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Simple Quantum Coins Enable Pretty Good State Transfer on Every Hypercube

Hanmeng Zhan

TL;DR

This work shows that every hypercube $Q_d$ admits pretty good state transfer between antipodal vertices with a real weighted Grover coin, by perturbing the arc weights on a single arc per vertex. The construction leverages a Cayley-graph based weighting $W_m$ to produce a Hermitian adjacency $H_m$ whose spectrum and strong cospectrality enable ab-PGST, aided by number-theoretic linear independence results to satisfy Kronecker-type conditions. A general sufficient condition for ab-PGST on other graphs is also provided, linking strong cospectrality, eigenvalue symmetry, and a prime spectral radius via a Kronecker framework. The results broaden the practical design space for quantum state transport on graphs, showing that simple coin weight tweaks can realize efficient transport on all hypercubes and suggesting a path to broader graph classes.

Abstract

We consider pretty good state transfer in coined quantum walks between antipodal vertices on the hypercube $Q_d$. When $d$ is a prime, this was proven to occur in the arc-reversal walk with Grover coins. We extend this result by constructing weighted Grover coins that enable pretty good state transfer on every $Q_d$. Our coins are real, and require modification of the weight on only one arc per vertex. We also generalize our approach and establish a sufficient condition for pretty good state transfer to occur on other graphs.

Simple Quantum Coins Enable Pretty Good State Transfer on Every Hypercube

TL;DR

This work shows that every hypercube admits pretty good state transfer between antipodal vertices with a real weighted Grover coin, by perturbing the arc weights on a single arc per vertex. The construction leverages a Cayley-graph based weighting to produce a Hermitian adjacency whose spectrum and strong cospectrality enable ab-PGST, aided by number-theoretic linear independence results to satisfy Kronecker-type conditions. A general sufficient condition for ab-PGST on other graphs is also provided, linking strong cospectrality, eigenvalue symmetry, and a prime spectral radius via a Kronecker framework. The results broaden the practical design space for quantum state transport on graphs, showing that simple coin weight tweaks can realize efficient transport on all hypercubes and suggesting a path to broader graph classes.

Abstract

We consider pretty good state transfer in coined quantum walks between antipodal vertices on the hypercube . When is a prime, this was proven to occur in the arc-reversal walk with Grover coins. We extend this result by constructing weighted Grover coins that enable pretty good state transfer on every . Our coins are real, and require modification of the weight on only one arc per vertex. We also generalize our approach and establish a sufficient condition for pretty good state transfer to occur on other graphs.
Paper Structure (6 sections, 15 theorems, 60 equations)

This paper contains 6 sections, 15 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. The quantum walk with respect to $(X,W)$
  3. Pretty good state transfer
  4. Hypercubes
  5. A sufficient condition for $ab$-PGST
  6. Future work

Key Result

Theorem 2.1

Chan2023 Let $U$ be the transition matrix with respect to $(X,W)$, and let $H$ be the Hermitian adjacency matrix of $X$ associated with $W$. 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_{\thet

Theorems & Definitions (15)

  • Theorem 2.1
  • Lemma 3.1
  • Corollary 3.2
  • Lemma 3.3
  • Corollary 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 5 more