Differentiable learning of quantum circuit Born machines

Quantum circuit Born machines are generative models which represent the probability distribution of classical dataset as quantum pure states. Computational complexity considerations of the quantum sampling problem suggest that the quantum circuits exhibit stronger expressibility compared to classical neural networks. One can efficiently draw samples from the quantum circuits via projective measurements on qubits. However, similar to the leading implicit generative models in deep learning, such as the generative adversarial networks, the quantum circuits cannot provide the likelihood of the generated samples, which poses a challenge to the training. We devise an efficient gradient-based learning algorithm for the quantum circuit Born machine by minimizing the kerneled maximum mean discrepancy loss. We simulated generative modeling of the BARS-AND-STRIPES dataset and Gaussian mixture distributions using deep quantum circuits. Our experiments show the importance of circuit depth and the gradient-based optimization algorithm. The proposed learning algorithm is runnable on near-term quantum device and can exhibit quantum advantages for probabilistic generative modeling.

Figure 1

Illustration of the differentiable QCBM training scheme. The top left is the quantum circuit which produces bit string samples. The dashed box on the right denotes the two-sample test on the generated samples and training samples with the loss function [Eq. (1)] and corresponding gradients [Eq. (2)] as outputs. $\mathrm{\Delta }\mathbf{\theta }$ is the amount of update applied to the circuit parameters, which are computed by a classical optimizer. The outcome of the training is to produce a quantum circuit which generates samples according to the learned probability distribution on the computational basis.

Figure 2

(a) Connectivity of the cnot gates for the $3\times 3$ BARS-AND-STRIPES dataset generated via the Chow-Liu tree algorithm. The qubits are arranged on a $3\times 3$ grid with some of them shifted a bit in order to visualize the edges clearly. (b) The Chow-Liu tree for the double-Gaussian peak model; the numbers represent the position of a bit in the digit, 0 for the big end and 9 for the little end. In this plot, the darkness of the edges indicates the amount of mutual information between two sites.

Figure 3

The MMD loss [Eq. (1)] as a function of training steps under different sampling errors that governed by batch size $N$. (a) Gradient-based training, solid colored lines are for Adam, and the dashed black line is for Limited-memory Broyden–Fletcher–Goldfarb–Shanno with Box constraints (L-BFGS-B) with $N=\infty$. The inset is a comparison of probability distribution between the training set and the QCBM output; the points on the dashed line mean exact matches. (b) The gradient-free CMA-ES training counterpart where each point on the graph represents a mean loss of its population.

Figure 4

$3\times 3$ BARS-AND-STRIPES samples generated from the QCBMs. The circuit parameters used here are from the final stages of Adam training with different batch sizes $N$ in Fig. 3. $\chi$ is the rate of generating valid samples in the training dataset. For illustrative purposes, we only show 12 samples for each situation with batch size $N$.

Figure 5

Losses as a function of training step for circuit depth $d=1,...,10$ (the darker the color of a line, the deeper the circuit). (a) The MMD loss Eq. (1) and (b) the corresponding Kullback-Leibler (KL) divergence. Here, we use the Adam optimizer with an exact gradient.

Figure 6

(a) The MMD loss as a function of the Adam training step. (b) Histogram for samples generated by a trained QCBM with a bin width of 20 (green bars) in comparison with the exact probability density function (black dashed line).

Figure 7

(a) The probability distribution of gradient elements in different layers grouped by depth. The circuit parameters are chosen randomly. (b) The variance of the MMD loss gradients as a function of sample size $N$.