Quantum optimization
Every decision is a field.
A week, a budget, a breeding program, a court calendar: each is a web of choices that strengthen or compete with one another. We cast the whole web as one field, let it settle into balance, and check the answer against a proof wherever one can be computed.
2 × 1031
ways to arrange one real working week of ours: 104 choices against 40 hours
13 seconds
for annealing to settle the whole week on one answer
Proven best
the answer it settled on, confirmed by two independent exact methods
The idea
Tension, settled into balance.
Each choice is a point in a field. Choices that strengthen each other pull together; choices that compete for the same cash, people or time push apart. The solver lets the whole field settle at once into its lowest-energy arrangement, the one where every pull and push is in balance.
One field, settled
What this shows
Ten choices an organization might weigh in one quarter. Solid lines join choices that strengthen each other; dashed lines join choices that compete for the same cash, people or time. The filled points are where the field settled: seven chosen, three left out.
What it means
Taking the five strongest choices one at a time scores 9. The settled field scores 29, 3.2 times better, because it weighs how the choices work together. Training is kept because hiring needs it. Expansion, the most valuable choice on its own, is left out because it competes with hiring, the reserve and the new market.
An illustration, solved exactly across all 1,024 combinations. Real problems carry hundreds of choices, far too many combinations to list.
The mathematics is the Ising model, written a century ago to describe how the atoms of a magnet settle into alignment. Annealing takes its name from metal cooled slowly so that its structure settles into its strongest form. The same form is native to quantum annealers and to gate-model quantum circuits, so one problem can be put to quantum and classical machines alike, and every answer compared.
Where it reaches
From one person's week to a whole enterprise.
Any decision made of many choices that strengthen or compete with one another can be cast as a field, at any scale.
A person
A week, in balance
A practice
A case, end to end
An organization
An enterprise or an administration
Research
A research program
The method
Four steps, every time.
Cast the choice
Solve it several ways
Certify the answer
Keep the ledger
The solvers
Four families of method, one ledger.
Each problem is put to every family that can take it, and each answer is scored against the same reference.
Exact proof
Annealing and tabu search
Gate-model circuits
Accelerated classical optimization
Measured
Exact where it can be proven. Fast where it cannot.
Seconds to solve, as the problem grows
What this shows
Each point is one problem solved. The grey line checks every possible answer, the most general way to prove which is best, and its work doubles with each added choice: twenty choices took 10.6 seconds, and at the same rate sixty would take about 370,000 years. The purple line is annealing, which settled every size from 8 to 60 choices in under a second. A filled purple point means it found the proven best answer.
What it means
Real decisions have far more than twenty choices, so proof by checking everything runs out early. Some have a structure that still allows a proof at full size, as the working week above does. Where none does, a method that settles in seconds is what makes the decision practical, and wherever a proof can run, it keeps the fast method honest.
Max-cut problems from our benchmark ledger, recorded June 2026: split a network so the most links cross the split.
Noise
Depth helps a perfect circuit. Noise takes it back.
With no noise, four layers did best under every profile, finding the best answer up to 72% of the time. Under noise, every profile did best at two layers. In the noisiest, four layers found it 4.1% of the time, against 3.1% for a random guess.
Chance of returning the best answer, by circuit depth
Profile A
Profile B
Profile C
What this shows
A quantum circuit works in layers, and more layers should find the best answer more often. The grey lines are perfect, noise-free circuits: four layers find it most often. The purple lines add the noise published for three superconducting processors: past two layers, the noise takes the gain back.
What it means
Under the noise these processors publish today, a circuit pays off only while it stays shallow. We measure where a circuit helps and where noise cancels it, and send every problem to the method that solves it best, quantum or classical, so the gains arrive as the hardware improves.
Six-variable max-cut on simulated circuits, 2,000 shots per run, recorded June 2026. Each panel applies the published noise profile of one superconducting processor. The dashed line is a random guess, 3.1%.
The ledger
Every run, on the record.
Each completed run is kept against a fingerprint of its problem, recorded June to September 2026.
| Problem | What it decides | Sizes tested, in variablesVariables | Completed runsRuns |
|---|---|---|---|
| From our own work | |||
| Authority selection | Which authorities a brief carries | 16, 24, 40 | 11 |
| Filing schedule | Which court day each filing lands on | 12, 73 | 8 |
| Caseload | Which matters receive finite hours | 104 | 2 |
| Treasury payables | Which payouts clear within the cash on hand | 13 | 6 |
| Plant breeding | Which plants form the most diverse breeding set | 12 | 6 |
| Feature selection | Which signals a model reads | 6 | 6 |
| Standard benchmarks | |||
| Max-cut | Split a network so the most links cross the split | 8 to 60 | 99 |
| Number partitioning | Divide numbers into two groups of equal sum | 14, 18, 30 | 43 |
| Knapsack | The most value that fits within a capacity | 16, 18, 21 | 37 |
Statute comparison, search memory and training curricula are certified against exact answers in their own test suites.