Quantum computational complexity cheriton school of computer. The complexity of quantum computation remains poorly understood. This section provides the schedule of lecture topics, notes taken by students from the fall 2008 version of the course, and a set of slides on. Lecture on quantum complexity and uncomplexity by leonard susskind given at the kavli institute for theoretical physics date. Quantum algorithms and quantum complexity theory introduction to quantum algorithms. Mixed states and density matrices pdf notes, video lecture 22. Pdf an introduction to quantum complexity theory semantic. The complexity of a quantum state may be closely related to the usefulness of the state for quantum computation. Quantum computation and quantum information michael a. Quantum complexity theory and adiabatic computation. A given complexity class consists of problems which all possess some similar charac. That quantum mechanics should have implications to computational complexity theory, how ever, is much less clear. The tools of computational complexity theory should come to bear on these important questions.
This section provides the schedule of lecture topics, notes taken by students from the fall 2008 version of the course, and a set of slides on quantum computing with noninteracting particles. Abstract we give a basic overview of computational complexity, query complexity, and communication complexity, with quantum information incorporated into each of these scenarios. Lecture notes quantum complexity theory electrical. We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. Quantum computation and complexity anonymous authors stanford university abstract this paper is a brief introduction to quantum computation and a survey of important results regarding quantum computation and quantum complexity theory. We consider three specific complexity scenarios, which illustrate different advantages of working with quantum information. In this paper we study quantum computation from a complexity theoretic viewpoint. A twoplayer game can reveal whether the universe has an infinite amount of complexity. Quantum computation and complexity theory course given at the institut f. Quantum complexity theory electrical engineering and. The theory of quantum information relies heavily on linear algebra in nitedimensional spaces. Quantum information theory pdf notes, video lecture 24. Collapsing recursive oracles for relativized polynomial hierarchies.
Complexity theory and quantum theory article pdf available in neuroquantology 153. While physicists attempt to nd ways to create quantum computers, we still do not have much evidence one way or the other as to how useful these machines will be. The role of complexity theory in quantum opticsa tutorial. However, interest in the eld increased tremendously after peter shors very surprising discovery of e cient quantum algorithms for the problems of integer factorization and discrete logarithms in 1994 129. In retrospect, the constantdepth quantum circuits considered by terhal and divincenzo td04 also have the property that fast exact classical simulation would collapse ph. We give a basic overview of computational complexity, query complexity, and communication complexity, with quantum information incorporated into each of. Quantum computation and complexity stanford cs theory. Also included is a glossary of complexity terms used throughout the tutorial. To learn the basics of quantum physics, this books is the recommendation of most experts and professors. This course is an introduction to quantum computational complexity theory, the study of the fundamental capabilities and limitations of quantum computers. We also discuss how the concept of span programs turns the quantum adversary method into an upper bound, giving optimal quantum algorithms for evaluating boolean formulas. It claims that there is a gap to be filled in the relationship between complexity theory ct and quantum theory qt.
Usually, ket notation will only be for unit vectors, but sometimes we might use it more generally. We discuss this link using the tree size of a multiqubit state, a complexity. In a different community, many complexity theory researchers have studied the power of quantum computation with a blackboxquantum oracle 38, 11,16,28, 31. Complexity theory classifies algorithmic problems according to the amount of ressources like time, space, hardware. Quantum computation cmu 18859bb, fall 2015 lecture 23. Quantum complexity theory siam journal on computing. Complexitytheoretic foundations of quantum supremacy. Chaos theory dissipative structure theory quantum physics complex adaptive system theory ie systems that are complex and also adapt. Complexity limitations on quantum computation sciencedirect. Our research is naturally in scope, spanning various areas of both quantum and complexity science.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. While it will turn out to be equivalent in power to the circuit model, it is interesting for two reasons. An introduction to quantum complexity theory richard cleve university of calgary. The prerequisites are that you should know something about quantum computing or about computational complexity theory. Bruton centennial professor of computer science at the university of texas at austin, and d. It studies the hardness of computational problems in relation to these complexity classes, as well as the relationship between quantum complexity classes and classical i. Rabani, proceedings of symposium on the theory of computing, 1994. Introduction to quantum complexity theory november 31, 2015 lecturer. Here is a list of all the features which make this book the best one to study the subject. Quantum complexity theory lecture 5 1 quantum circuits. Cs 2942 quantum complexity theory 107 spring 2007 lecture 5 1 quantum circuits a quantum circuit implements a unitary operator in a hilbert space c2n, given as primitive a usually. In other words, jvistands in for the column vector a b. Special issue on quantum computation of the siam journal of computing, oct.
Quantum computing in complexity theory and theory of computation austin mohr southern illinois university at carbondale carbondale, il 62901 email. Within the last four years, quantum supremacy via sampling has made the leap from complexity theory to a serious experimental prospect. The subsections that follow present an overview of the aspects of this subject that are most relevant within the theory of quantum information. Topics include complexity classes, lower bounds, communication complexity, proofs, advice, and interactive proof systems in the quantum world.
The names of scribes for the lectures are noted in the table below. An invitation to quantum computation a rosetta stone for quantum mechanics with an introduction to quantum computation samuel j. Bell developed the games to test one of the strangest propositions about the physical world made by the theory of quantum mechanics. We studied the general theoretical foundations for demonstrating quantum supremacy with nearterm. Bernstein, proceedings of symposium on the theory of computing, 1993. Complexitytheoretic foundations of quantum supremacy experiments scott aaronson, lijie chen computational complexity conference ccc 2017 invited to the toc special issue for ccc 2017 conference version summary. Provides a strong, evidentiary foundation for leadership as well as effective translations of theory into contemporary practice. Our rst result is the existence of an ecient universal quantum turing machine in deutschs model of a quantum turing machine qtm proc. An introduction to quantum complexity theory richard cleve university of calgary abstract we give a basic overview of computational complexity, query complexity, and communication complexity, with quantum information incorporated into each of these scenarios.
The objective is to bring students to the research frontier. Given a quantum state, how many operations do we need to prepare it. In section 1, we give an overview of quantum mechanics. Our first result is the existence of an efficient universal quantum turing machine in deutschs model of a quantum turing machine qtm proc. Author links open overlay panel lance fortnow a 1 john rogers b 2. In doing so, we aim to build a truly interdisciplinary group where quantum information and complexity theorists can interact, and thus pioneer. To be a bit more speci c, well be interested in the following sorts of questions. Quantum leadership a unique combination of text, audio, and case study resources focusing on complexity leadership, principlebased leadership, the cybernetic dynamic of innovation and evidence and accountability. December 14, 2012 abstract a brief survey of the role of fermions in quantum computation is presented, focusing on the terhaldivincenzo result for noninteracting fermions. Fermions in quantum complexity theory edwin ng department of physics, massachusetts institute of technology, cambridge, ma 029 dated. An introduction to complexity theory jun park medium. Scott aaronson on computational complexity theory and. In this lecture, we study the most important complexity classes for deterministic, nondeterministic, parallel, and probabilistic computations.
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