# Exact sampling hardness of Ising spin models

## TL;DR

This paper demonstrates that classically sampling from the output distribution of certain Ising spin models is computationally hard, linking quantum dynamics to complex computational problems and extending the understanding of quantum sampling hardness.

## Contribution

The paper constructs a specific Ising Hamiltonian whose exact output sampling is classically hard, advancing the classification of quantum sampling problems and connecting to complexity theory.

## Key findings

- Exact sampling from the Ising model is computationally hard.
- The results extend the classification of two-qubit commuting Hamiltonians.
- Physical implementations may soon reach problem sizes where classical sampling is infeasible.

## Abstract

We study the complexity of classically sampling from the output distribution of an Ising spin model, which can be implemented naturally in a variety of atomic, molecular, and optical systems. In particular, we construct a specific example of an Ising Hamiltonian that, after time evolution starting from a trivial initial state, produces a particular output configuration with probability very nearly proportional to the square of the permanent of a matrix with arbitrary integer entries. In a similar spirit to BosonSampling, the ability to sample classically from the probability distribution induced by time evolution under this Hamiltonian would imply unlikely complexity theoretic consequences, suggesting that the dynamics of such a spin model cannot be efficiently simulated with a classical computer. Physical Ising spin systems capable of achieving problem-size instances (i.e. qubit numbers) large enough so that classical sampling of the output distribution is classically difficult in practice may be achievable in the near future. Unlike BosonSampling, our current results only imply hardness of exact classical sampling, leaving open the important question of whether a much stronger approximate-sampling hardness result holds in this context. As referenced in a recent paper of Bouland, Mancinska, and Zhang (A. Bouland, L. Mancinska, and X. Zhang, CCC 2016, pp. 28:1-28:33), our result completes the sampling hardness classification of two-qubit commuting Hamiltonians.