QCBM and quantum generative models

Sampling from a distribution learned from scratch

A Born machine is the simplest generative model a quantum computer supports. The circuit's measurement probabilities are the distribution, so sampling it is a single execution rather than a decoding step, and training means moving those probabilities toward a target by adjusting rotation angles. Nothing is fed in: the model has no input, only parameters.

The distance reported throughout is total variation distance (TVD), half the sum of the absolute differences between the learned and target probabilities, so 0 is an exact match and 1 is no overlap. The circuit was trained on exact.cpu, where every gradient evaluation is its own job, and the converged circuit was then submitted once to a Rigetti superconducting QPU.

The ZKSF console showing a circuit above the job history, each row naming the engine it ran on and what it cost — A Born machine is an ordinary gate circuit, submitted like any other program. Open the console

Method: QCBM against its own untrained start
Who runs this: Generative modelling research, synthetic data programmes
Classical baseline: The untrained circuit, at TVD 0.459
Structural limit: 2^n outcomes against a fixed shot budget: at 20 qubits most outcomes are never sampled once

Run on our engines

A quantum circuit Born machine (QCBM) learning a two-qubit target distribution from a random start, then sampled. Submitted to each kind of compute we offer, on 18 September 2026. Every figure below is a real job on the service, priced as any customer would be priced.

Engine	Kind	Qubits	Result	Cost
exact.cpu	CPU	2	TVD 0.459 untrained, 0.0117 after 60 Adam steps	$0.0806
exact.gpu	GPU	2	TVD 0.041, the same converged circuit at 1,000 shots certificate	$0.0001
qpu.rigetti	QPU	2	TVD 0.102, the converged circuit re-run at 1,000 shots	$0.7250
qpu.iqm.garnet	QPU	2	TVD 0.104, the same circuit and shot count on a second superconducting device certificate	$1.7500
qpu.aqt.ibex	QPU	2	3 of 100 shots outside the target, against 102 of 1,000 on Rigetti and 38 of 1,000 on Garnet certificate	$2.6500
neural.tpu	TPU	—	a Born machine is a circuit, not a Hamiltonian, so it does not reach the neural tier	—

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.

Training took 806 circuits across 60 Adam steps and moved the distribution from 0.459 to 0.0117. The same converged circuit on Rigetti returned 0.102 at 1,000 shots, against 0.068 for the noiseless reference at the same shot count.

The learned distribution puts 0.496 on 00 and 0.492 on 11 against a target of 0.5 and 0.5, with the two remaining outcomes at 0.006 each.

The same converged circuit has now run on four engines. The two superconducting devices land together, 0.102 on Rigetti and 0.104 on IQM Garnet, and two independent devices agreeing that closely on the same circuit is the useful part of running it twice.

This target forbids two of its four outcomes, so shots landing on 01 or 10 are a direct fidelity measure rather than a distance: 102 of 1,000 on Rigetti, 38 of 1,000 on Garnet, 3 of 100 on AQT IBEX Q1. At 100 shots the AQT figure carries about half its own width in uncertainty, enough to place it with Garnet and below Rigetti, not enough to separate the two. Its distance column is not comparable to the rows above it either, because the sampling floor at 100 shots is near the whole gap being measured.

That shot count is a purchase, not a shortcut. A trapped-ion shot is $0.0235 against Rigetti's $0.000425, so the same 1,000-shot run is $23.80 there and $0.7250 on Rigetti, and every device on this table charges the same $0.30 to accept the job. Which way that trade falls depends on whether the question is how good each shot is or how many of them you need.

The two simulator rows cost the same $0.0001, and that figure is the platform floor rather than a measurement: the GPU tier bills at $3.00 an hour against the CPU tier's $0.69, and a two-qubit run finishes long before either rate reaches the floor. The floor is hiding a thirty-fold difference in rate, which starts to show once a circuit is wide enough to run for real time. They also agree with each other to within 1.3 standard deviations of sampling noise, which is what confirms a circuit rebuilds identically on both.

What the makers say it is for

Rigetti, whose processor the hardware row below ran on, publishes quantum kernel and quanvolutional neural network methods aimed at classification and regression, and a technique intended to help with the difficulty of training generative adversarial networks for financial applications. A circuit Born machine is the generative end of that same family. Instead of scoring or labelling, it samples.

IonQ has published work it presents as quantum-enhanced applications advancing AI, so the direction is not one vendor's alone.

What a vendor publishes is a direction of travel. What the table below carries is one instance, at one size, with the bill attached. Both are worth having and they answer different questions.

Where this stops

The outcome space is 2^n and the shot budget is fixed, so the number of shots per outcome falls exponentially with width. At 4 qubits, 16 outcomes from 512 shots is already thin
Every gradient evaluation is a separate job, so a training loop is hundreds of circuits and training on hardware is priced per task
TVD is measured against a known target, which is available here and is not available for a distribution you are trying to learn from data

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: Quantum machine learning on simulators and real quantum hardware.

Read the docs Benchmark methodology

Other AI use cases

QGAN and synthetic data