Obstructions To Classically Simulating The Quantum Adiabatic Algorithm

TL;DR

This paper demonstrates that quantum Monte Carlo methods can fail to efficiently simulate certain adiabatic quantum algorithms due to topological obstructions, even when the spectral gap is only polynomially small.

Contribution

The paper constructs specific Hamiltonians with topological features that cause QMC to fail in equilibration, revealing fundamental obstructions to classical simulation of quantum adiabatic algorithms.

Findings

QMC does not equilibrate in certain topologically nontrivial Hamiltonians.

Failure of QMC occurs even with polynomially small spectral gaps.

Topological properties can prevent classical simulation of quantum algorithms.

Abstract

We consider the adiabatic quantum algorithm for systems with "no sign problem", such as the transverse field Ising mode, and analyze the equilibration time for quantum Monte Carlo (QMC) on these systems. We ask: if the spectral gap is only inverse polynomially small, will equilibration methods based on slowly changing the Hamiltonian parameters in the QMC simulation succeed in a polynomial time? We show that this is not true, by constructing counter-examples. Some examples are Hamiltonians where the space of configurations where the wavefunction has non-negligible amplitude has a nontrivial fundamental group, causing the space of trajectories in imaginary time to break into disconnected components, with only negligible probability outside these components. For the simplest example we give with an abelian fundamental group, QMC does not equilibrate but still solves the optimization problem. More severe effects leading to failure to solve the optimization can occur when the fundamental group is a free group on two generators. Other examples where QMC fails have a trivial fundamental group, but still use ideas from topology relating group presentations to simplicial complexes. We define gadgets to realize these Hamiltonians as the effective low-energy dynamics of a transverse field Ising model. We present some analytic results on equilibration times which may be of some independent interest in the theory of equilibration of Markov chains. Conversely, we show that a small spectral gap implies slow equilibration at low temperature for some initial conditions and for a natural choice of local QMC updates.