Quantum Information Chapter 6. Quantum Algorithms
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What Is Quantum Information?
What Is Quantum Information? Robert B. Griffiths Carnegie-Mellon University Pittsburgh, Pennsylvania Research supported by the National Science Foundation References (work by R. B. Griffiths) • \Nature and location of quantum information." Ph◦ys. Rev. A 66 (2002) 012311; quant-ph/0203058 \Channel kets, entangled states, and the location of quan◦ tum information," Phys. Rev. A 71 (2005) 042337; quant-ph/0409106 Consistent Quantum Theory (Cambridge 2002) http://quan◦ tum.phys.cmu.edu/ What Is Quantum Information? 01. 100.01.tex Introduction What is quantum information? Precede by: • What is information? ◦ What is classical information theory? ◦ What is information? Example, newspaper • Symbolic representation of some situation ◦ Symbols in newspaper correlated with situation ◦ Information is about that situation ◦ What Is Quantum Information? 04. 100.04.tex Classical Information Theory Shannon: • \Mathematical Theory of Communication" (1948) ◦ One of major scientific developments of 20th century ◦ Proposed quantitative measure of information ◦ Information entropy • H(X) = p log p − X i i i Logarithmic measure of missing information ◦ Probabilistic model: pi are probabilities ◦ Applies to classical (macroscopic)f g signals ◦ Coding theorem: Bound on rate of transmission of information• through noisy channel What Is Quantum Information? 05. 100.05.tex Quantum Information Theory (QIT) Goal of QIT: \Quantize Shannon" • Extend Shannon's ideas to domain where quantum effects◦ are important Find quantum counterpart of H(X) ◦ We live in a quantum world, so • QIT should be the fundamental info theory ◦ Classical theory should emerge from QIT ◦ Analogy: relativity theory Newton for v c ◦ ! What Is Quantum Information? 06. 100.06.tex QIT: Current Status Enormous number of published papers • Does activity = understanding? ◦ Some topics of current interest: • Entanglement ◦ Quantum channels ◦ Error correction ◦ Quantum computation ◦ Decoherence ◦ Unifying principles have yet to emerge • At least, they are not yet widely recognized ◦ What Is Quantum Information? 07. -
Quantum Computing Overview
Quantum Computing at Microsoft Stephen Jordan • As of November 2018: 60 entries, 392 references • Major new primitives discovered only every few years. Simulation is a killer app for quantum computing with a solid theoretical foundation. Chemistry Materials Nuclear and Particle Physics Image: David Parker, U. Birmingham Image: CERN Many problems are out of reach even for exascale supercomputers but doable on quantum computers. Understanding biological Nitrogen fixation Intractable on classical supercomputers But a 200-qubit quantum computer will let us understand it Finding the ground state of Ferredoxin 퐹푒2푆2 Classical algorithm Quantum algorithm 2012 Quantum algorithm 2015 ! ~3000 ~1 INTRACTABLE YEARS DAY Research on quantum algorithms and software is essential! Quantum algorithm in high level language (Q#) Compiler Machine level instructions Microsoft Quantum Development Kit Quantum Visual Studio Local and cloud Extensive libraries, programming integration and quantum samples, and language debugging simulation documentation Developing quantum applications 1. Find quantum algorithm with quantum speedup 2. Confirm quantum speedup after implementing all oracles 3. Optimize code until runtime is short enough 4. Embed into specific hardware estimate runtime Simulating Quantum Field Theories: Classically Feynman diagrams Lattice methods Image: Encyclopedia of Physics Image: R. Babich et al. Break down at strong Good for binding energies. coupling or high precision Sign problem: real time dynamics, high fermion density. There’s room for exponential -
Key Concepts for Future QIS Learners Workshop Output Published Online May 13, 2020
Key Concepts for Future QIS Learners Workshop output published online May 13, 2020 Background and Overview On behalf of the Interagency Working Group on Workforce, Industry and Infrastructure, under the NSTC Subcommittee on Quantum Information Science (QIS), the National Science Foundation invited 25 researchers and educators to come together to deliberate on defining a core set of key concepts for future QIS learners that could provide a starting point for further curricular and educator development activities. The deliberative group included university and industry researchers, secondary school and college educators, and representatives from educational and professional organizations. The workshop participants focused on identifying concepts that could, with additional supporting resources, help prepare secondary school students to engage with QIS and provide possible pathways for broader public engagement. This workshop report identifies a set of nine Key Concepts. Each Concept is introduced with a concise overall statement, followed by a few important fundamentals. Connections to current and future technologies are included, providing relevance and context. The first Key Concept defines the field as a whole. Concepts 2-6 introduce ideas that are necessary for building an understanding of quantum information science and its applications. Concepts 7-9 provide short explanations of critical areas of study within QIS: quantum computing, quantum communication and quantum sensing. The Key Concepts are not intended to be an introductory guide to quantum information science, but rather provide a framework for future expansion and adaptation for students at different levels in computer science, mathematics, physics, and chemistry courses. As such, it is expected that educators and other community stakeholders may not yet have a working knowledge of content covered in the Key Concepts. -
Quantum Clustering Algorithms
Quantum Clustering Algorithms Esma A¨ımeur [email protected] Gilles Brassard [email protected] S´ebastien Gambs [email protected] Universit´ede Montr´eal, D´epartement d’informatique et de recherche op´erationnelle C.P. 6128, Succursale Centre-Ville, Montr´eal (Qu´ebec), H3C 3J7 Canada Abstract Multidisciplinary by nature, Quantum Information By the term “quantization”, we refer to the Processing (QIP) is at the crossroads of computer process of using quantum mechanics in order science, mathematics, physics and engineering. It con- to improve a classical algorithm, usually by cerns the implications of quantum mechanics for infor- making it go faster. In this paper, we initiate mation processing purposes (Nielsen & Chuang, 2000). the idea of quantizing clustering algorithms Quantum information is very different from its classi- by using variations on a celebrated quantum cal counterpart: it cannot be measured reliably and it algorithm due to Grover. After having intro- is disturbed by observation, but it can exist in a super- duced this novel approach to unsupervised position of classical states. Classical and quantum learning, we illustrate it with a quantized information can be used together to realize wonders version of three standard algorithms: divisive that are out of reach of classical information processing clustering, k-medians and an algorithm for alone, such as being able to factorize efficiently large the construction of a neighbourhood graph. numbers, with dramatic cryptographic consequences We obtain a significant speedup compared to (Shor, 1997), search in a unstructured database with a the classical approach. quadratic speedup compared to the best possible clas- sical algorithms (Grover, 1997) and allow two people to communicate in perfect secrecy under the nose of an 1. -
Free-Electron Qubits
Free-Electron Qubits Ori Reinhardt†, Chen Mechel†, Morgan Lynch, and Ido Kaminer Department of Electrical Engineering and Solid State Institute, Technion - Israel Institute of Technology, 32000 Haifa, Israel † equal contributors Free-electron interactions with laser-driven nanophotonic nearfields can quantize the electrons’ energy spectrum and provide control over this quantized degree of freedom. We propose to use such interactions to promote free electrons as carriers of quantum information and find how to create a qubit on a free electron. We find how to implement the qubit’s non-commutative spin-algebra, then control and measure the qubit state with a universal set of 1-qubit gates. These gates are within the current capabilities of femtosecond-pulsed laser-driven transmission electron microscopy. Pulsed laser driving promise configurability by the laser intensity, polarizability, pulse duration, and arrival times. Most platforms for quantum computation today rely on light-matter interactions of bound electrons such as in ion traps [1], superconducting circuits [2], and electron spin systems [3,4]. These form a natural choice for implementing 2-level systems with spin algebra. Instead of using bound electrons for quantum information processing, in this letter we propose using free electrons and manipulating them with femtosecond laser pulses in optical frequencies. Compared to bound electrons, free electron systems enable accessing high energy scales and short time scales. Moreover, they possess quantized degrees of freedom that can take unbounded values, such as orbital angular momentum (OAM), which has also been proposed for information encoding [5-10]. Analogously, photons also have the capability to encode information in their OAM [11,12]. -
Quantum Information Science, NIST, and Future Technological Implications National Institute of Standards & Technology Http
Quantum Information Science, NIST, and Future Technological Implications Carl J. Williams, Chief Atomic Physics Division National Institute of Standards & Technology http://qubit.nist.gov NIST – Atomic Physics Division Carl J. Williams, Chief What is Quantum Information? “Quantum information is a radical departure in information technology, more fundamentally different from current technology than the digital computer is from the abacus.” W. D. Phillips, 1997 Nobel Prize Winner in Physics A convergence of two of the 20th Century’s great revolutions A new science! Quantum Mechanics Information Science (i.e. computer science, (i.e. atoms, photons, JJ’s, communications, physics of computation) cryptology) The second quantum revolution NIST – Atomic Physics Division Carl J. Williams, Chief Second Quantum Revolution We are witnessing the second quantum revolution • First quantum revolution (1920’s - 1980’s) – Described how nature works at the quantum level – Weirdness of QM discovered and debated – Wave-particle duality -> Wavefunctions – Spooky action at a distance -> Entanglement – Technology uses semiclassical properties – quantization & wave properties • Second quantum revolution (starts ~1980’s - ) – Exploits how nature works at the quantum level – Interactions are how Nature computes! – Weirdness of QM exploited – Information storage capacity of superposed wavefunctions – Information transmittal accomplished by entanglement, teleportation – Information exchange accomplished by interactions – Technology uses the weird properties of quantum mechanics NIST – Atomic Physics Division Carl J. Williams, Chief Classical Bits vs. Quantum Bits • Classical Bits: two-state systems Classical bits: 0 (off) or 1 (on) (switch) • Quantum Bits are also two-level(state) systems ⎜↑〉 ⎜1〉 Atom ⎜0〉 ⎜↓〉 Internal State Motional State Ö But: Quantum Superpositions are possible Ψ= α|↓〉 + β|↑〉 = α|0〉 + β|1〉 NIST – Atomic Physics Division Carl J. -
Quantum Computation and Complexity Theory
Quantum computation and complexity theory Course given at the Institut fÈurInformationssysteme, Abteilung fÈurDatenbanken und Expertensysteme, University of Technology Vienna, Wintersemester 1994/95 K. Svozil Institut fÈur Theoretische Physik University of Technology Vienna Wiedner Hauptstraûe 8-10/136 A-1040 Vienna, Austria e-mail: [email protected] December 5, 1994 qct.tex Abstract The Hilbert space formalism of quantum mechanics is reviewed with emphasis on applicationsto quantum computing. Standardinterferomeric techniques are used to construct a physical device capable of universal quantum computation. Some consequences for recursion theory and complexity theory are discussed. hep-th/9412047 06 Dec 94 1 Contents 1 The Quantum of action 3 2 Quantum mechanics for the computer scientist 7 2.1 Hilbert space quantum mechanics ..................... 7 2.2 From single to multiple quanta Ð ªsecondº ®eld quantization ...... 15 2.3 Quantum interference ............................ 17 2.4 Hilbert lattices and quantum logic ..................... 22 2.5 Partial algebras ............................... 24 3 Quantum information theory 25 3.1 Information is physical ........................... 25 3.2 Copying and cloning of qbits ........................ 25 3.3 Context dependence of qbits ........................ 26 3.4 Classical versus quantum tautologies .................... 27 4 Elements of quantum computatability and complexity theory 28 4.1 Universal quantum computers ....................... 30 4.2 Universal quantum networks ........................ 31 4.3 Quantum recursion theory ......................... 35 4.4 Factoring .................................. 36 4.5 Travelling salesman ............................. 36 4.6 Will the strong Church-Turing thesis survive? ............... 37 Appendix 39 A Hilbert space 39 B Fundamental constants of physics and their relations 42 B.1 Fundamental constants of physics ..................... 42 B.2 Conversion tables .............................. 43 B.3 Electromagnetic radiation and other wave phenomena ......... -
Models of Quantum Complexity Growth
Models of quantum complexity growth Fernando G.S.L. Brand~ao,a;b;c;d Wissam Chemissany,b Nicholas Hunter-Jones,* e;b Richard Kueng,* b;c John Preskillb;c;d aAmazon Web Services, AWS Center for Quantum Computing, Pasadena, CA bInstitute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125 cDepartment of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125 dWalter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125 ePerimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5 *Corresponding authors: [email protected] and [email protected] Abstract: The concept of quantum complexity has far-reaching implications spanning theoretical computer science, quantum many-body physics, and high energy physics. The quantum complexity of a unitary transformation or quantum state is defined as the size of the shortest quantum computation that executes the unitary or prepares the state. It is reasonable to expect that the complexity of a quantum state governed by a chaotic many- body Hamiltonian grows linearly with time for a time that is exponential in the system size; however, because it is hard to rule out a short-cut that improves the efficiency of a computation, it is notoriously difficult to derive lower bounds on quantum complexity for particular unitaries or states without making additional assumptions. To go further, one may study more generic models of complexity growth. We provide a rigorous connection between complexity growth and unitary k-designs, ensembles which capture the randomness of the unitary group. This connection allows us to leverage existing results about design growth to draw conclusions about the growth of complexity. -
Lecture 6: Quantum Error Correction and Quantum Capacity
Lecture 6: Quantum error correction and quantum capacity Mark M. Wilde∗ The quantum capacity theorem is one of the most important theorems in quantum Shannon theory. It is a fundamentally \quantum" theorem in that it demonstrates that a fundamentally quantum information quantity, the coherent information, is an achievable rate for quantum com- munication over a quantum channel. The fact that the coherent information does not have a strong analog in classical Shannon theory truly separates the quantum and classical theories of information. The no-cloning theorem provides the intuition behind quantum error correction. The goal of any quantum communication protocol is for Alice to establish quantum correlations with the receiver Bob. We know well now that every quantum channel has an isometric extension, so that we can think of another receiver, the environment Eve, who is at a second output port of a larger unitary evolution. Were Eve able to learn anything about the quantum information that Alice is attempting to transmit to Bob, then Bob could not be retrieving this information|otherwise, they would violate the no-cloning theorem. Thus, Alice should figure out some subspace of the channel input where she can place her quantum information such that only Bob has access to it, while Eve does not. That the dimensionality of this subspace is exponential in the coherent information is perhaps then unsurprising in light of the above no-cloning reasoning. The coherent information is an entropy difference H(B) − H(E)|a measure of the amount of quantum correlations that Alice can establish with Bob less the amount that Eve can gain. -
Quantum Algorithms
An Introduction to Quantum Algorithms Emma Strubell COS498 { Chawathe Spring 2011 An Introduction to Quantum Algorithms Contents Contents 1 What are quantum algorithms? 3 1.1 Background . .3 1.2 Caveats . .4 2 Mathematical representation 5 2.1 Fundamental differences . .5 2.2 Hilbert spaces and Dirac notation . .6 2.3 The qubit . .9 2.4 Quantum registers . 11 2.5 Quantum logic gates . 12 2.6 Computational complexity . 19 3 Grover's Algorithm 20 3.1 Quantum search . 20 3.2 Grover's algorithm: How it works . 22 3.3 Grover's algorithm: Worked example . 24 4 Simon's Algorithm 28 4.1 Black-box period finding . 28 4.2 Simon's algorithm: How it works . 29 4.3 Simon's Algorithm: Worked example . 32 5 Conclusion 33 References 34 Page 2 of 35 An Introduction to Quantum Algorithms 1. What are quantum algorithms? 1 What are quantum algorithms? 1.1 Background The idea of a quantum computer was first proposed in 1981 by Nobel laureate Richard Feynman, who pointed out that accurately and efficiently simulating quantum mechanical systems would be impossible on a classical computer, but that a new kind of machine, a computer itself \built of quantum mechanical elements which obey quantum mechanical laws" [1], might one day perform efficient simulations of quantum systems. Classical computers are inherently unable to simulate such a system using sub-exponential time and space complexity due to the exponential growth of the amount of data required to completely represent a quantum system. Quantum computers, on the other hand, exploit the unique, non-classical properties of the quantum systems from which they are built, allowing them to process exponentially large quantities of information in only polynomial time. -
SECURING CLOUDS – the QUANTUM WAY 1 Introduction
SECURING CLOUDS – THE QUANTUM WAY Marmik Pandya Department of Information Assurance Northeastern University Boston, USA 1 Introduction Quantum mechanics is the study of the small particles that make up the universe – for instance, atoms et al. At such a microscopic level, the laws of classical mechanics fail to explain most of the observed phenomenon. At such a state quantum properties exhibited by particles is quite noticeable. Before we move forward a basic question arises that, what are quantum properties? To answer this question, we'll look at the basis of quantum mechanics – The Heisenberg's Uncertainty Principle. The uncertainty principle states that the more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa. For instance, if you measure the position of an electron revolving around the nucleus an atom, you cannot accurately measure its velocity. If you measure the electron's velocity, you cannot accurately determine its position. Equation for Heisenberg's uncertainty principle: Where σx standard deviation of position, σx the standard deviation of momentum and ħ is the reduced Planck constant, h / 2π. In a practical scenario, this principle is applied to photons. Photons – the smallest measure of light, can exist in all of their possible states at once and also they don’t have any mass. This means that whatever direction a photon can spin in -- say, diagonally, vertically and horizontally -- it does all at once. This is exactly the same as if you constantly moved east, west, north, south, and up-and-down at the same time. -
Computational Complexity: a Modern Approach
i Computational Complexity: A Modern Approach Sanjeev Arora and Boaz Barak Princeton University http://www.cs.princeton.edu/theory/complexity/ [email protected] Not to be reproduced or distributed without the authors’ permission ii Chapter 10 Quantum Computation “Turning to quantum mechanics.... secret, secret, close the doors! we always have had a great deal of difficulty in understanding the world view that quantum mechanics represents ... It has not yet become obvious to me that there’s no real problem. I cannot define the real problem, therefore I suspect there’s no real problem, but I’m not sure there’s no real problem. So that’s why I like to investigate things.” Richard Feynman, 1964 “The only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative..” Richard Feynman, in “Simulating physics with computers,” 1982 Quantum computing is a new computational model that may be physically realizable and may provide an exponential advantage over “classical” computational models such as prob- abilistic and deterministic Turing machines. In this chapter we survey the basic principles of quantum computation and some of the important algorithms in this model. One important reason to study quantum computers is that they pose a serious challenge to the strong Church-Turing thesis (see Section 1.6.3), which stipulates that every physi- cally reasonable computation device can be simulated by a Turing machine with at most polynomial slowdown. As we will see in Section 10.6, there is a polynomial-time algorithm for quantum computers to factor integers, whereas despite much effort, no such algorithm is known for deterministic or probabilistic Turing machines.