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.
Highlights
A sampling problem is the task of drawing samples from an implicitly defined probability distribution, which may, for example, be the Gibbs distribution of a classical system at a fixed temperature
This implies that it is easy to sample from many Gibbs distributions at high temperature, while no efficient, general-purpose algorithm is known below the ordering transition temperature of the underlying physical system
The possibility of ballistic propagation in a coherent quantum system, as evidenced by the dynamical critical exponent z = 1 away from the tricritical point, suggests that adiabatic state preparation could achieve a quadratic speedup over the classical Markov chain for sampling from the Gibbs distribution of the Ising chain at zero temperature. We show that this simple argument captures the relevant physics
Summary
A sampling problem is the task of drawing samples from an implicitly defined probability distribution, which may, for example, be the Gibbs distribution of a classical system at a fixed temperature. [12], we introduced a family of quantum algorithms that provide unbiased samples by preparing a state that encodes the entire Gibbs distribution We showed that this approach leads to a speedup over a classical Markov chain algorithm for several examples. The gap between the ground state and the first excited state of the parent Hamiltonian can be related to the mixing time of the Markov chain, thereby establishing a connection between quantum phases and the classical complexity of the sampling problem. Analyzing in detail several examples, we find for each that adiabatic state preparation along the one-parameter family Hq(β ), which may be viewed as a quantum analog of simulated annealing [13], exhibits the same time complexity as sampling by means of the Markov chain used to construct the parent Hamiltonian.
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