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README.md

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This is the repository for [QuantumAlgorithms.org](https://quantumalgorithms.org).
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This webiste is meant to be a set of lecture notes for students in quantum algorithms and quantum machine learning.
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It will be updated regularly with new research material. The scope is to bridge the gap between introductory material in quantum computing and research-grade papers, standardize notation, and be an overfiew on the state of useful algorithms for quantum machine learning and information processing.
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It will be updated regularly with new research material. The scope is to bridge the gap between introductory material in quantum computing and research-grade papers, standardize notation, and be an overview on the state of useful algorithms for quantum machine learning and information processing.
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index.Rmd

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# Preface
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This is an [open source project accessible on GitHub](https://github.com/Scinawa/quantumalgorithms.org/) - only possible thanks to its many [contributors.](contributions-and-acknowledgements.html) The website is licensed under CC BY-NC-SA 4.0. We are searching for talented people and researchers to contribute: you can find a list of issues and enhancements that would improve this book further on the [issues list](https://github.com/Scinawa/quantumalgorithms.org/issues) for the GitHub repo, making recent material that keeps popping up in quantum information accessible to a larger audience.
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This [open source project accessible on GitHub](https://github.com/Scinawa/quantumalgorithms.org/) is only possible thanks to its many [contributors.](contributions-and-acknowledgements.html) The website is licensed under CC BY-NC-SA 4.0. We are searching for talented people and researchers to [contribute](https://github.com/Scinawa/quantumalgorithms.org/issues).
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**MEMO: we have some funding for motivated contributors**
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The aim of this book is twofold:
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- First, we want to bridge the gap between introductory material in quantum computation and research material.
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- Secondarily, you should be able to use this book as a resource for state-of-the-art algorithms. Readers and scholars should find statements of theorems (along with their citations) and runtimes of the best quantum subroutines in literature, ready to be used in new quantum algorithms or applications.
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- Second, you should be able to use this book as a resource for state-of-the-art algorithms. Readers and scholars should find statements of theorems (along with their citations) and runtimes of the best quantum subroutines in literature, ready to be used in new quantum algorithms or applications.
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These lecture notes were used to teach at:
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- Politecnico di Milano (2019) - Quantum machine learning in practice.
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- Politecnico di Milano (2021) - Applied quantum computing.
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Are you using these lecture notes as support for your course? [Write us an email!](mailto://[email protected])
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Are you using these lecture notes to support your course? [Write us an email!](mailto://[email protected])
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## Abstract
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In these lecture notes, we explore how we can leverage quantum computers and quantum algorithms for information processing. It has long been known that quantum computation can offer computational advantages with respect to classical computation, and in this place we explore more the consequences of this intuition in current domains of computer sciences.
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In these lecture notes, we explore how we can leverage quantum computers and quantum algorithms for information processing. It has long been known that quantum computation can offer computational advantages over classical computation, and in this book we explore the consequences of this fact in current research areas of computer science.
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Why are we studying quantum algorithms? Studying how to use quantum mechanical
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systems is already fascinating in itself, but we argue that having faster
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algorithms is not the only reason for studying quantum computing. Studying
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quantum computation might also reveal profound insights into new ways to process
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Are there other reasons, besides getting a practical computational advantage for studying quantum algorithms? We argue that having faster algorithms is not the only reason for studying quantum computing.
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One --- perhaps shallow --- reason is to satisfy our curiosity by studying how to use quantum mechanical systems for doing computation, and challenging yourself in finding a faster-then-classical algorithms. Studying quantum computation might also reveal profound insights into new ways to process
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information. For instance, it can give us ideas on processing data
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in a secure way (though, quantum cryptography is not discussed in these notes).
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Understanding the computational capabilities of quantum machines
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is certainly an interesting thing to do. This might lead to understanding the
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computational limits of nature: what can be computed in this world? Last
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but not least, because of the interplay between classical and quantum
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A better understanding of quantum computing might lead to understanding the
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computational limits of nature: what can be computed in this world? What can be computed with classical computers? As an example, because of the interplay between research in classical and quantum
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computation, many new *classical* algorithms have been invented (i.e. the
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dequantizations of quantum machine learning algorithms, the classical algorithms
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for Gibbs sampling, simulations of QAOA, etc..). This, in turn,
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dequantizations of quantum machine learning algorithms, new classical algorithms
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for Gibbs sampling, classical simulations of quantum circuits, etc..). This, in turn,
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improved our understanding of physics, and ultimately of the world itself.
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<!-- As some scientists believe\footnote{A friend told me this was originally an idea -->
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<!-- of Miklos Santha}, quantum computing is the occasion for Theoretical Computer -->
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<!-- is basically a branch of Mathematics. With quantum computing, TCS can welcome -->
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<!-- also contribution from physics, perhaps going towards the direction of a -->
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Another reason for studying quantum algorithms is that quantum computers are posing
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a significant challenge to the *strong* Church-Turing thesis, which says that any "reasonable"
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model of computation can be *efficiently* simulated on a probabilistic Turing machine (i.e. a Turing machine which has access to randomness).
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However, there are some physical processes that we do not know how to simulate efficiently on classical computers, but for which we have efficient quantum algorithms! This is strong evidence that the strong Church-Turing thesis might be false!
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One last reason for studying quantum algorithms --- which a computer scientist can surely appreciate --- is that quantum computers are posing a significant challenge to the [*extended* Church-Turing thesis](https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis), which states that any "reasonable"
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model of computation can be *efficiently* simulated on a probabilistic Turing machine.
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However, there are many physical processes that we do not know how to simulate efficiently on classical computers, but for which we have efficient quantum algorithms! This is strong evidence that the strong Church-Turing thesis might be false!
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<!-- Algorithms are often overlooked when thinking about speed of calculations. But they deserve more credit for the performance gains than the well known Moore's Law. As we can read [here](https://www.nitrd.gov/pubs/PCAST-NITRD-report-2010.pdf), some researcher tried to quantify the performance gains for some specific problem (linear programming and mixed integer programming), and they found that in 15 years we had an improvement of a factor of 43millions: a factor of roughly 1000 was due to increased processor speed, and a factor of 43.000 was due to algorithmic improvements for the problem. -->
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You might often hear that there are only two real quantum algorithms: phase estimation and the Grover's algorithm. This is somewhat true, but it is true in the same way that we have only 12 notes in the western temperate scale, and yet Pink Floyd were able to write [The Dark Side of the Moon](https://www.youtube.com/playlist?list=OLAK5uy_l1x-JAx0w53suECoCI0YJtW6VB8DBQWRQ) (and the other musicians came up with "the rest" of the music).
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You might often hear that there are only two real quantum algorithms: phase estimation and the Grover's algorithm. This is true in the same way that we have only 12 notes in the western temperate scale, yet Pink Floyd was able to write [The Dark Side of the Moon](https://www.youtube.com/playlist?list=OLAK5uy_l1x-JAx0w53suECoCI0YJtW6VB8DBQWRQ) (and other musicians came up with "the rest" of the music).
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The common thread of these algorithms is that they are faster than their best classical counterpart. Oftentimes, (especially for ML) the runtime will depend only [poly-logarithmically](https://en.wikipedia.org/wiki/Polylogarithmic_function) on the number of elements of the dataset, and it is usually only linear in the number of features (classical algorithms are often either linear in the number of elements and quadratic in the number of features, or depend on the number of nonzero components of the matrix and depend polynomially on other parameters of the matrix). The runtime of a quantum machine learning algorithm also often depends on characteristics of the matrix that represents the data under analysis, such as its rank, the Frobenius norm (or other matrix norms), the sparsity, the condition number, and the error we tolerate in the analysis. For this, along with an error-corrected quantum computer, we assume to have quantum access to a dataset. In other words, we assume that the data is stored in a quantum memory: the corresponding quantum version of the classical random-access memory.
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We will see that, for a new QML algorithm, one often needs to make sure that the real performances of the quantum algorithms offer concrete advantages with respect to the effective runtime and the accuracy that is offered by the best classical algorithms. As we don't have access to big-enough quantum computers *yet*, we can only assess the performance of these quantum algorithms via a classical simulation.
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These lecture notes should prepare the future quantum data analyst to understand the potential and limitations of quantum computers, so as to unlock new capabilities in information processing and machine learning. The hope is that this kind of technology can foster further technological advancements that benefit society and humankind, as soon as the hardware that supports this kind of computation becomes ready.
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Last but not least, we will also cover important algorithms that are not necessarily related to machine learning, but are the quantum counterpart of important classical algorithms. Don't get swayed by the "lack" of exponential speedups. Remember: the square root of 365 days is a little less than 3 weeks. Besides this, big polynomial speedups, small polynomial speedups in important problems, or polynomial speedups proposing new algorithmic techniques are all much welcome in quantum computer science. All in all, quantum algorithms (to me) should be seen as a way for making **impossible things possible**.
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Last but not least, we will also cover important algorithms that are not necessarily related to machine learning, but are the quantum counterpart of important classical algorithms. Don't get swayed by the "lack" of exponential speedups. Remember: the square root of 365 days is a little less than 3 weeks. Besides this, big polynomial speedups, small polynomial speedups in important problems, or polynomial speedups proposing new algorithmic techniques are all much welcome in quantum computer science. All in all, quantum algorithms can be seen as a way for making **impossible things possible**.
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While reading these lecture notes you should always remember the good [Simon Martiel](https://scholar.google.fr/citations?user=upaq0vIAAAAJ&hl=en):
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While reading these lecture notes you should always remember a quote from the good Simon Martiel:
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<p style="text-align:center">
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"(quantum) Theoretical computer science is the fun part of mathematics."
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</p>

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