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Peak state transfer in continuous quantum walks

Gabriel Coutinho, Krystal Guo, Vincent Schmeits

TL;DR

Peak state transfer reframes quantum state transfer by relaxing perfect fidelity to enable explicit, computable transfer times. The authors provide a spectral characterization based on mutual eigenvalue supports and quadratic-integer eigenvalues, enabling deterministic timing conditions; they construct a family of weighted paths where peak transfer across distance grows arbitrarily large with transfer probability approaching $${\pi\over 4} \approx 0.78$$. They also show peak transfer can be robust to timing errors through spectral spread and present a variety of examples for both adjacency and Laplacian dynamics, including small graphs and infinite families. The work suggests peak state transfer is widespread and practical for quantum wires, and it poses open questions on trees, sparsity, and deeper spectral structure.

Abstract

We introduce and study peak state transfer, a notion of high state transfer in qubit networks modeled by continuous-time quantum walks. Unlike perfect or pretty good state transfer, peak state transfer does not require fidelity arbitrarily close to 1, but crucially allows for an explicit determination of the time at which transfer occurs. We provide a spectral characterization of peak state transfer, which allows us to find many examples of peak state transfer, and we also establish tight lower bounds on fidelity and success probability. As a central example, we construct a family of weighted path graphs that admit peak state transfer over arbitrarily long distances with transfer probability approaching $π/4 \approx 0.78$. These graphs offer exponentially improved sensitivity over known perfect state transfer examples such as the weighted paths related to hypercubes, making them practical candidates for efficient quantum wires.

Peak state transfer in continuous quantum walks

TL;DR

Peak state transfer reframes quantum state transfer by relaxing perfect fidelity to enable explicit, computable transfer times. The authors provide a spectral characterization based on mutual eigenvalue supports and quadratic-integer eigenvalues, enabling deterministic timing conditions; they construct a family of weighted paths where peak transfer across distance grows arbitrarily large with transfer probability approaching . They also show peak transfer can be robust to timing errors through spectral spread and present a variety of examples for both adjacency and Laplacian dynamics, including small graphs and infinite families. The work suggests peak state transfer is widespread and practical for quantum wires, and it poses open questions on trees, sparsity, and deeper spectral structure.

Abstract

We introduce and study peak state transfer, a notion of high state transfer in qubit networks modeled by continuous-time quantum walks. Unlike perfect or pretty good state transfer, peak state transfer does not require fidelity arbitrarily close to 1, but crucially allows for an explicit determination of the time at which transfer occurs. We provide a spectral characterization of peak state transfer, which allows us to find many examples of peak state transfer, and we also establish tight lower bounds on fidelity and success probability. As a central example, we construct a family of weighted path graphs that admit peak state transfer over arbitrarily long distances with transfer probability approaching . These graphs offer exponentially improved sensitivity over known perfect state transfer examples such as the weighted paths related to hypercubes, making them practical candidates for efficient quantum wires.
Paper Structure (13 sections, 8 theorems, 103 equations, 10 figures, 2 tables)

This paper contains 13 sections, 8 theorems, 103 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Continuous-time quantum walks
  3. Peak state transfer
  4. Fidelity of peak state transfer
  5. Spectral characterization
  6. Examples of peak state transfer
  7. Small sporadic examples
  8. Family of graphs with peak state transfer going to 1
  9. High peak state transfer in weighted path graphs
  10. Peak state transfer in small graphs
  11. Conclusions and open problems
  12. Nomenclature
  13. Proof of Lemma \ref{['lem:weighted_adj_eigenvalues']}

Key Result

Lemma 4.1

Let $H \in \{H_{XY},H_{XYZ}\}$ be such that there is peak state transfer from vertex $1$ to vertex $n$ at time $\tau$ in the underlying graph. Moreover, let the initial state of the system be for some arbitrary qubit state $| \psi \rangle = \alpha| 0 \rangle + \beta| 1 \rangle$. Measuring the first $n-1$ qubits yields the state $| 0^{n-1} \rangle \otimes | \phi \rangle$ with probability where $M

Figures (10)

  • Figure 1: The probability of measuring at vertex 8, having started at vertex 0, from time $0$ to $100$, in the path on $9$ vertices. This graph has pretty good state transfer between vertices $0$ and $8$, thus the curve will come arbitrarily close to the $y=1$ line. However, as we see, it is not approaching $1$ very quickly and we cannot easily find, for example, a time when the state probability exceeds 0.9.
  • Figure 2: Left: an example of a weighted graph with peak state transfer between vertices at distance $4$ from Section \ref{['sec:fam-high-peak-wtd-path']}, with its realization as an unweighted graph under it. Right: the weighted path coming from the hypercube of dimension $4$, with the hypercube under it.
  • Figure 3: From left to right: a weighted graph on vertices $0,1,2,3$ with edge weights, the weighted adjacency matrix, and the corresponding bounding matrix.
  • Figure 4: The Petersen graph and the probability of measuring at $v$ when starting at vertex $u$, for any pair of adjacent vertices $u$ and $v$.
  • Figure 5: $G_{11}$ and the probability of measuring at $v$ when starting at vertex $u$.
  • ...and 5 more figures

Theorems & Definitions (18)

  • Remark 3.1
  • Lemma 4.1
  • proof
  • Lemma 5.1
  • proof
  • Corollary 5.2
  • proof
  • Theorem 5.3
  • proof
  • Corollary 5.4
  • ...and 8 more