Applications / AI / AI for quantum
NNQS for materials and chemistry
Ground states where tensor networks stop
A machine learning model solving a quantum problem, rather than a quantum computer solving a machine learning one: the wavefunction is a neural network, trained by variational Monte Carlo until the energy stops falling. It reaches states where a tensor network's area-law assumption does not hold.
- Method
- Neural network quantum states on a Google TPU
- Who runs this
- Condensed matter, quantum chemistry, materials research
- Classical baseline
- Exact diagonalisation, where the system still fits in memory
- Structural limit
- 40 spins on the neural tier; exact diagonalisation stops near 30
Run on our engines
A spin Hamiltonian solved on the TPU tier, and the H2 molecule solved on both neural engines for comparison. On 25 September the identical spin request ran on neural.gpu, an NVIDIA GPU, reaching energy -6.647038 with a ceiling of -6.630361. Submitted to each kind of compute we offer, on 16, 18 and 25 September 2026. Every figure below is a real job on the service, priced as any customer would be priced.
| Device | Engine | Kind | Qubits | Result | Cost |
|---|---|---|---|---|---|
 | neural.tpu | TPU | 6 | energy -6.918861, ceiling -6.886729, against an exact ground state of -7.048804 | $0.0395 |
 | neural.gpu | GPU | 6 | energy -6.647038, ceiling -6.630361, against an exact ground state of -7.048804 certificate | $0.0253 |
 | neural.cpu | CPU | 2 | H2 at -1.116981 Ha, 0.0203 Ha above exact certificate | $0.0001 |
| neural.tpu | TPU | 2 | the same H2 molecule, -1.116981 Ha certificate | $0.0740 |
The same problem is yours to run: every instance here is seeded, so it rebuilds exactly. Open the console and a cost estimate is free before anything executes.
What makes these numbers usable is the ceiling rather than the energy. The variational principle puts the true ground state at or below whatever the optimiser reports, so adding the run's own error bound gives a number the true answer cannot exceed, and that ceiling is unconditional. The floor beside it comes from the measured energy variance and holds only while the trial state sits nearer the ground state than the first excited state, which a run cannot verify about itself.
The TPU and CPU agree on H2 to five decimal places. The certified variational energies differ by 1.5e-05 Ha, the CPU's being the lower of the two, which follows from the TPU narrowing to complex64 because it has no complex128.
At two qubits the TPU spends more of the job being provisioned than computing. The tier is built for widths where dense matrix multiplication against batches of sampled configurations dominates the run.
What the makers say it is for
This is the one case on these pages where the deep learning research came first and the hardware question came second. Google DeepMind's FermiNet represents the wavefunction of a system of electrons as a neural network trained by variational Monte Carlo, and its successor Psiformer, built on self-attention, was published as the most accurate method of its kind for ground-state energies.
The runs below are the same idea on a spin lattice rather than on electrons, which is what puts them on a tensor processing unit rather than a quantum one.
Where this stops
- The engine takes a Hamiltonian rather than a circuit, so it is reached through /solve and refuses a circuit at the door
- A run whose Markov chains did not mix is refused rather than returned, because an estimate from an unmixed chain is not an expectation value at all
- The floor is conditional and the result says so; only the ceiling holds unconditionally
The structural limit above does not move when a benchmark is re-run. Everything else on this page is a measurement, and a measurement can be repeated.
Run it yourself
Every figure above is from a job billed on the production service. Load the same circuits into the console, change the instance to your own data, and export a certificate for your own run rather than citing ours.
The argument behind these numbers, at length: Neural network quantum states, with a bound.
Other AI use cases
QCBM and quantum generative models
Sampling from a distribution learned from scratch
QCBM against its own untrained start
QGAN and synthetic data
Adversarial generation against a learning discriminator
QGAN with a classical discriminator
Quantum AI image generation
Writing an image into a circuit and reading it back
Patch GAN and NTQIP, both on Rigetti