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arXiv:2103.05708

Machine Learning the period finding algorithm

John George Francis, Anil Shaji

Published Mar 9, 2021• 0 citations
arXivHTMLPDF

Abstract

We use differentiable programming and gradient descent to find unitary matrices that can be used in the period finding algorithm to extract period information from the state of a quantum computer post application of the oracle. The standard procedure is to use the inverse quantum Fourier transform. Our findings suggest that that this is not the only unitary matrix appropriate for the period finding algorithm, There exist several unitary matrices that can affect out the same transformation and they are significantly different from each other as well. These unitary matrices can be learned by an algorithm. Neural networks can be applied to differentiate such unitary matrices from randomly generated ones indicating that these unitaries do have characteristic features that cannot otherwise be discerned easily.

DOI
10.48550/arxiv.2103.05708

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We use differentiable programming and gradient descent to find unitary matrices that can be used in the period finding algorithm to extract period information from the state of a quantum computer post application of the oracle. The standard procedure is to use the inverse quantum Fourier transform. Our findings suggest that that this is not the only unitary matrix appropriate for the period finding algorithm, There exist several unitary matrices that can affect out the same transformation and they are significantly different from each other as well. These unitary matrices can be learned by an algorithm. Neural networks can be applied to differentiate such unitary matrices from randomly generated ones indicating that these unitaries do have characteristic features that cannot otherwise be discerned easily.
This dissertation presents a study of quantum algorithms for problems that can be posed as matrix function tasks. In Chapter 1 we demonstrate a simple unifying framework for implementing of smooth functions of matrices on a quantum computer. This framework captures a variety of problems that can be solved by evaluating properties of some function of a matrix, and we identify speedups over classical algorithms for some problem classes. The analysis combines ideas from the classical theory of function approximation with the quantum algorithmic primitive of implementing linear combinations of unitary operators. In Chapter 2 we continue this study by looking at the role of sparsity of input matrices in constructing efficient quantum algorithms. We show that classically pre-processing an input matrix by spectral sparsification can be profitable for quantum Hamiltonian simulation algorithms, without compromising the simulation error or complexity. Such preprocessing incurs a one time cost linear in the size of the matrix, but can be exploited to exponentially speed up subsequent subroutines such as inversion. In Chapter 3, we give an application of this theory of matrix functions to the problem of estimating the Renyi entropy of an unknown quantum state. We combine matrix function techniques with mixed state quantum computation in the one-clean qubit model, and are able to bound of the expected runtime of our algorithm in terms of the unknown target quantity. In addition to the theme of analysing the complexity of our algorithms, we also identify instances that are of practical relevance, leading us to some problems of machine learning. In Chapter 4 we investigate kernel based learning methods using random features. We work with the QRAM input model suitable for big data, and show how matrix functions and the quantum Fourier transform can be used to devise a quantum algorithm for sampling random features that are optimised for given input data and choice of kernel. We obtain a potential exponential speedup over the best known classical algorithm even without explicit assumptions of sparsity or low rank. Finally in Chapter 5 we consider the technique of beamsearch decoding used in natural language processing. We work in the query model, and show how quantum search with advice can be used to construct a quantum search decoder that can find the optimal parse (which may for instance be a best translation, or text-to-speech transcript) at least quadratically faster than the best known classical algorithms, and obtain super-quadratic speedups in the expected runtime.
Quantum computing is certainly one of the greatest advances in the computer science field. Thanks to the parallelism and entanglement properties, it has proved to offer several advantages compared to the classical algorithms especially in the great reduction of the processing time. Quantum phase estimation (QPE) is one of the most important algorithms for quantum computing. It is known as the eigenvalue finding module for unitary operators. The Fourier transform is the key to this procedure. It has been researched and used to solve many problems such as the order finding problem, and the factoring problem. It was also applied for quantum sampling algorithms and the calculation of the eigenvalues of unitary matrices. In this paper, we study three important quantum algorithms for machine learning that use the QPE algorithm as a subroutine: the quantum principal components analysis (PCA) for data visualization, the Harrow-Hassidim-Lloyd (HHL) algorithm for solving linear systems, and the quantum singular value thresholding (SVT) for matrix completion in recommender systems. We also discuss the advantages and limits of such algorithms compared to their classical versions. Then we discuss potential ways of amelioration of such algorithms, and end with a proposed approach for further improvement.
Quantum computers are here, and the search for applications and use of these allow us to overcome the limits that today’s hardware information processing gives us is constantly going on. Quantum machine learning is one of the many emerging fields that use quantum computers to process information. In this paper, we present a method and a set of experiments where we see the potential and capacity of the Noisy intermediate-scale quantum hardware for the execution of different models having as the basis in some of them the quantum algorithm corresponding to the Quantum Fourier Transform. With this, we demonstrate the effectiveness of how this algorithm can enhance the performance of quantum computations in quantum machine learning models in near-term applications. We used the systems offered by IBM Quantum and the cross-platform Python library for quantum differentiable programming Pennylane by Xanadu Quantum Technologies Inc.
We investigate learning models for implementing arbitrary unitary operations and perform quantum machine learning tasks. First, we discuss efficient constructions for building unitary transformations in U(d), thus providing upper bounds on the minimum time required to implement them. Second, we study the hardness of learning unitary operations in U(d) via gradient descent on the time parameters of an alternating operator sequence, and numerically find that, despite the nonconvexity of the loss landscape, gradient descent always converges to the target unitary when the sequence contains d2 or more parameters. Third, we introduce the first quantum boosting algorithm for converting a weak quantum learner into a strong accurate quantum learner for a Boolean concept class C; thereby achieving a quadratic quantum improvement over classical AdaBoost in terms of VC(C).
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