Hardness of efficiently generating ground states in postselected quantum computation

TL;DR

This paper provides evidence that efficiently generating ground states of local Hamiltonians with postselected quantum computation is likely impossible, linking it to a major complexity class collapse.

Contribution

It demonstrates that if such ground states could be generated efficiently with postselection, it would imply ${ m PP}={ m PSPACE}$, which is considered highly unlikely.

Findings

Generating ground states in polynomial time with postselection implies ${ m PP}={ m PSPACE}$.

The result reduces the hardness of ground state generation to a classical complexity class relation.

Discusses the fundamental reasons behind the difficulty of this problem for postselected quantum computers.

Abstract

Generating ground states of any local Hamiltonians seems to be impossible in quantum polynomial time. In this paper, we give evidence for the impossibility by applying an argument used in the quantum-computational-supremacy approach. More precisely, we show that if ground states of any $3$-local Hamiltonians can be approximately generated in quantum polynomial time with postselection, then ${\sf PP}={\sf PSPACE}$. Our result is superior to the existing findings in the sense that we reduce the impossibility to an unlikely relation between classical complexity classes. We also discuss what makes efficiently generating the ground states hard for postselected quantum computation.