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Improved quantum circuit modelling based on Heisenberg representation

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Abstract

Heisenberg model allows a more compact representation of certain quantum states and enables efficient modelling of stabilizer gates operation and single-qubit measurement in computational basis on classical computers. Since generic quantum circuit modelling appears intractable on classical computers, the Heisenberg representation that makes the modelling process at least practical for certain circuits is crucial. This paper proposes efficient algorithms to facilitate accurate global phase maintenance for both stabilizer and non-stabilizer gates application that play a vital role in the stabilizer frames data structure, which is based on the Heisenberg representation. The proposed algorithms are critical as maintaining global phase involves compute-intensive operations that are necessary for the modelling of each quantum gate. In addition, the proposed work overcomes the limitations of prior work where the phase factors due to non-stabilizer gates application was not taken into consideration. The verification of the proposed algorithms is made against the golden reference model that is constructed based on the conventional state vector approach.

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Notes

  1. The terms Heisenberg model and stabilizer formalism are used interchangeably in this paper.

  2. Global phase is important in quantum mechanics when the differences in phase factors between two interacting quantum states are measurable [21].

  3. A three-dimensional sphere of radius 1 that provides a way of visualizing a single-qubit state.

  4. Recall that the eigenvalue \(\lambda \) and eigenvector v of an n-by-n square matrix A are defined as \(Av = \lambda v\).

  5. The complexities stated in the caption of Algorithms 2–4 are for the computation of a complete set of N basis amplitudes. In the case of global phase maintenance, it only requires to obtain the amplitude of one or more basis indexes depending on the quantum gate type, regardless of the qubit size of the quantum circuit.

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Authors and Affiliations

  1. School of Electrical and Information Engineering, The University of Sydney, Darlington, NSW, 2006, Australia

    Y. H. Lee

  2. VeCAD Research Laboratory, Faculty of Electrical Engineering, Universiti Teknologi Malaysia, 81310 UTM, Skudai, Johor Bahru, Malaysia

    M. Khalil-Hani & M. N. Marsono

Authors
  1. Y. H. Lee
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  2. M. Khalil-Hani
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  3. M. N. Marsono
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Correspondence to Y. H. Lee.

Additional information

This work is supported by the Ministry of Higher Education (MOHE) and Universiti Teknologi Malaysia (UTM) under Fundamental Research Grant Scheme (FRGS) Vote No. 4F422.

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Lee, Y.H., Khalil-Hani, M. & Marsono, M.N. Improved quantum circuit modelling based on Heisenberg representation. Quantum Inf Process 17, 36 (2018). https://doi.org/10.1007/s11128-017-1806-5

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