Near-Term Quantum Algorithms for Classical Sampling

Abstract

In the current era of noisy intermediate-scale quantum devices, quantum sampling algorithms have been of great interest as they permit errors in their execution while maintaining their advantage over classical counterparts [26]. However, the sampling problems considered often do not possess immediate practical relevance. This thesis explores two quantum algorithms for applicable classical sampling problems that can be implemented on today’s quantum devices. Specifically, we are considering algorithms to sample from a Boltzmann distribution of a classical Hamiltonian. This sampling task is of significant importance in the fields of statistical physics, machine learning, and optimization.

The first such algorithm adiabatically prepares a quantum state which encodes the desired Boltzmann distribution [44]. Projectively measuring this state then produces uncorrelated samples from the desired distribution. The state preparation time scaling of this algorithm can be related to the properties of quantum phase transitions, giving physical insights into the mechanism of speedups found. Numerical investigations of the algorithmic performance on the Ising chain are reproduced, showing a quadratic improvement over a classical Markov chain Monte Carlo (MCMC) method. On the same model, counterdiabatic driving protocols are explored with the limitation of local driving terms. It is shown numerically this restriction of local driving terms leads to unfavourable scaling of the state preparation time.

Next, the quantum-enhanced Markov Chain Monte Carlo algorithm is explored [23]. This hybrid algorithm creates a Markov chain over the classical configuration space, where new configurations are proposed through a projectively measured quantum evolution. This algorithm has guaranteed convergence, independent of the quality of the evolution, making it an algorithm suited for near-term implementation. The performance of this algorithm on the Sherrington-Kirkpatrick model is numerically reproduced, showing faster mixing time than classical MCMC in the low-temperature limit. Bottlenecks of this chain are then explored for the Ising chain, giving an analytic bound on performance showing the algorithmic advantage found for small systems numerically persists for larger system sizes. Finally, this algorithm is tested numerically on the maximum independent set problem, which is native to an array of Rydberg atoms and has been experimentally realized on current quantum devices [12]. Our findings did not indicate any advantage of the quantumenhanced MCMC algorithm over classical algorithms for the limited number of numerically accessible system sizes.