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State transfer analysis for linear spin chains with non-uniform on-site energies

Chad C. Nelmes, Irene D'Amico, Timothy P. Spiller

TL;DR

This work analyzes high-fidelity state transfer in linear spin chains with non-uniform on-site energies, bridging two complementary strategies: a genetic-algorithm approach that enforces uniform couplings while optimizing $\varepsilon_i$ to achieve quasi-perfect state transfer (QPST), and a persymmetric inverse-eigenvalue method that reconstructs a unique Jacobi matrix yielding perfect state transfer (PST) from a carefully chosen spectrum. The authors illuminate the physical intuition via a discrete-potential analogy, show that parabolic on-site energy profiles yield near-equally spaced spectra, and demonstrate that PST can be realized with only small adjustments to couplings through persymmetric reconstruction. They provide explicit constructions (including N=3 example) and discuss experimental platforms where these control schemes are viable, highlighting the practical trade-offs between maintaining uniform couplings and achieving robust PST/QPST in larger systems. Overall, the paper advances methods to tailor spectral and spatial properties of spin chains to enable reliable quantum information transfer in realistic, imperfect hardware.

Abstract

High fidelity state transfer is an important ingredient of distributed quantum information processing. We present and analyse results on perfect and quasi-perfect state transfer with linear spin chains incorporating non-uniform on-site energies. The motivation is maintenance of coupling uniformity, which could be beneficial for some physical implementations. We relate this coupling uniformity to a particle in a discrete potential analogue. Our analysis further considers the statistical variation in couplings and on-site energies, as a function of increasing chain site number N.

State transfer analysis for linear spin chains with non-uniform on-site energies

TL;DR

This work analyzes high-fidelity state transfer in linear spin chains with non-uniform on-site energies, bridging two complementary strategies: a genetic-algorithm approach that enforces uniform couplings while optimizing to achieve quasi-perfect state transfer (QPST), and a persymmetric inverse-eigenvalue method that reconstructs a unique Jacobi matrix yielding perfect state transfer (PST) from a carefully chosen spectrum. The authors illuminate the physical intuition via a discrete-potential analogy, show that parabolic on-site energy profiles yield near-equally spaced spectra, and demonstrate that PST can be realized with only small adjustments to couplings through persymmetric reconstruction. They provide explicit constructions (including N=3 example) and discuss experimental platforms where these control schemes are viable, highlighting the practical trade-offs between maintaining uniform couplings and achieving robust PST/QPST in larger systems. Overall, the paper advances methods to tailor spectral and spatial properties of spin chains to enable reliable quantum information transfer in realistic, imperfect hardware.

Abstract

High fidelity state transfer is an important ingredient of distributed quantum information processing. We present and analyse results on perfect and quasi-perfect state transfer with linear spin chains incorporating non-uniform on-site energies. The motivation is maintenance of coupling uniformity, which could be beneficial for some physical implementations. We relate this coupling uniformity to a particle in a discrete potential analogue. Our analysis further considers the statistical variation in couplings and on-site energies, as a function of increasing chain site number N.

Paper Structure

This paper contains 13 sections, 47 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The spin chain model
  3. Non-uniform on-site energies ($\varepsilon_i$) in High fidelity QPST solutions
  4. Genetic Algorithm
  5. Specific Examples
  6. Particle in a Discrete Potential analogue
  7. PST vs. QPST: Persymmetric Matrix Reconstruction
  8. Background
  9. Generating unique $\varepsilon_i$ and $J_{i,i+1}$ values from a chosen spectrum $\{\lambda_k\}$
  10. Transitioning from QPST to PST
  11. Experimental Implementations
  12. Conclusions
  13. The Persymmetric Matrix Spectral Reconstruction Method: An $N=3$ Example

Figures (4)

  • Figure 2: Negative coupling eigenstate amplitude shape, drawn through the actual amplitude points at the discrete sites, (in ascending order, centered about their eigenvalue in the style of an energy-level diagram) of the $N = 5$ high-fidelity solution, plotted against their corresponding eigenvalue, in an energy level diagram-like fashion, along the site indices ($i$). It can be seen that the eigenstates exhibit forms analogous to those for a harmonic oscillator or other potential well, with $k$ nodes present in the state with energy $\lambda_k$. Correspondingly, the eigenstates alternate in even and odd symmetry (with respect to mirror symmetry about the center of the chain), moving upwards in energy from the even-symmetry ground state. The solid dots serve as a marker for the amplitudes of the eigenstates on site $i$.
  • Figure 3: High-fidelity dynamics of the 5-site chain described by Equation \ref{['N = 5p3']} (top plot) plotted over an extended window. The transfer fidelity tends towards (approximately) $80\%$ within a window of $t \cdot J_{max} = 400$. Transfer fidelity over extended time window for $H^*_{XY}$ (lower plot) reconstructed using the inverse eigenvalue method. It can be seen through plotting over larger time scales that the transfer fidelity associated with the correctly chosen spectral data for the persymmetric matrix reconstruction remains arbitrarily close to 1 and periodic.
  • Figure 4: The standard and maximum deviation numerical values (y-axis) in the couplings as a function of $N$, derived from inverse spectral persymmetric matrix construction.
  • Figure : (a) $N = 4; p = 3$.