Quantum Adiabatic Computation With a Constant Gap is Not Useful in One Dimension

TL;DR

This paper demonstrates that in one-dimensional quantum systems with a constant spectral gap, classical algorithms can efficiently simulate adiabatic evolution, indicating such systems are not suitable for universal quantum computation.

Contribution

It proves that adiabatic evolution with a constant gap in one dimension can be classically simulated, challenging their usefulness for universal quantum computation.

Findings

Classical simulation of 1D adiabatic evolution with constant gap is efficient.

Ground states have a good matrix product state representation due to the area law.

Adiabatic algorithms with a constant gap in 1D are not universal for quantum computation.

Abstract

We show that it is possible to use a classical computer to efficiently simulate the adiabatic evolution of a quantum system in one dimension with a constant spectral gap, starting the adiabatic evolution from a known initial product state. The proof relies on a recently proven area law for such systems, implying the existence of a good matrix product representation of the ground state, combined with an appropriate algorithm to update the matrix product state as the Hamiltonian is changed. This implies that adiabatic evolution with such Hamiltonians is not useful for universal quantum computation. Therefore, adiabatic algorithms which are useful for universal quantum computation either require a spectral gap tending to zero or need to be implemented in more than one dimension (we leave open the question of the computational power of adiabatic simulation with a constant gap in more than one dimension).