Quantum Ml

Quantum Circuit Born Machines

A quantum circuit Born machine is a generative model whose output distribution is the measurement probabilities of a trained parameterized circuit.

The Born rule as a model

A quantum circuit Born machine (QCBM) prepares a state |psi(theta)> with a parameterized circuit and defines a probability distribution over bitstrings by the Born rule: p_theta(x) = ||^2. Measuring the circuit in the computational basis draws a sample from p_theta. Training adjusts theta so this distribution matches a target dataset. There is no separate sampler; the physics of measurement does the sampling.

Training without explicit likelihoods

Kronos motion — three machines

A QCBM is an implicit model: it can produce samples but cannot cheaply report the probability of a given bitstring, because that would require knowing an amplitude. So the standard maximum-likelihood loss is impractical. Instead training uses losses computable from samples, most commonly the squared maximum mean discrepancy (MMD) with a classical kernel, which compares model samples to data samples through their kernel means.

python

# MMD loss between model and data samples (schematic)
def mmd_loss(model_samples, data_samples, kernel):
    def mean_kernel(A, B):
        return sum(kernel(a, b) for a in A for b in B) / (len(A) * len(B))
    return (mean_kernel(model_samples, model_samples)
            - 2 * mean_kernel(model_samples, data_samples)
            + mean_kernel(data_samples, data_samples))

Gradients

Gradients of the MMD loss with respect to circuit parameters can be obtained with the parameter-shift rule, evaluating the circuit at shifted angles. Each gradient component costs several batches of shots, so training is measurement-hungry. Like all variational models, QCBMs can hit barren plateaus when the ansatz is too expressive.

Expressive power

A QCBM can represent distributions arising from quantum interference that are believed hard to sample classically, the source of any potential advantage.
Entanglement in the ansatz lets it capture correlations between bits that a product model cannot.
Its capacity is bounded by circuit depth and connectivity, so shallow near-term circuits model only limited distributions.

Uses and caution

QCBMs have been demonstrated on small synthetic distributions and as components in optimization and finance-style sampling tasks (without any economic claim here). They are a clean testbed for quantum generative modeling. As with the whole field, small demonstrations do not establish an advantage over strong classical generators on real data; the case rests on the sampling-complexity argument, which applies only to suitably structured distributions.