How to Know If Your Quantum Chemistry Experiment Will Fail Before You Run It

We spent days optimizing HeH+ on quantum hardware. Same 2-qubit circuit as H2. Same error mitigation. Same backends. We expected similar results.

H2 on IBM Torino: 0.22 kcal/mol. Chemical accuracy, first try.

HeH+ on IBM Torino: 4.45 kcal/mol. Not even close. Twenty times worse.

We tried everything. Different qubit pairs. Different mitigation layers. Different backends. Nothing worked. HeH+ wouldn't budge below ~4.4 kcal/mol on any hardware.

Then we looked at the math and realized: the answer was in the Hamiltonian the whole time. We could have predicted this failure before submitting a single job.

The One-Number Pre-Flight Check

When you reduce a molecule to a 2-qubit Hamiltonian for VQE, it takes this form:

H = g0·I + g1·Z0 + g2·Z1 + g3·Z0Z1 + g4·X0X1 + g5·Y0Y1

The g4 and g5 terms are where the quantum magic lives — they encode the electron correlation that a classical computer can't efficiently capture. The g1 and g2 terms are basically classical energy offsets that happen to be measured on a quantum chip.

Here's the problem: on real hardware, every measurement has error. If g1 is large and g4 is small, hardware noise on the classical terms drowns out the quantum signal. It's like trying to hear a whisper at a rock concert.

The ratio |g1| / |g4| tells you the concert-to-whisper ratio before you walk in the door:

1.8x larger ratio → 20x worse energy. The relationship is nonlinear because errors compound through the g1−g2 difference.

We Predicted It, Then Confirmed It

Once we spotted the pattern, we made a prediction: any molecule with |g1|/|g4| > ~5 will fail to achieve chemical accuracy on current NISQ hardware, regardless of the backend or mitigation strategy.

The confirmation came from an unexpected direction. IBM Torino (133 qubits, TREX mitigation) and Tuna-9 (9 qubits, confusion matrix + post-selection) gave nearly identical HeH+ errors:

• IBM Torino: 4.45 kcal/mol
• Tuna-9: 4.44 kcal/mol

Different hardware, different continents, different mitigation strategies — same error floor. The Hamiltonian, not the hardware, is the bottleneck.

On a noiseless emulator, HeH+ is actually easier than H2 (0.20 vs 0.75 kcal/mol). The molecule isn't hard. It's just noise-fragile.

The Sweet Spot for Bond Distance

The ratio changes as you stretch the molecule. We mapped the full H2 potential energy surface on Tuna-9 and found two competing failure modes:

At short distances, the ratio is high and noise amplification kills accuracy. At long distances, the ratio is low but the circuit needs more entanglement (larger rotation angle), which adds gate noise. The best accuracy lives in the middle — where neither failure mode dominates.

This is a real constraint on which parts of a potential energy surface you can trust from NISQ hardware.

What to Do About It

Before you commit QPU time to a new molecule:

1. Compute the sector-projected Hamiltonian. Tools like OpenFermion or PySCF give you the coefficients in seconds on a laptop.
2. Check |g1|/|g4|. If it's below 5, you have a shot at chemical accuracy with current mitigation. If it's above 5, you're fighting the math.
3. If the ratio is bad, change the encoding. Different qubit mappings (Jordan-Wigner vs Bravyi-Kitaev), tapering, or symmetry reduction can change the coefficients. Find an encoding that shrinks the ratio.
4. If nothing works, wait. Better hardware (lower readout error) effectively reduces the noise that g1 amplifies. Or use a classical method — for these small molecules, CCSD(T) gives exact answers anyway.

The deeper lesson: not all molecules are created equal for NISQ. The hardware is getting better every year, but the Hamiltonian structure creates a ceiling that no amount of error mitigation can break through. Knowing where that ceiling is before you start saves time, QPU credits, and frustration.

Full analysis: amplification-threshold-analysis.json. HeH+ replication: Peruzzo 2014. H2 mitigation results: error mitigation showdown.