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This research demonstrates how the rooted product graph construction can be leveraged to engineer efficient, high-fidelity quantum state transfer in sparse networks, even when individual components lack perfect state transfer properties. By controlling the condition number through pendant subgraphs, the study offers a novel mathematical framework for optimizing quantum communication protocols without relying on dense tensor structures.
Controlling quantum state transfer in rooted products
arXiv:2609.14537v1 Announce Type: new 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 $X^Y$ 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.
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 $X^Y$ 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.
Quantum Physics (quant-ph); Combinatorics (math.CO)
arXiv:2609.14537 [quant-ph]
(or arXiv:2609.14537v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2609.14537 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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