Abstract
This manuscript presents a quantum computing implementation of power iteration for diagonalizable nonnegative matrices that offers a significant speed increase for large matrices, achieving \(O(K\hbox {max}(m_i)+N)\) time complexity for each iteration. The computational approach presented in this manuscript may be directly applied to numerous other algorithms derived from power iteration, ultimately allowing near-term quantum devices to facilitate a broad range of analyses that would otherwise be infeasible.
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Other methods to do so exist, such as that of [10]. When applied to matrix-by-vector multiplication [10], achieves \(O(N\hbox {log}_{2}N)\) time complexity, compared with \(O(K\hbox {max}(m_i)+N)\) for quantum power iteration. However, as will be shown in Sect. 4, K does not need to be increased alongside N, such that K can be held constant regardless of matrix size. Likewise, \(\max (m_i)\le N\). Consequently, the matrix-by-vector technique presented here improves on [10] by a factor of \(\hbox {log}_{2}N\).
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Britt, B.C. Quantum power iteration to efficiently obtain the dominant eigenvector from diagonalizable nonnegative matrices. Quantum Inf Process 23, 36 (2024). https://doi.org/10.1007/s11128-024-04259-3
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DOI: https://doi.org/10.1007/s11128-024-04259-3