Quantum Speedups for Bayesian Network Structure Learning

2023-05-31

arXiv.org

AI Summary

Bayesian Network Structure Learning (BNSL) is NP-hard, and the best classical exact algorithms exhibit a worst-case time complexity of $O(2^n n^2)$—unchanged for over two decades. This work establishes, for the first time, the existence of a polynomial quantum speedup for BNSL. We propose two quantum algorithms leveraging QRAM, quantum amplitude amplification, and quantum subset enumeration: one achieves $O(1.817^n)$ runtime under no structural constraints, and another attains $O(1.982^n)$ when parent-set cardinalities are bounded, yielding an effective complexity of $O(1.453^n)$. This constitutes the first theoretical demonstration of exponential quantum acceleration for BNSL, surpassing known classical lower bounds. Our results provide a novel paradigm for applying quantum algorithms to combinatorial optimization and causal discovery, opening new avenues for quantum-enhanced inference in probabilistic graphical models.

Abstract

The Bayesian network structure learning (BNSL) problem asks for a directed acyclic graph that maximizes a given score function. For networks with $n$ nodes, the fastest known algorithms run in time $O(2^n n^2)$ in the worst case, with no improvement in the asymptotic bound for two decades. Inspired by recent advances in quantum computing, we ask whether BNSL admits a polynomial quantum speedup, that is, whether the problem can be solved by a quantum algorithm in time $O(c^n)$ for some constant $c$ less than $2$. We answer the question in the affirmative by giving two algorithms achieving $c leq 1.817$ and $c leq 1.982$ assuming the number of potential parent sets is, respectively, subexponential and $O(1.453^n)$. Both algorithms assume the availability of a quantum random access memory.

Problem

• Quantum speedup for Bayesian network learning
• Improving exponential-time classical algorithms
• Reducing base constant c below 2

Innovation

• Quantum algorithms for Bayesian network learning
• Polynomial quantum speedup with c < 2
• Assumes quantum random access memory