Classical training of conditional instantaneous quantum polynomial-time Born machines

Instantaneous quantum polynomial-time (IQP) Born machines are attractive quantum generative models, as certain IQP circuit families are conjectured to be hard to sample, yet they can be trained classically. Previously proposed conditional IQP Born machines make the parameters of the diagonal layer condition dependent. We introduce conditional IQP Born machines, in which the condition is encoded in the input state and a single IQP unitary is shared across conditions. Efficient sampling and amplitude-ratio evaluation for the reference state entering the diagonal layer suffice to obtain unbiased classical estimates of a conditional maximum mean discrepancy objective and its gradients. We show that input conditioning can represent conditional families that no diagonally conditioned IQP model with the same number of qubits can reproduce. For the sixteen-qubit Curie-Weiss spin model, a single conditional model captures the magnetization transition at held-out temperatures. Its mean total variation distance is six times lower than that of diagonal conditioning under the same protocol and comparable to that of classical generative models with up to seven times as many parameters. We also train conditional models classically with up to 2048 qubits, which reproduce the mean squared magnetization of the target away from the crossover. These results extend classically trained IQP Born machines to conditional generation.