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On the Computational Complexity of Schrödinger Operators

Yufan Zheng, Jiaqi Leng, Yizhou Liu, Xiaodi Wu

TL;DR

It is proved that simulating the dynamics generated by the Schrodinger operator implements universal quantum computation, i.e., it is StoqMA-complete and estimating the ground energy of the Schrodinger operator is as hard as estimating that of local Hamiltonians with no sign problem.

Abstract

We study computational problems related to the Schrödinger operator $H = -Δ+ V$ in the real space under the condition that (i) the potential function $V$ is smooth and has its value and derivative bounded within some polynomial of $n$ and (ii) $V$ only consists of $O(1)$-body interactions. We prove that (i) simulating the dynamics generated by the Schrödinger operator implements universal quantum computation, i.e., it is BQP-hard, and (ii) estimating the ground energy of the Schrödinger operator is as hard as estimating that of local Hamiltonians with no sign problem (a.k.a. stoquastic Hamiltonians), i.e., it is StoqMA-complete. This result is particularly intriguing because the ground energy problem for general bosonic Hamiltonians is known to be QMA-hard and it is widely believed that $\texttt{StoqMA}\varsubsetneq \texttt{QMA}$.

On the Computational Complexity of Schrödinger Operators

TL;DR

It is proved that simulating the dynamics generated by the Schrodinger operator implements universal quantum computation, i.e., it is StoqMA-complete and estimating the ground energy of the Schrodinger operator is as hard as estimating that of local Hamiltonians with no sign problem.

Abstract

We study computational problems related to the Schrödinger operator in the real space under the condition that (i) the potential function is smooth and has its value and derivative bounded within some polynomial of and (ii) only consists of -body interactions. We prove that (i) simulating the dynamics generated by the Schrödinger operator implements universal quantum computation, i.e., it is BQP-hard, and (ii) estimating the ground energy of the Schrödinger operator is as hard as estimating that of local Hamiltonians with no sign problem (a.k.a. stoquastic Hamiltonians), i.e., it is StoqMA-complete. This result is particularly intriguing because the ground energy problem for general bosonic Hamiltonians is known to be QMA-hard and it is widely believed that .

Paper Structure

This paper contains 23 sections, 24 theorems, 92 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of Results
  3. Simulating the Schrödinger operator is BQP-hard.
  4. Determining the ground energy of the Schrödinger operator is StoqMA-complete.
  5. Technical Contributions
  6. Encoding qubit states into bosonic modes
  7. Substitute of Schrieffer--Wolff theory
  8. Organization
  9. Complexity of the transverse field Ising model
  10. Algorithms for Schrödinger Operators
  11. Estimating the Ground Energy is in StoqMA
  12. Time-Independent Simulation is in BQP
  13. Hardness of Schrödinger operators
  14. Proof Idea
  15. 1-Dimensional Double-Well Function
  16. ...and 8 more sections

Key Result

Theorem 2.4

Determining the ground energy of TIM Hamiltonian is $\normalfont \textsf{StoqMA}$-complete. More specifically, estimating within error $1/\mathop{\mathrm{poly}}\nolimits(n)$ the smallest eigenvalue of the TIM Hamiltonian with $|h_u|,|g_u|,|g_{u,v}| \leq \mathop{\mathrm{poly}}\nolimits(n)$ given explicitly, is complete for the complexity class StoqMA under P reductions.

Figures (1)

  • Figure 1: First two eigenfunctions $\psi_0(x),\psi_1(x)$ and their linear combination $\widehat{s_0} (x) = (\psi_0(x)+\psi_1(x))/\sqrt{2}$ for the operator $\widehat{X} = -\frac{ \mathrm{d} ^2}{ \mathrm{d} x^2} + f_{\rm dw}$ with parameter $w = 0.05$ and $\ell = 0.5$, where $f_{\rm dw}$ is defined in \ref{['sec:proof-outline']}.

Theorems & Definitions (74)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: bravyi2017complexity
  • Definition 2.5
  • Theorem 2.6
  • proof
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • ...and 64 more