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Subspace State Transfer in Coined Quantum Walks

Yichi Xu, Hanmeng Zhan

TL;DR

The paper investigates subspace state transfer in discrete-time coined quantum walks using reflection coins. It develops a spectral framework based on Hermitian weighted digraphs, providing complete characterizations for perfect and pretty good subspace transfer and an efficient polynomial-time test for integer-step transfer when subspaces and coins are rational. It introduces notions of (gamma-)strong-cospectrality, rational subspaces, and pointwise transfer, tying transfer properties to eigenvalue supports and pole structures of associated rational functions. The authors also construct several infinite families of graphs that admit pointwise perfect or pretty good d-dimensional subspace transfer, broadening multi-state quantum transport beyond single-state transfer and suggesting applications in quantum routing on networks.

Abstract

We study a transport phenomenon in certain coined quantum walks where a subspace of states localized at a vertex gets transferred to another vertex. We first develop characterizations for perfect and pretty good subspace state transfer using the spectral properties of a Hermitian weighted digraph obtained from the underlying graph. We then provide a polynomial-time algorithm that tests whether pointwise perfect subspace state transfer occurs at an integer step, given that the subspace and coins are rational. Finally, we construct several infinite families of examples that admit pointwise perfect $d$-dimensional subspace state transfer where $d\ge 2$.

Subspace State Transfer in Coined Quantum Walks

TL;DR

The paper investigates subspace state transfer in discrete-time coined quantum walks using reflection coins. It develops a spectral framework based on Hermitian weighted digraphs, providing complete characterizations for perfect and pretty good subspace transfer and an efficient polynomial-time test for integer-step transfer when subspaces and coins are rational. It introduces notions of (gamma-)strong-cospectrality, rational subspaces, and pointwise transfer, tying transfer properties to eigenvalue supports and pole structures of associated rational functions. The authors also construct several infinite families of graphs that admit pointwise perfect or pretty good d-dimensional subspace transfer, broadening multi-state quantum transport beyond single-state transfer and suggesting applications in quantum routing on networks.

Abstract

We study a transport phenomenon in certain coined quantum walks where a subspace of states localized at a vertex gets transferred to another vertex. We first develop characterizations for perfect and pretty good subspace state transfer using the spectral properties of a Hermitian weighted digraph obtained from the underlying graph. We then provide a polynomial-time algorithm that tests whether pointwise perfect subspace state transfer occurs at an integer step, given that the subspace and coins are rational. Finally, we construct several infinite families of examples that admit pointwise perfect -dimensional subspace state transfer where .

Paper Structure

This paper contains 10 sections, 35 theorems, 155 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Reflection coins and weighted digraphs
  3. Subspace state transfer
  4. A characterization
  5. Pointwise subspace state transfer
  6. $\gamma$-strong-cospectrality
  7. Rational subspaces
  8. Transfer between marked vertices
  9. Infinite families
  10. Discussion

Key Result

Theorem 2.1

Godsil2023 Let $U=R(2NN^*-I)$ be the transition matrix where $N$ has orthonormal columns. Let $H=N^*RN$. The eigenvalues of $U$ are $1$, $-1$ and $e^{\pm i\theta}$, where $\theta=\arccos\lambda$ for some eigenvalue $\lambda\in(-1,1)$ of $H$. Moreover, if $F_{\theta}$ denotes the $e^{i\theta}$-eigenp

Figures (11)

  • Figure 1: At time $0$
  • Figure 2: At time $4$
  • Figure 3: At time $0$
  • Figure 4: At time $4$
  • Figure 5: Reflection coins and Hermitian weighted digraphs; $\omega=e^{2\pi i/3}$
  • ...and 6 more figures

Theorems & Definitions (35)

  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Corollary 4.4
  • Corollary 4.5
  • Theorem 4.6
  • ...and 25 more