Quantum Sampling Algorithms, Phase Transitions, and Computational Complexity

TL;DR

This paper explores the complexity of quantum algorithms for sampling from classical distributions, revealing phase transition effects and proposing physically realizable Hamiltonians with potential quantum speedups.

Contribution

It introduces a framework linking quantum state preparation complexity to phase transitions and demonstrates feasible implementations with Rydberg atoms.

Findings

Quantum speedup identified for certain sampling problems.

Parent Hamiltonians can be realized with Rydberg atom systems.

Adiabatic paths can be optimized to improve quantum sampling efficiency.

Abstract

Drawing independent samples from a probability distribution is an important computational problem with applications in Monte Carlo algorithms, machine learning, and statistical physics. The problem can in principle be solved on a quantum computer by preparing a quantum state that encodes the entire probability distribution followed by a projective measurement. We investigate the complexity of adiabatically preparing such quantum states for the Gibbs distributions of various classical models including the Ising chain, hard-sphere models on different graphs, and a model encoding the unstructured search problem. By constructing a parent Hamiltonian, whose ground state is the desired quantum state, we relate the asymptotic scaling of the state preparation time to the nature of transitions between distinct quantum phases. These insights enable us to identify adiabatic paths that achieve a quantum speedup over classical Markov chain algorithms. In addition, we show that parent Hamiltonians for the problem of sampling from independent sets on certain graphs can be naturally realized with neutral atoms interacting via highly excited Rydberg states.