# Perfect Sampling for Quantum Gibbs States

## TL;DR

This paper introduces a quantum algorithm for perfect sampling from quantum Gibbs states using a quantum adaptation of Coupling from the Past, with runtime depending on Hamiltonian degeneracy and stability under noise.

## Contribution

It adapts classical perfect sampling algorithms to the quantum setting, enabling sampling without prior mixing time knowledge and analyzing its efficiency and noise stability.

## Key findings

- Efficient for Hamiltonians with high degeneracy, polylogarithmic in dimension.
- Runtime comparable to classical algorithms for non-degenerate spectra.
- Algorithm is stable under implementation noise.

## Abstract

We show how to obtain perfect samples from a quantum Gibbs state on a quantum computer. To do so, we adapt one of the `Coupling from the Past'-algorithms proposed by Propp and Wilson. The algorithm has a probabilistic run-time and produces perfect samples without any previous knowledge of the mixing time of a quantum Markov chain. To implement it, we assume we are able to perform the phase estimation algorithm for the underlying Hamiltonian and implement a quantum Markov chain such that the transition probabilities between eigenstates only depend on their energy. We provide some examples of quantum Markov chains that satisfy these conditions and analyze the expected run-time of the algorithm, which depends strongly on the degeneracy of the underlying Hamiltonian. For Hamiltonians with highly degenerate spectrum, it is efficient, as it is polylogarithmic in the dimension and linear in the mixing time. For non-degenerate spectra, its runtime is essentially the same as its classical counterpart, which is linear in the mixing time and quadratic in the dimension, up to a logarithmic factor in the dimension. We analyze the circuit depth necessary to implement it, which is proportional to the sum of the depth necessary to implement one step of the quantum Markov chain and one phase estimation. This algorithm is stable under noise in the implementation of different steps. We also briefly discuss how to adapt different `Coupling from the Past'-algorithms to the quantum setting.