Quantum Sampling Algorithms, Phase Transitions, and Computational Complexity

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...

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Published in:arXiv.org
Main Authors: Wild, Dominik S, Sels, Dries, Pichler, Hannes, Zanoci, Cristian, Lukin, Mikhail D
Format: Paper
Language:English
Published: Ithaca Cornell University Library, arXiv.org 07.09.2021
Subjects:
Adiabatic flow
Algorithms
Complexity
Graphs
Hamiltonian functions
Ising model
Machine learning
Markov chains
Neutral atoms
Phase transitions
Probability distribution
Quantum computers
Rydberg states
Sampling
Statistical analysis
Statistical methods
ISSN:2331-8422
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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.
AbstractList 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.
Author Lukin, Mikhail D
Wild, Dominik S
Sels, Dries
Pichler, Hannes
Zanoci, Cristian
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SubjectTerms Adiabatic flow
Algorithms
Complexity
Graphs
Hamiltonian functions
Ising model
Machine learning
Markov chains
Neutral atoms
Phase transitions
Probability distribution
Quantum computers
Rydberg states
Sampling
Statistical analysis
Statistical methods
Title Quantum Sampling Algorithms, Phase Transitions, and Computational Complexity
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