Variational Inference with a Quantum Computer

Inference is the task of drawing conclusions about unobserved variables given observations of related variables. Applications range from identifying diseases from symptoms to classifying economic regimes from price movements. Unfortunately, performing exact inference is intractable in general. One alternative is variational inference, where a candidate probability distribution is optimized to approximate the posterior distribution over unobserved variables. For good approximations, a flexible and highly expressive candidate distribution is desirable. In this work, we use quantum Born machines as variational distributions over discrete variables. We apply the framework of operator variational inference to achieve this goal. In particular, we adopt two specific realizations: one with an adversarial objective and one based on the kernelized Stein discrepancy. We demonstrate the approach numerically using examples of Bayesian networks, and implement an experiment on an IBM quantum computer. Our techniques enable efficient variational inference with distributions beyond those that are efficiently representable on a classical computer.

Figure 1

Some applications of inference on Bayesian networks. Filled (empty) circles indicate observed (unobserved) random variables. (a) “Sprinkler” network. (b) Regime switching in time series. (c) “Lung cancer” network [53]. (d) Error correction codes.

Figure 2

Probabilistic models used in our methods. The classical model comprises a prior over unobserved discrete variables and a likelihood over observed variables. The quantum model approximates the posterior distribution of the unobserved variables given observed data. All distributions can be sampled efficiently as long as one follows the arrows and uses a suitable computer.

Figure 3

Adversarial variational inference with a Born machine. Step 1 optimizes a classifier $d_{\mathit{\varphi }}$ to output the probabilities that the observed samples come from the Born machine rather than from the prior. Step 2 optimizes the Born machine $q_{\mathit{\theta }}$ to better match the true posterior. The updated Born machine is fed back into step 1 and the process repeats until convergence of the Born machine.

Figure 4

Kernelized Stein variational inference with a Born machine. Step 1 computes the Stein discrepancy (SD) between the Born machine and the true posterior using samples from the Born machine alone. By choosing a reproducing kernel Hilbert space $\mathcal{H}$ with kernel $k$, the kernelized Stein discrepancy (KSD) can be calculated in closed form. Step 2 optimizes the Born machine to reduce the discrepancy. The process repeats until convergence.

Figure 5

Example of hardware-efficient ansatz for the Born machine used in the experiments. All rotations $R_x$, $R_z$ are parameterized by individual angles. The layer denoted as $S(\mathbf{x})$ encodes the values of the observed variables $\mathbf{x}$. The particular choice of $S(\mathbf{x})$ depends on the application (see the main text). The corresponding classical random variables are indicated below the circuit.

Figure 6

Total variation distance of the approximate and true posteriors. Median (lines) and $68\mathrm{\%}$ confidence interval (shaded area) of $30$ instances for (a) KL and (b) KSD objective functions for a different number of layers in the ansatz (cf. Fig. 5). Each instance is a “sprinkler” Bayesian network as in Fig. 1 where the conditional probabilities are chosen at random. (a) Adversarial objective. (b) Kernelized Stein objective.

Figure 7

Truncated, ordered histograms of the posteriors for two observed samples (a) ${\mathbf{x}}^{(1)}$ and (b) ${\mathbf{x}}^{(2)}$ of the hidden Markov model in Eqs. (15)–(16). The histograms are sorted by probability of the true posterior. The blue bars are the probabilities of the corresponding approximate posterior. The $x$ axis shows the latent state for each bar and the observed data point $\mathbf{x}$. The lower panels are the time series of the data (c) ${\mathbf{x}}^{(1)}$ and (d) ${\mathbf{x}}^{(2)}$, as well as the corresponding modes of the true posterior and Born machine posterior as indicated with stars in the upper panel. (a) Posterior histogram given data ${\mathbf{x}}^{(1)}$. (b) Posterior histogram given data ${\mathbf{x}}^{(2)}$. (c) Time series and posterior modes for ${\mathbf{x}}^{(1)}$. (d) Time series and posterior modes for ${\mathbf{x}}^{(2)}$.

Figure 8

Histograms of true versus the learned posteriors with a simulated and hardware trained Born machine for the “lung cancer” network in Fig. 1. Here we use the ibmq_rome quantum processor (whose connectivity is shown in the inset). We condition on $X=\mathsf{f}\mathsf{a},D=\mathsf{f}\mathsf{a},I=\mathsf{t}\mathsf{r}$. The $x$ axis shows the configuration of observed (gray represents $X$, $D$, $I$) and unobserved variables (blue represents $A$, $S$, $T$, $L$, $B$) corresponding to each probability (filled circles $=\mathsf{t}\mathsf{r}$, empty circles $=\mathsf{f}\mathsf{a}$). For hardware results, the histogram is generated using $1024$ samples.

Figure 9

Examples of successful (a) and unsuccessful (b) VI on the “sprinkler” Bayesian network in Sec. 5a using the KL objective with the adversarial method. (a) Successful example. (b) Unsuccessful example.