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Efficient Verification of Ground States of Frustration-Free Hamiltonians

Huangjun Zhu, Yunting Li, Tianyi Chen

Abstract

Ground states of local Hamiltonians are of key interest in many-body physics and also in quantum information processing. Efficient verification of these states are crucial to many applications, but very challenging. Here we propose a simple, but powerful recipe for verifying the ground states of general frustration-free Hamiltonians based on local measurements. Moreover, we derive rigorous bounds on the sample complexity by virtue of the quantum detectability lemma (with improvement) and quantum union bound. Notably, the number of samples required does not increase with the system size when the underlying Hamiltonian is local and gapped, which is the case of most interest. As an application, we propose a general approach for verifying Affleck-Kennedy-Lieb-Tasaki (AKLT) states on arbitrary graphs based on local spin measurements, which requires only a constant number of samples for AKLT states defined on various lattices. Our work is of interest not only to many tasks in quantum information processing, but also to the study of many-body physics.

Efficient Verification of Ground States of Frustration-Free Hamiltonians

Abstract

Ground states of local Hamiltonians are of key interest in many-body physics and also in quantum information processing. Efficient verification of these states are crucial to many applications, but very challenging. Here we propose a simple, but powerful recipe for verifying the ground states of general frustration-free Hamiltonians based on local measurements. Moreover, we derive rigorous bounds on the sample complexity by virtue of the quantum detectability lemma (with improvement) and quantum union bound. Notably, the number of samples required does not increase with the system size when the underlying Hamiltonian is local and gapped, which is the case of most interest. As an application, we propose a general approach for verifying Affleck-Kennedy-Lieb-Tasaki (AKLT) states on arbitrary graphs based on local spin measurements, which requires only a constant number of samples for AKLT states defined on various lattices. Our work is of interest not only to many tasks in quantum information processing, but also to the study of many-body physics.
Paper Structure (26 sections, 11 theorems, 108 equations, 2 figures)

This paper contains 26 sections, 11 theorems, 108 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum state verification
  4. Subspace verification
  5. Hypergraphs
  6. Frustration-free Hamiltonians
  7. A stronger detectability lemma
  8. Improvement of the detectability lemma
  9. Proofs of Lemmas \ref{['lem:DL']} and \ref{['lem:DLnorm']}
  10. Efficient verification of ground states
  11. Matching and coloring protocols
  12. Sample complexity
  13. Proof of Theorem \ref{['thm:SpectralGap']}
  14. Proof of Theorem \ref{['thm:SpectralGap2']}
  15. Efficient verification of AKLT states
  16. ...and 11 more sections

Key Result

Lemma 1

Let $\{P_k\}_{k=1}^q$ be a set of projectors on a given Hilbert space $\mathcal{H}$, $Q_k=1-P_k$, and $H=\sum_k P_k$. Let $|\psi\rangle$ be any normalized ket in $\mathcal{H}$, $|\varphi\rangle=Q_1Q_2\cdots Q_q|\psi\rangle$, and $\varepsilon_\varphi =\langle\varphi|H|\varphi\rangle/\|\varphi\|^2$ wi with $\zeta=\max_{j} \zeta_j$ and $s=\max_{j<k}s_{jk}$, where $s_{jk}$ is the largest singular valu

Figures (2)

  • Figure 1: Optimal edge colorings of the square lattice and honeycomb lattice. These optimal colorings can be used to construct efficient protocols for verifying ground states of frustration-free Hamiltonians, including AKLT states.
  • Figure 2: Comparison of sample costs in the verification of the AKLT state on the even closed chain with $n$ nodes within precision $\epsilon=\delta=0.01$. Here the coloring protocol is a special matching protocol based on the optimal edge coloring, and the number of tests is determined by the first upper bound in Eq. (\ref{['eq:TestNumUB']}) with $m=g=2$, $s=1/2$, and $\nu_E=2/5$. The HKSE protocol is proposed in Refs. HangKSE17, and the number of tests is determined by Eq. (\ref{['eq:HKSE']}) with $|E|=n$. The BHSRE protocol is proposed in Ref. BermHSR18, and the number of tests is determined by the lower bound in Eq. (\ref{['eq:BHSRElb']}) with $\kappa=2$.

Theorems & Definitions (21)

  • Lemma 1
  • Lemma 2
  • proof : Proof of Lemma \ref{['lem:DL']}
  • Lemma 3
  • proof : Proof of Lemma \ref{['lem:DLnorm']}
  • Theorem 1
  • Theorem 2
  • proof : Proof of Theorem \ref{['thm:SpectralGap']}
  • Lemma 4
  • proof : Proof of Lemma \ref{['lem:SpectralGap']}
  • ...and 11 more