Qvine: Vine Structured Quantum Circuits for Loading High Dimensional Distributions
Loading high dimensional distributions is an important task for utilizing quantum computers on applications ranging from machine learning to finance. The high dimensionality leads to a curse of dimensionality, representing a d-dimensional distribution with k resolution requires dk qubits and an unstructured parameterized circuit would express a unitary in an exponential operator space in the number of qubits, leading to vanishing gradients and poor convergence guarantees even at high depth. Vine copula decompositions are widely used to represent high dimensional distributions classically, showing high quality approximation in many important applications, such as financial modeling. We present Qvine, a vine structured ansatz for quantum circuits, that mirrors the vine decomposition to construct scalable quantum circuits with efficient trainability while achieving similarly high quality approximation for amplitude encoding distributions. For regular vines (R-vines), we show that the circuit depth scales at most quadratic in the dimension of the distribution, while for D-vines, as well as many practical R-vines, the circuit depth scales linear in the dimension. For 3-dimensional and 4-dimensional Gaussians and empirical joint stock price return distributions for selected stocks, our experiments show Qvines achieve high quality loading.