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.
This paper characterize the conjugation-closed connection sets of connected Cayley graphs over dihedral groups that admit perfect state transfer by applying Ramanujan sums, Mobius inversion, and arguments based on the rational numbers that yields a complete and practical characterization of these connection sets.
Graph theory is a powerful tool in quantum computing, with applications ranging from quantum circuit synthesis and optimization to entanglement mapping. Recent work has shown how one can use edge-colored graphs to model photonic experiments that generate GHZ and W states. However, the latter work also proved that verif...
Let $U_X(t)$ be the transition matrix of a quantum walk on a graph $X$ relative to its adjacency matrix $A$ or the Laplacian matrix $L$. This paper investigates the behavior of quantum walks under Cartesian products, joins, and graph complements. We have two main goals. First, we characterize the conditions such that p...
Joy Cooper, Homer De Vera, Hermie Monterde et al.· 0 citations
Optimal approximate state conversion is a fundamental task in quantum resource theories, yet existing solutions are largely tailored to specific distinguishability measures and resource theories. We show that every optimal approximate-conversion problem in the three paradigmatic majorization-based quantum resource theo...
F. Meroi, C. Massri, M. Losada et al.· 0 citations
Quantum state preparation is a basic primitive of quantum computation and control, but its resource cost is usually analyzed for one target state or under an independent-copy construction. Here we study the many-copy problem under weighted Pauli control, where a Pauli-product rotation $e^{i\phi P}$ is charged by its ro...
Lin Zhu, Ranyiliu Chen, Xin Wang et al.· 0 citations
Decoded quantum interferometry (DQI) and its Hamiltonian extension (HDQI) prepare states whose amplitudes are low-degree polynomials of an objective, using Fourier transforms and coherent decoding. We recast this approach as quantum state transduction through algebra homomorphisms. We transfer an efficiently preparable...
Zhong-Xia Shang, Daniel Stilck França· 0 citations
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