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Fast state transfer via loop weights

Gabor Lippner, Yujia Shi

TL;DR

This work addresses robust quantum state transfer along a uniform XX spin chain by introducing loop weights implemented as a magnetic field of strength $Q$ at the 2nd and $(n-1)$-st sites, yielding the Hamiltonian $H = A_{\text{path}} + Q D_{2,n-1}$ in the 1-excitation subspace. The authors prove that for chain length $n = \Omega(1/\\varepsilon)$ there exists $Q = O(\\sqrt{n}/\\varepsilon)$ enabling end-to-end fidelity $> 1-\\varepsilon$ in time $t_0 < O(n/\\varepsilon)$, with the transfer governed by $t_0 = \pi/|\\lambda_1 - \\lambda_2|$. A detailed eigenvector analysis partitions eigenvectors into symmetric and alternating families, showing that two low-lying eigenvectors can dominate end weights when $Q$ and a parameter $k = n-3$ are tuned so that the eigenvalue gap yields the desired $t_0$. The result provides a rigorous, near-linear-time protocol for high-fidelity state transfer that is more robust in timing than prior approaches that place inhomogeneities at or near the ends.

Abstract

We prove that almost-linear-time high-fidelity state transfer is achievable in a quantum spin chain using loop weights at the second and second-to-last nodes. We provide specific parameter values, and using a careful analysis of the eigenvectors we make precise quantitative estimates of the transfer time and strength.

Fast state transfer via loop weights

TL;DR

This work addresses robust quantum state transfer along a uniform XX spin chain by introducing loop weights implemented as a magnetic field of strength at the 2nd and -st sites, yielding the Hamiltonian in the 1-excitation subspace. The authors prove that for chain length there exists enabling end-to-end fidelity in time , with the transfer governed by . A detailed eigenvector analysis partitions eigenvectors into symmetric and alternating families, showing that two low-lying eigenvectors can dominate end weights when and a parameter are tuned so that the eigenvalue gap yields the desired . The result provides a rigorous, near-linear-time protocol for high-fidelity state transfer that is more robust in timing than prior approaches that place inhomogeneities at or near the ends.

Abstract

We prove that almost-linear-time high-fidelity state transfer is achievable in a quantum spin chain using loop weights at the second and second-to-last nodes. We provide specific parameter values, and using a careful analysis of the eigenvectors we make precise quantitative estimates of the transfer time and strength.
Paper Structure (10 sections, 6 theorems, 62 equations, 3 figures)

This paper contains 10 sections, 6 theorems, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Eigenvectors
  4. Symmetric eigenvectors
  5. Alternating eigenvectors
  6. Ensuring strong state transfer
  7. Transfer fidelity
  8. Calculus
  9. Putting it all together
  10. Acknowledgements:

Key Result

Theorem 1.2

Fix $\varepsilon > 0$. Then for $n = \Omega(1/\varepsilon)$ there is a $Q = O(\sqrt{n}/\varepsilon)$ such that we get quantum state transfer between the chains ends with fidelity at least $1-\varepsilon$ and in time $t_0 < O(n/\varepsilon)$.

Figures (3)

  • Figure 1: Transfer fidelity as a function of time in the Chen et al. protocol for a chain of length $N=501$. The second image is zoomed in to near the optimal transfer time.
  • Figure 2: Transfer fidelity as a function of time in our protocol for a chain of length $N=501$ and $Q = 80$ The second image is zoomed in to near the optimal transfer time. Note that our protocol has transfer fidelity consistently above 0.95 in a large time window.
  • Figure 3: $\varrho^j - \varrho^{k-j}$ vs $\mu^k$

Theorems & Definitions (19)

  • Definition 1.1
  • Theorem 1.2
  • Claim 2.1
  • proof
  • Lemma 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • ...and 9 more