Hardness of efficiently generating ground states in postselected quantum computation

Yuki Takeuchi,Yasuhiro Takahashi,Seiichiro Tani

doi:10.1103/physrevresearch.3.013213

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.

Highlights

Quantum computing is expected to outperform classical computing
We say that a ground state is specified by using a polynomial number of bits if there exists a polynomial-size quantum circuit that outputs the ground state given the success of the postselection
We have shown that in the worst case, efficient generation of ground states of a given 3-local Hamiltonian is impossible for postselected quantum computation under a plausible assumption, i.e., the infiniteness of counting hierarchy (CH)

Summary

INTRODUCTION

Quantum computing is expected to outperform classical computing. quantum advantages have already been shown in terms of query complexity [1] and communication complexity [2]. By using a similar argument, we show that if the ground states of any 3-local Hamiltonians can be specified by using polynomial numbers of bits, NPPP = PSPACE This leads to the second-level collapse of CH, i.e., CH = PPPP. Our results are different from the existing ones on the impossibility of efficient groundstate generation in a sense that we reduce the impossibility to unlikely relations between classical complexity classes as in the quantum-computational-supremacy approach.

PRELIMINARIES

HARDNESS OF APPROXIMATELY GENERATING

CONCLUSION AND DISCUSSION