Controlling quantum state transfer in rooted products
Abstract
Godsil and McKay (1978) showed that the rooted product is a powerful tool for constructing non-isomorphic cospectral pairs of graphs. Despite lacking a convenient tensor product structure, we show that the rooted product is useful for constructing graphs with good quantum state transfer properties. In particular, we prove a simple transference principle: if a graph X has quantum state transfer and Y is a controllable graph, their rooted product XY has quantum state transfer (inherited from X). This complements a folklore property of Cartesian product which preserves perfect state transfer. However, the rooted product is a significantly sparser graph and, more importantly, can be easily used to construct efficient high-fidelity state transfer even if X has no quantum state transfer. Our proof exploits the fact that a rooted product creates a large number of strongly cospectral pairs of vertices and that its condition number can be controlled by its pendant subgraph.
Related
- Vaught's Conjecture for Sums of Products of Rooted Trees
- Pretty good quantum state transfer via transcendental edge weights
- Quantum State Routing and Perfect State Transfer on Signed Graphs under Environmental Noise
- Quantum-State Oblivious Transfer over Quantum Erasure Channels
- Perfect and peak state transfer on IBM quantum hardware
BibTeX
@misc{ballif2026controlling,
title = {{Controlling quantum state transfer in rooted products}},
author = {Addison Ballif and Christino Tamon and Gabriel Tucker},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14537},
eprint = {2609.14537},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14537},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Controlling quantum state transfer in rooted products
AU - Addison Ballif
AU - Christino Tamon
AU - Gabriel Tucker
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14537
UR - https://arxiv.org/abs/2609.14537
AB - Godsil and McKay (1978) showed that the rooted product is a powerful tool for constructing non-isomorphic cospectral pairs of graphs. Despite lacking a convenient tensor product structure, we show that the rooted product is useful for constructing graphs with good quantum state transfer properties. In particular, we prove a simple transference principle: if a graph X has quantum state transfer and Y is a controllable graph, their rooted product X^Y has quantum state transfer (inherited from X). This complements a folklore property of Cartesian product which preserves perfect state transfer. However, the rooted product is a significantly sparser graph and, more importantly, can be easily used to construct efficient high-fidelity state transfer even if X has no quantum state transfer. Our proof exploits the fact that a rooted product creates a large number of strongly cospectral pairs of vertices and that its condition number can be controlled by its pendant subgraph.
ER -