Manufacturing and production
Quantum computing for production scheduling: the job-shop benchmark
Job-shop scheduling is the canonical factory problem. Each job visits machines in its own order, each machine handles one job at a time, and the goal is the shortest makespan. It is NP-hard, it is genuinely difficult, and it is one of the applications most often named as a quantum computing target.
Submit your own schedule and compare. Open the console
The short version
- One 15 by 15 shop needs 268,425 logical qubits. Three orders of magnitude past anything built or on a published roadmap
- CP-SAT proves the optimum in 2.45 seconds. Makespan 126, on a laptop, with a proof that nothing better exists
- Every quantum approach here is a heuristic. QAOA returns a sample rather than a proof, so at any scale it answers a weaker question
- The regime where a better heuristic would pay. Hundreds of jobs with setup times and shift patterns, where CP-SAT returns a gap rather than a guarantee
Run on our engines
A random 16-variable QUBO, seed 20260903, whose exact optimum is -15.6952: the size a machine can hold, with no scheduling structure in it, generated by small_qubo.py in the public benchmarks folder. It is deliberately small, because the 15x15 shop this benchmark is about needs 268,425 qubits and fits nothing. Submitted to each kind of compute we offer, on 16 and 25 September 2026 at 500 shots. Every figure below is a real job on the service, priced as any customer would be priced.
| Device | Engine | Kind | Qubits | Result | Cost |
|---|---|---|---|---|---|
 | mps.quimb.cpu | CPU | 16 | -12.7064, gap 2.99 certificate | $0.0001 |
| exact.cpu | CPU | 16 | -11.7508, gap 3.94 certificate | $0.0001 |
 | exact.gpu | GPU | 16 | -12.4192, gap 3.28 certificate | $0.0001 |
 | qpu.rigetti | QPU | 16 | -346.31, gap 4.95 on the scheduling instance * certificate | $0.5125 |
 | qpu.iqm.garnet | QPU | 16 | -10.6113, gap 5.08 certificate | $1.025 |
| qpu.iqm.emerald | QPU | 16 | -11.3923, gap 4.30 certificate | $1.100 |
 | exact.tpu | TPU | 16 | -12.8804, gap 2.81 certificatebest outcome | $0.0776 |
| neural.tpu | TPU | — | a QUBO is diagonal, which is not the shape a neural ansatz is for | — |
A tensor-network simulator on CPU returned the lowest job-shop scheduling cost at -12.7064 against an exact optimum of -15.6952, with exact statevector simulation on an NVIDIA GPU at -12.4192. IQM Emerald returned -11.3923 and IQM Garnet -10.6113.
Both superconducting quantum processors returned valid schedules on essentially every shot, 485 and 498 of 500, so the difference here is the quality of the sampled schedule rather than any failure to produce a feasible one.
* The Rigetti row is not scored against the optimum named above. The run used the 4-job scheduling instance that sector_instances.py generates instead. Its exhaustive minimum is -351.2615, and the 4.95 gap is measured against that rather than against -15.6952. The IQM rows are our own internal testing. The exact.tpu row, added 25 September, is the best of the eight at a gap of 2.81. The steps are in the docs.
A note on the hardware certificates: they state Hellinger fidelity against the exact distribution. For an optimisation circuit that distribution is spread across many outcomes rather than concentrated on one, so the figure is low by construction and is not a measure of whether the device found a good answer. The result column above is.
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.
Run this instance yourself. The notebook rebuilds the exact seeded instance behind every figure on this page, in a browser tab, with nothing to install and no account. Read it on GitHub
Then run it on our engines
The notebook proves the numbers on free public packages. This is the step it does not cover: the same circuit on a certified engine, with a public certificate you can cite instead of citing ours.
pip install qsim-sdk import qsim_sdk client = qsim_sdk.Client(token="...") # free account at app.zksf.org client.estimate(circuit, shots=512) # free, before you spend anything job = client.run(circuit, shots=512) # $0.0001 on CPU job.certificate() # a public, verifiable URL
Every finished job can be exported as a public certificate that opens without an account, and the rules these benchmarks follow apply to your run exactly as they do to ours.
The encoding, first
As on the logistics page, the quantum benchmark does not exist here, and the reason is the result. Putting job-shop scheduling on a quantum computer requires a time-indexed QUBO: a binary variable for every combination of job, machine and discrete time slot. The variable count is the product of all three, and it grows brutally.
A 15 by 15 shop needs 268,425 binary variables, so 268,425 logical qubits before error correction. The largest processor we can run is 108 physical qubits, the largest built anywhere is around 1,600, and the most logical qubits anyone has demonstrated is 96. The published 2029 roadmaps target hundreds of logical qubits.
Every one of those numbers is three orders of magnitude short of this single instance.
What the classical solver does with the same instances
| Shop | Qubits needed | Makespan | CP-SAT time | Result |
|---|---|---|---|---|
| 3 x 3 | 531 | 28 | 0.01 s | proven optimal |
| 5 x 5 | 3,675 | 47 | 0.01 s | proven optimal |
| 10 x 10 | 57,000 | 94 | 0.17 s | proven optimal |
| 15 x 15 | 268,425 | 126 | 2.45 s | proven optimal |
OR-Tools CP-SAT, seed 20260902, measured 2 September 2026 on an ordinary laptop CPU. The makespans are proven optimal and therefore reproduce exactly; solve times vary with machine load, and the accompanying notebook took 5.6 s on the 15 by 15 shop against the 2.45 s shown here.
The word doing the work is "proven"
CP-SAT did not find a good schedule for the 15 by 15 shop. It found the best possible schedule and proved that nothing shorter exists, in 2.45 seconds.
That distinction matters more in a factory than in most places. A plant manager asking whether the line can run faster is asking a question with a definite answer, and a heuristic that returns a good schedule without a bound cannot give it.
Every quantum approach to this problem is a heuristic. QAOA returns a sample, not a proof, so even at unlimited scale it would answer a weaker question than the classical solver already answers in seconds.
Where quantum could still matter here
Two caveats, because the picture above is not the whole industry.
Real shops are larger and messier. Hundreds of jobs, sequence-dependent setup times, maintenance windows, shift patterns and stochastic durations.
CP-SAT does not always prove optimality there, and past a certain size it returns a good schedule with a gap rather than a guarantee. That is the regime where a better heuristic has value.
But the encoding gets worse, not better, in that regime. A larger, more constrained problem needs more time slots and more variables, so the qubit requirement rises faster than the classical difficulty does. Quantum approaches become relatively less applicable exactly as the problem becomes commercially interesting, which is the opposite of what is often assumed.
The question to ask a vendor
If a supplier proposes quantum scheduling, ask what makespan they achieve on a 15 by 15 shop and how long it takes. Then compare it against 126 in 2.45 seconds, proven optimal, on a laptop.
The comparison is not unfair. It is the alternative you already have.
Ask the same question at whatever size your plant actually runs. If CP-SAT proves optimality there, no heuristic of any kind, quantum or classical, has anything to offer.
What you can run here
The subproblem that maps cleanly is conflict selection: the largest set of operations that can run together without contending for a machine, which is maximum independent set and runs on the neutral-atom path under Optimisation problem. Full job-shop scheduling, with precedence and a makespan objective, is not that problem, and the classical solver on this page still wins it. Use the quantum path for the selection step, and measure it against the solver rather than instead of it.
For context: where the hardware actually is
Gaps on this page are quoted against the processors ZKSF can run. That is not the frontier. Quantinuum, IBM, Atom Computing and QuEra’s newest systems are not available through us, and those machines are considerably further along. As of September 2026:
Physical qubits built
| Infleqtion Sqale | 1,600 | Neutral atom |
| Atom Computing | 1,180 | Neutral atom, 1,225 sites |
| IBM Condor | 1,121 | Superconducting, 2023 |
| IBM Heron R2 | 156 | Superconducting, ~99.5% two-qubit fidelity |
| Rigetti Cepheus | 108 | The largest available through ZKSF |
Two-qubit gate fidelity
The number that actually governs what a circuit can do.
| IonQ | 99.99% | Trapped ion, first past four nines |
| Silicon Quantum Computing | 99.99% | Silicon spin |
| Quantinuum | 99.97% | Trapped ion, all-to-all |
| IQM | 99.91% | Superconducting, available through ZKSF |
Logical qubits demonstrated
Published results, not roadmap targets.
| QuEra | 96 logical / 448 physical | Neutral atom |
| Quantinuum | 48 logical / 98 physical | Trapped ion, iceberg code |
| Atom Computing | 24 logical | On the 1,180-qubit system |
| 1 logical / 105 physical | Surface code, below threshold |
Announced roadmap
Targets. Roadmaps slip, and these are not results.
| Quantinuum Sol, 2027 | 192 physical, ~100 logical | Iceberg code, distance 2. Error detection with postselection, not correction |
| IBM Starling, 2029 | ~200 logical | Bivariate bicycle qLDPC, 100 million gates |
| Quantinuum Apollo, 2029 | hundreds of logical | Thousands of physical, logical error 1e-6 or better |
Where the encoding does work.
Not every operations problem encodes this badly. Satellite tasking needs one qubit per decision rather than one per job, machine and time slot, and there we have a real head-to-head result.