Quantum Enhanced Inference in Markov Logic Networks

Abstract

Markov logic networks (MLNs) reconcile two opposing schools in machine learning and artificial intelligence: causal networks, which account for uncertainty extremely well, and first-order logic, which allows for formal deduction. An MLN is essentially a first-order logic template to generate Markov networks. Inference in MLNs is probabilistic and it is often performed by approximate methods such as Markov chain Monte Carlo (MCMC) Gibbs sampling. An MLN has many regular, symmetric structures that can be exploited at both first-order level and in the generated Markov network. We analyze the graph structures that are produced by various lifting methods and investigate the extent to which quantum protocols can be used to speed up Gibbs sampling with state preparation and measurement schemes. We review different such approaches, discuss their advantages, theoretical limitations, and their appeal to implementations. We find that a straightforward application of a recent result yields exponential speedup compared to classical heuristics in approximate probabilistic inference, thereby demonstrating another example where advanced quantum resources can potentially prove useful in machine learning.

Figure 1. An example of a first-order knowledge base, a matching MLN, and the corresponding concepts of a thermal state and a local Hamiltonian.

The knowledge base has only two formulas, and the variables range over a finite domain of two elements, {A, B}. Grounding out all formulas in all possible way, we obtain the MLN of maximal size (i.e., lifted inference is not used). The maximum of absolute value of the weights w₁ and w₂ defines the inverse temperature β in the thermal state. Since all ground atoms are binary valued, the local space is , and thus the thermal state is in , where n is the total number of nodes.