Quantum principal component analysis

By Seth Lloyd, Masoud Mohseni, Patrick Rebentrost. Nature Physics, 2013

799 citations recorded by the source corpus.

Characterizing an unknown quantum state typically relies on analysing the outcome of a large set of measurements. Certain quantum-processing tasks are now shown to be realizable using only approximate knowledge of the state, which can be gathered with exponentially fewer resources. 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 analyse the results statistically1,2. For non-sparse but low-rank quantum states, revealing eigenvectors and corresponding eigenvalues in classical form scales super-linearly with the system dimension3,4,5,6. Here we show that multiple copies of a quantum system with density matrix ρ can be used to construct the unitary transformation e−iρt. As a result, one can perform quantum principal component analysis of an unknown low-rank density matrix, revealing in quantum form the eigenvectors corresponding to the large eigenvalues in time exponentially faster than any existing algorithm. We discuss applications to data analysis, process tomography and state discrimination.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/faddf7f2fcac43032645bd7f562ef21ddddb71fb

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Citations in the corpus

Quantum Generative Adversarial Learning. (2018, positive): “matrix exponentiation [15] allow one to implement ρ → ρþ αR by the modified swap operations of [15].”
Quantum computing for finance: Overview and prospects (2018, methodology): “Alternatively, the training process could be sped-up exponentially (compared to classical training methods) by using quantum PCA methods to implement iterative gradient descent methods for network training [22].”
Variational quantum algorithms (2020, methodology): “This idea was exploited in [124] to propose a quantum algorithm for PCA.”
Dual Principal Component Pursuit (2015, background): “d most fundamental techniques in data analysis, with ubiquitous applications in engineering (Moore, 1981), economics and sociology (Vyas and Kumaranayake, 2006), chemistry (Ku et al., 1995), physics (Loyd et al., 2014) and genetics (Price et al., 2006) to name a few; see Jolliffe (2002) for more app”
Adiabatic Quantum Computing (2016, background): “Quantum machine learning is currently an exciting and rapidly moving frontier in the context of the circuit model (Lloyd et al., 2014; Rebentrost et al., 2014; Wiebe et al., 2014), though it must be evaluated carefully (Aaronson, 2015).”
Machine learning \& artificial intelligence in the quantum domain (2017, methodology): “also used for the problem of k-nearest neighbour algorithms for supervised and unsupervised learning (Wiebe et al.,2015). Quantum Principal Component Analysis The ideas of DME were in the same paper (Lloyd et al., 2014) immediately applied to a quantum version of principal component analysis (PCA). ”
Sample-Optimal Tomography of Quantum States (2015, methodology): “There is a method to extract the eigenvectors of a small-rank density matrix on a quantum computer efficiently [56], but it remains challenging to convert the eigenvector into a classical description.”
Entanglement Induced Barren Plateaus (2020, background): “The QML ﬁeld has enjoyed increased attention for quantum algorithms for principal component analysis [1], support vector machines [2], kernel meth-ods [3, 4], and quantum neural networks (QNN) [3, 5–8] but experiences setbacks in the form of dequantization techniques [9–11].”
Continuous-variable quantum neural networks (2018, background): “Sufficiently powerful quantum computers can in principle provide computational speedups for key machine learning algorithms and subroutines such as data fitting [4], principal component analysis [5], Bayesian inference [6, 7], Monte Carlo methods [8], support vector machines [9, 10], Boltzmann machi”
Quantum convolutional neural network for classical data classification (2021, background): “[9] Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost.”
Quantum Generative Adversarial Networks for learning and loading random distributions (2019, background): “[2] S.”
Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics (2018, methodology): “This technique can be used in combination with the optimal block-Hamiltonian simulation result Corollary 32, in order to get much better simulation performance, compared to density matrix exponentiation techniques [38, 40] which does not use purification.”