Case Study 36.1: The Nine-Decimal Result

The situation

A materials group has a quantum computing pilot. Six months in, they have a result: a VQE calculation of a lithium hydride binding energy, converged to $2\times10^{-9}$ Hartree, run on a simulator and reproduced on hardware with error mitigation. The slide reads:

VQE achieves chemical accuracy on LiH — energy error $2.04\times10^{-9}$ Ha, more than five orders of magnitude inside the $1.6\times10^{-3}$ Ha chemical accuracy threshold.

Every number on that slide is correct. The calculation is correct. The convergence is real. The hardware run reproduced the simulator to within mitigation error.

The chemistry director asks one question: "Compared to what?"

What the slide is actually claiming

The claim decomposes into two, and the slide only supports one:

Claim A (supported). The variational optimizer found the lowest eigenvalue of the Hamiltonian it was given to $2.04\times10^{-9}$ Ha.

Claim B (not supported, and not stated). That eigenvalue is the LiH binding energy to $2.04\times10^{-9}$ Ha.

The gap between them is the active space. §36.4 measured it: the (2e,2o) space that reduced LiH from 12 qubits to 4 — the reduction that made the hardware run possible at all — is wrong by 0.0201 Hartree against the full (4e,6o) Hamiltonian. That is 12.62 kcal/mol, about 12.6× chemical accuracy.

  VQE's error vs the active space's exact answer:   2.04e-09 Ha
  The active space's error vs the full Hamiltonian: 2.01e-02 Ha
  ratio: 9,870,104x

The slide reports the smaller number, to nine decimal places, and the third decimal place is already wrong.

The uncomfortable part

Nobody lied. This is worth dwelling on, because the failure is structural rather than dishonest.

The active-space reduction was made by the computational chemist in week two, for a completely sound reason: 12 qubits of chemistry does not run usefully on current hardware, and (2e,2o) does. It was documented in the setup notes. Everyone involved knew about it.

The VQE error was computed by the quantum engineer in month five, against the exact diagonalization of the Hamiltonian on their desk — which is the correct reference for the question they were asking, namely whether the optimizer converged.

The slide was assembled from both, and the composition is where the claim broke. Neither person made an error. The error is in the seam, and the seam is exactly where Chapter 26 §26.6 warned that verification tends not to reach.

What the group should have reported

Three sentences instead of one:

Within a (2e,2o) active space, VQE converges to the exact eigenvalue to $2\times10^{-9}$ Ha from three parameters. That active space is itself wrong by 0.0201 Ha (12.6 kcal/mol) against the full (4e,6o) Hamiltonian, measured by exact diagonalization. The calculation's total error is dominated by the active-space truncation and is about 12.6× chemical accuracy.

This is a better result to present, not a worse one, and the reason is instructive. The one-sentence version claims something implausible and invites the audience to find the hole. The three-sentence version claims something true, demonstrates that the group knows where its error budget lives, and identifies the actual research problem — which is the active space, not the optimizer.

What vqelab.chemistry does about it

The module makes the composition failure impossible to reproduce by accident:

>>> from vqelab.chemistry import ActiveSpace, total_error
>>> space = ActiveSpace(n_electrons=2, n_orbitals=2)     # nobody measured its error
>>> total_error(2.04e-09, space)
UnvalidatedActiveSpaceError: (2e,2o) -> 4 qubits, approximation error NEVER
MEASURED. The VQE error 2.04e-09 Ha is the error against THIS SPACE's exact
answer, which is not the error of the calculation. Measure the space against a
larger one, or report the VQE error with an explicit 'within this active space'.

The refusal is the point. A default of zero would be silently wrong by seven orders of magnitude, and it would be wrong in the direction that flatters the result. Once the space is validated:

>>> space = ActiveSpace(2, 2, error_ha=0.0201, validated_against="(4e,6o)")
>>> print(total_error(2.04e-09, space))
VQE 2.04e-09 Ha + active space 2.01e-02 Ha = 2.01e-02 Ha (12.61 kcal/mol),
dominated by active_space by 9,852,941x
>>> chemical_accuracy_claim(2.04e-09, space)
'REACHED_WITHIN_ACTIVE_SPACE'

REACHED_WITHIN_ACTIVE_SPACE is a different string from REACHED because it is a different claim.

The generalization

The pattern is not specific to chemistry, and you have seen it four times already:

  • Chapter 27 §27.4 — a test suite that passes because its tolerance is below the shot-noise floor.
  • Chapter 29 §29.3 — a layout optimized against calibration data that is a day old.
  • Chapter 30 §30.5 — a fidelity quoted from a distribution's best statistic.
  • Chapter 33 §33.6 — an accuracy from one train/test split.

Each is a correct measurement of the wrong quantity, reported as the right one. The number that is easy to get is not the number that answers the question, and in every case the easy number is the flattering one — which is not a coincidence, because the flattering number is the one that stops the search.

Questions

  1. The group's setup notes documented the active-space choice. Why did documentation not prevent the overstatement, and what would have?
  2. The quantum engineer's error was correct for the question they were asking. Is there a version of their job description under which they should have caught this?
  3. total_error raises rather than warning. Argue for a warning instead, then argue against — and say which side Chapter 27 §27.7's two-error-rates framing supports.
  4. Suppose the group re-runs in a (4e,4o) space. What must they measure before reporting anything, and what is the reference for that measurement?
  5. The hard one. The full (4e,6o) LiH Hamiltonian is itself in the STO-3G minimal basis, which is a crude basis set. So the 0.0201 Ha is measured against a reference that is itself wrong against experiment. Does that undermine this case study's argument, strengthen it, or neither?