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A graph theoretical approach to states and unitary operations

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Abstract

Building upon our previous work, on graphical representation of a quantum state by signless Laplacian matrix, we pose the following question. If a local unitary operation is applied to a quantum state, represented by a signless Laplacian matrix, what would be the corresponding graph and how does one implement local unitary transformations graphically? We answer this question by developing the notion of local unitary equivalent graphs. We illustrate our method by a few, well known, local unitary transformations implemented by single-qubit Pauli and Hadamard gates. We also show how graph switching can be used to implement the action of the \(C_\mathrm{NOT}\) gate, resulting in a graphical description of Bell state generation.

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Correspondence to Supriyo Dutta.

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This work is supported by CSIR (Council of Scientific and Industrial Research) Grant No. 25(0210)/13/EMR-II, New Delhi, India.

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Dutta, S., Adhikari, B. & Banerjee, S. A graph theoretical approach to states and unitary operations. Quantum Inf Process 15, 2193–2212 (2016). https://doi.org/10.1007/s11128-016-1250-y

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  • DOI: https://doi.org/10.1007/s11128-016-1250-y

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