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How we benchmark quantum against classical
Every sector page on this site compares a quantum method against a classical one on the same problem. Those comparisons are only worth reading if the rules are fixed in advance and stated in public. These are the rules.
Reproducing any benchmark here costs cents. Open the console
1. Every comparison uses a strong classical baseline
A comparison is only as informative as the classical method it is measured against, so each one uses the strongest baseline that fits the problem: exhaustive enumeration where the instance is small enough that it returns the provable optimum, exact diagonalisation for molecular ground states, and established solvers such as OR-Tools CP-SAT or CVXPY where the problem is too large to enumerate.
Where enumeration is used, the classical result is the exact optimum rather than an estimate, so the quantum method is measured against the best answer that exists.
Classical simulators
Run the classical side yourself, from $0.0001 a circuit
CPU simulation is billed by the second at $0.69 per hour: exact statevector, tensor network, Clifford and Pauli propagation engines, each with a free estimate before anything runs and a certificate for every result.
2. Instances are generated from a published seed
A benchmark on a hand-picked instance proves nothing, because the instance can be chosen after the fact to produce the desired result. Every problem instance on these pages is generated by a documented procedure from a stated random seed, so anyone can regenerate the exact same instance and check the numbers.
3. The variational loop is counted in full
Variational algorithms such as QAOA and VQE do not run once. They run inside an optimiser that evaluates the circuit hundreds of times while it searches for parameters. A benchmark that reports the cost of the final circuit, and leaves out the search that found it, understates the true cost by two orders of magnitude.
Every page reports the total number of circuit evaluations and the total wall-clock time for the whole optimisation, not for the last run.
4. The quantum method is given a fair optimiser
This cuts the other way, and it matters just as much. A QAOA circuit at higher depth contains the lower-depth circuit as a special case, so a deeper run scoring worse is evidence that the classical optimiser failed to converge, not that the quantum method is worse. Under-optimising the quantum side produces a flattering result for classical computing and a benchmark that does not survive scrutiny.
We use multiple restarts from different initial parameters and check that results improve monotonically with circuit depth. Where they do not, the run is treated as a bug in the benchmark rather than as a finding.
5. Scoring is scale-free
Percentage gap against the optimum is the obvious metric and it breaks whenever the optimum sits near zero, where a negligible absolute miss reads as a thousand-percent error. We report instead where the answer sits between the best and worst possible solution, and what fraction of all feasible solutions it beats. Both stay meaningful regardless of scale.
6. Approximate results carry a bound
Where the quantum result comes from an approximate simulation method, it is accompanied by a certified error statement under the ZCC-v0.1 protocol, and hardware runs carry ZHF-v0.1. An approximate number without a bound is an assertion rather than a measurement, and it has no place in a comparison.
7. Results are published whichever side wins
We publish benchmarks where the classical method wins outright, and we expect to keep publishing them. Classical computing is extremely good at most of the problems industry actually has, and a page claiming otherwise would be false.
Publishing them is deliberate. A set of results that contains only wins cannot show what was left out, so the pages where quantum does not win are what make the pages where it does worth believing.
8. Every run is dated, and re-run
Quantum hardware improves and classical solvers improve. A benchmark without a date is a claim about an unspecified moment. Every result states when it was measured, on what hardware, and with which software versions.
What we are not claiming
None of these pages assert that quantum computing is ready to replace a classical method in production today. At the sizes current hardware can run, it generally is not. What the pages provide is the comparison at the size you can actually test, the point at which the classical method stops being tractable, and what hardware would have to look like to change the answer.
The conclusion is yours to draw. Ours is only that the numbers should be real.
What you can run here
Every comparison on these pages is reproducible on the same engines, at the same prices. The free estimate returns the engine and the cost before anything executes, each variational step goes as one sweep so the evaluation count is exactly the number reported here, and every run carries a certificate whose URL resolves without an account. If a figure on these pages is wrong, that is the machinery for proving it.
Run any of these benchmarks yourself.
Each sector page links a preloaded circuit into the console, so you can change the instance to your own numbers and export a certificate for your own run rather than citing ours.
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