Quantum principal component analysis

Abstract

The usual way to reveal properties of an unknown quantum state, given many copies of a system in that state, is to perform measurements of different observables and to analyze the measurement results statistically. Here we show that the unknown quantum state can play an active role in its own analysis. In particular, given multiple copies of a quantum system with density matrix \rho, then it is possible to perform the unitary transformation e^{-i\rho t}. As a result, one can create quantum coherence among different copies of the system to perform quantum principal component analysis, revealing the eigenvectors corresponding to the large eigenvalues of the unknown state in time exponentially faster than any existing algorithm.Comment: 9 pages, Plain Te

Quantum Machine Learning over Infinite Dimensions

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“…These discrete-variable schemes have observed a performance that scales logarithmically in the vector dimension, such as supervised and unsupervised learning [9], support vector machine [10], cluster assignment [11] and others [12][13][14][15][16][17][18]. Initial proof-of-principle experimental demonstrations have also been performed [19][20][21][22].…”

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“…As suggested in Ref. [11], such an operation can be implemented by repeatedly applying the exponential swap operation and tracing out the auxiliary mode, i.e.,…”

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“…Quantum Machine Learning is a recent area of research initiated by the demonstration of a quantum Support Vector Machine (SVM) by Rebentrost, Mohseni & Lloyd [1] and the k-means algorithm by Aïmeur, Brassard & Gambs [2] (cf also [3][4][5][6][7][8]). The development of the quantum SVM can be regarded as particularly significant in that the classical SVM constitutes perhaps the exemplar instance of a supervised binary classifier, i.e.…”