Quantum Generative Models

A generative model learns a probability distribution and then samples new data from it, rather than mapping inputs to labels. Quantum circuits are natural generators: measuring a prepared state produces a sample, and the Born rule already gives the distribution. The cleanest realization is the quantum circuit Born machine (QCBM).

The Born machine

Prepare a state with a parameterized circuit and read off the measurement statistics. For an $n$ -qubit circuit $U(\boldsymbol{\theta})$ acting on $|0\rangle^{\otimes n}$ , the probability of observing bitstring $\boldsymbol{x} \in \{0,1\}^n$ is given directly by the Born rule:

p_{\boldsymbol{\theta}}(\boldsymbol{x}) = \bigl|\langle \boldsymbol{x} | U(\boldsymbol{\theta}) | 0\rangle^{\otimes n}\bigr|^2.

This is the model distribution. There is no separate likelihood to compute — the circuit is the distribution, and sampling it is just repeated measurement. The name comes from this reliance on Born's rule (Benedetti et al., 2019).

Training without an explicit likelihood

The catch is that we usually cannot write down $p_{\boldsymbol{\theta}}(\boldsymbol{x})$ for every $\boldsymbol{x}$ — there are $2^n$ of them. QCBMs are therefore implicit models: we can draw samples but not cheaply evaluate the full probability vector. Training uses a loss that needs only samples. A standard choice is the maximum mean discrepancy (MMD), which compares the model and data distributions through a kernel $k$ :

\mathcal{L}_{\text{MMD}} = \mathbb{E}_{\boldsymbol{x},\boldsymbol{x}'\sim p_{\boldsymbol{\theta}}}\,k(\boldsymbol{x},\boldsymbol{x}') - 2\,\mathbb{E}_{\boldsymbol{x}\sim p_{\boldsymbol{\theta}},\,\boldsymbol{y}\sim q}\,k(\boldsymbol{x},\boldsymbol{y}) + \mathbb{E}_{\boldsymbol{y},\boldsymbol{y}'\sim q}\,k(\boldsymbol{y},\boldsymbol{y}'),

where $q$ is the target data distribution. The loss is zero exactly when the two distributions match under the kernel. Crucially, every term is an expectation over samples, so the gradient can be estimated with the parameter-shift rule even though the likelihood is intractable (Liu & Wang, 2018).

Why a quantum generator might help

Two arguments recur in the literature:

Expressive distributions. A shallow quantum circuit can produce distributions that are conjectured hard to sample classically (the same complexity intuition behind random-circuit sampling). A Born machine inherits that expressivity essentially for free.
Correlation structure. Entangling layers create correlations between output bits that a factorized classical model would need many parameters to capture.

Neither argument is a proof of practical advantage, and on small problems a classical model often wins. But QCBMs are a clean, fully quantum generative primitive and a common benchmark target.

Relation to the rest of the module

A QCBM is the generative cousin of the variational classifier: same parameterized circuit and same parameter-shift training, but the output is a sample, not a label, and the loss compares distributions, not predictions. The next lesson on quantum GANs replaces the fixed MMD kernel with a learned discriminator, turning the Born machine into a generator inside an adversarial game.