofactor (FeMo-co) of nitrogenase is the molecule that launched a field. It catalyses nitrogen fixation — converting atmospheric N₂ to ammonia — and understanding its mechanism could transform fertiliser production, one of the most energy-intensive industrial processes on Earth.
FeMo-co has ~54 active electrons in ~108 active spin-orbitals. Classical methods cannot accurately compute its electronic structure because the iron centres are strongly correlated: many electron configurations contribute comparably to the ground state, defeating perturbation theory and single-reference methods like coupled cluster.
The Numbers
| Quantity | Value |
|---|---|
| Active electrons | ~54 |
| Active spin-orbitals | ~108 |
| Qubits (JW) | ~108 |
| JW max Pauli weight | 108 |
| TT max Pauli weight | ~5 |
| JW CNOTs per worst-case rotation | 214 |
| TT CNOTs per worst-case rotation | 8 |
| Ratio | 27× |
At 108 qubits, the JW encoding produces Pauli strings where nearly every qubit participates. Each off-diagonal rotation requires a CNOT staircase spanning the entire register. The ternary tree encoding compresses the worst-case weight to ~5 — a chain of 4 CNOTs per direction.
How Far Away Is the Hardware?
For FeMo-co at chemical accuracy via QPE:
- Qubits needed: ~108 system + ~10 ancilla ≈ 120 logical qubits
- Trotter steps: ~$10^4$ (for the required precision)
- CNOTs per Trotter step (TT): ~$10^5$
- Total CNOTs: ~$10^9$
At current error rates, this requires full quantum error correction — perhaps 1,000–10,000 physical qubits per logical qubit, depending on the code and hardware. That puts the total physical qubit count at $10^5$–$10^6$.
No device available today can do this. But the same pipeline we developed for H₂ (4 qubits, 15 terms, 12 CNOTs) is the same code that would generate the FeMo-co circuit. The bottleneck is hardware, not software. When the hardware arrives, the pipeline is ready.
Where Classical Methods Still Win
It’s worth being honest about the landscape. Quantum simulation has a theoretical advantage for strongly correlated systems, but classical methods are formidable:
- Density functional theory (DFT) handles hundreds of atoms routinely. It’s not exact, but it’s remarkably accurate for weakly correlated systems and costs $O(n^3)$.
- Coupled cluster CCSD(T) — the “gold standard” of quantum chemistry — handles up to ~50 atoms with chemical accuracy for single-reference systems.
- DMRG and tensor network methods exploit the entanglement structure of 1D and quasi-1D systems, often matching quantum simulation accuracy for those geometries.
Quantum simulation’s niche is the strongly correlated regime: transition-metal complexes, open-shell systems, conical intersections, and exotic electronic states where no classical method converges reliably. FeMo-co is the poster child, but the real impact may be in catalysis design, high-temperature superconductors, and photochemistry — areas where electron correlation defies classical approximation.
Key Takeaways
- Circuit cost grows with system size, but encoding choice determines the growth rate: linear (JW) vs logarithmic (TT) Pauli weight.
- The encoding crossover becomes meaningful around 12–14 qubits — exactly the range of near-term quantum hardware.
- Tapering compounds with encoding choice, reducing both qubit count and term count multiplicatively.
- FeMo-co (~108 qubits) is the grand challenge: classically intractable, but requiring fault-tolerant hardware that doesn’t yet exist.
- The same pipeline code works at every scale — from H₂ to FeMo-co. The bottleneck is hardware, not software.