A Grand Unification of Quantum Algorithms

Martyn, John M.; Rossi, Zane M.; Tan, Andrew K.; Chuang, Isaac L.

doi:10.48550/arxiv.2105.02859

Cited by 15 publications

(32 citation statements)

References 30 publications

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“…Recently, manipulation of matrices and their spectra by block-encoding into submatrices of unitary matrices has become a topic of great interest due to its versatility in quantum algorithm design [32,33], which may provide an alternative route to qDM algorithm.…”

Section: Discussionmentioning

confidence: 99%

Quantum diffusion map for nonlinear dimensionality reduction

Sornsaeng,

Dangniam,

Palittapongarnpim

et al. 2021

Preprint

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Inspired by random walk on graphs, diffusion map (DM) is a class of unsupervised machine learning that offers automatic identification of low-dimensional data structure hidden in a highdimensional dataset. In recent years, among its many applications, DM has been successfully applied to discover relevant order parameters in many-body systems, enabling automatic classification of quantum phases of matter. However, classical DM algorithm is computationally prohibitive for a large dataset, and any reduction of the time complexity would be desirable. With a quantum computational speedup in mind, we propose a quantum algorithm for DM, termed quantum diffusion map (qDM). Our qDM takes as an input N classical data vectors, performs an eigen-decomposition of the Markov transition matrix in time O(log 3 N ), and classically constructs the diffusion map via the readout (tomography) of the eigenvectors, giving a total runtime of O(N 2 polylog N ). Lastly, quantum subroutines in qDM for constructing a Markov transition operator, and for analyzing its spectral properties can also be useful for other random walk-based algorithms.

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Section: Discussionmentioning

confidence: 99%

Quantum diffusion map for nonlinear dimensionality reduction

Sornsaeng,

Dangniam,

Palittapongarnpim

et al. 2021

Preprint

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“…2 Solving Problem 1 with these parameter values also solves the "eigenvalue threshold problem" [29,30], which, unlike eigenvalue thresholding, is a promise problem and consequently cannot be used as a subroutine for phase estimation in the same manner. with t j := −jτ λ and Ĥ := H/λ.…”

Section: Eigenvalue Thresholdingmentioning

confidence: 99%

A randomized quantum algorithm for statistical phase estimation

Wan,

Berta,

Campbell

2021

Preprint

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Phase estimation is a quantum algorithm for measuring the eigenvalues of a Hamiltonian. We propose and rigorously analyse a randomized phase estimation algorithm with two distinctive features. First, our algorithm has complexity independent of the number of terms L in the Hamiltonian. Second, unlike previous L-independent approaches, such as those based on qDRIFT, all sources of error in our algorithm can be suppressed by collecting more data samples, without increasing the circuit depth.

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“…Quantum singular value transformation (QSVT) [4] is its generalization to perform nonlinear functions on singular values of a matrix encoded in a unitary. Existing quantum algorithms can be reconstructed in terms of QSVT * u645239h@ecs.osaka-u.ac.jp † mitarai@qc.ee.es.osaka-u.ac.jp ‡ fujii@qc.ee.es.osaka-u.ac.jp [4,5], which unifies the disparate problems like Hamiltonian simulation [3,[6][7][8], linear equation solving [2,9] and amplitude amplification techniques [10][11][12], implying the importance of developing tools to introduce nonlinearity on quantum computers.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear transformation of complex amplitudes via quantum singular value transformation

Guo¹,

Mitarai²,

Fujii³

2021

Preprint

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Due to the linearity of quantum operations, it is not straightforward to implement nonlinear transformations on a quantum computer, making some practical tasks like a neural network hard to be achieved. In this work, we define a task called nonlinear transformation of complex amplitudes and provide an algorithm to achieve this task. Specifically, we construct a block-encoding of complex amplitudes from a state preparation oracle. This allows us to transform the complex amplitudes by using quantum singular value transformation. We evaluate the required overhead in terms of input dimension and precision, which reveals that the algorithm depends on the roughly square root of input dimension and achieves an exponential speedup on precision compared with previous work. We also discuss its possible applications to quantum machine learning, where complex amplitudes encoding classical or quantum data are processed by the proposed method. This paper provides a promising way to introduce highly complex nonlinearity of the quantum states, which is essentially missing in quantum mechanics.

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A Grand Unification of Quantum Algorithms

Cited by 15 publications

References 30 publications

Quantum diffusion map for nonlinear dimensionality reduction

Quantum diffusion map for nonlinear dimensionality reduction

A randomized quantum algorithm for statistical phase estimation

Nonlinear transformation of complex amplitudes via quantum singular value transformation

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