Quantum Information Theory by Michael Aaron Nielsen
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
On the Tightness of the Buhrman-Cleve-Wigderson Simulation
On the tightness of the Buhrman-Cleve-Wigderson simulation Shengyu Zhang Department of Computer Science and Engineering, The Chinese University of Hong Kong. [email protected] Abstract. Buhrman, Cleve and Wigderson gave a general communica- 1 1 n n tion protocol for block-composed functions f(g1(x ; y ); ··· ; gn(x ; y )) by simulating a decision tree computation for f [3]. It is also well-known that this simulation can be very inefficient for some functions f and (g1; ··· ; gn). In this paper we show that the simulation is actually poly- nomially tight up to the choice of (g1; ··· ; gn). This also implies that the classical and quantum communication complexities of certain block- composed functions are polynomially related. 1 Introduction Decision tree complexity [4] and communication complexity [7] are two concrete models for studying computational complexity. In [3], Buhrman, Cleve and Wigderson gave a general method to design communication 1 1 n n protocol for block-composed functions f(g1(x ; y ); ··· ; gn(x ; y )). The basic idea is to simulate the decision tree computation for f, with queries i i of the i-th variable answered by a communication protocol for gi(x ; y ). In the language of complexity measures, this result gives CC(f(g1; ··· ; gn)) = O~(DT(f) · max CC(gi)) (1) i where CC and DT are the communication complexity and the decision tree complexity, respectively. This simulation holds for all the models: deterministic, randomized and quantum1. It is also well known that the communication protocol by the above simulation can be very inefficient. -
Thesis Is Owed to Him, Either Directly Or Indirectly
UvA-DARE (Digital Academic Repository) Quantum Computing and Communication Complexity de Wolf, R.M. Publication date 2001 Document Version Final published version Link to publication Citation for published version (APA): de Wolf, R. M. (2001). Quantum Computing and Communication Complexity. Institute for Logic, Language and Computation. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:30 Sep 2021 Quantum Computing and Communication Complexity Ronald de Wolf Quantum Computing and Communication Complexity ILLC Dissertation Series 2001-06 For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam phone: +31-20-525 6051 fax: +31-20-525 5206 e-mail: [email protected] homepage: http://www.illc.uva.nl/ Quantum Computing and Communication Complexity Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magni¯cus prof.dr. -
The Grand Challenges in the Chemical Sciences
The Israel Academy of Sciences and Humanities Celebrating the 70 th birthday of the State of Israel conference on THE GRAND CHALLENGES IN THE CHEMICAL SCIENCES Jerusalem, June 3-7 2018 Biographies and Abstracts The Israel Academy of Sciences and Humanities Celebrating the 70 th birthday of the State of Israel conference on THE GRAND CHALLENGES IN THE CHEMICAL SCIENCES Participants: Jacob Klein Dan Shechtman Dorit Aharonov Roger Kornberg Yaron Silberberg Takuzo Aida Ferenc Krausz Gabor A. Somorjai Yitzhak Apeloig Leeor Kronik Amiel Sternberg Frances Arnold Richard A. Lerner Sir Fraser Stoddart Ruth Arnon Raphael D. Levine Albert Stolow Avinoam Ben-Shaul Rudolph A. Marcus Zehev Tadmor Paul Brumer Todd Martínez Reshef Tenne Wah Chiu Raphael Mechoulam Mark H. Thiemens Nili Cohen David Milstein Naftali Tishby Nir Davidson Shaul Mukamel Knut Wolf Urban Ronnie Ellenblum Edvardas Narevicius Arieh Warshel Greg Engel Nathan Nelson Ira A. Weinstock Makoto Fujita Hagai Netzer Paul Weiss Oleg Gang Abraham Nitzan Shimon Weiss Leticia González Geraldine L. Richmond George M. Whitesides Hardy Gross William Schopf Itamar Willner David Harel Helmut Schwarz Xiaoliang Sunney Xie Jim Heath Mordechai (Moti) Segev Omar M. Yaghi Joshua Jortner Michael Sela Ada Yonath Biographies and Abstracts (Arranged in alphabetic order) The Grand Challenges in the Chemical Sciences Dorit Aharonov The Hebrew University of Jerusalem Quantum Physics through the Computational Lens While the jury is still out as to when and where the impressive experimental progress on quantum gates and qubits will indeed lead one day to a full scale quantum computing machine, a new and not-less exciting development had been taking place over the past decade. -
Efficient Quantum Algorithms for Simulating Lindblad Evolution
Efficient Quantum Algorithms for Simulating Lindblad Evolution∗ Richard Cleveyzx Chunhao Wangyz Abstract We consider the natural generalization of the Schr¨odingerequation to Markovian open system dynamics: the so-called the Lindblad equation. We give a quantum algorithm for simulating the evolution of an n-qubit system for time t within precision . If the Lindbladian consists of poly(n) operators that can each be expressed as a linear combination of poly(n) tensor products of Pauli operators then the gate cost of our algorithm is O(t polylog(t/)poly(n)). We also obtain similar bounds for the cases where the Lindbladian consists of local operators, and where the Lindbladian consists of sparse operators. This is remarkable in light of evidence that we provide indicating that the above efficiency is impossible to attain by first expressing Lindblad evolution as Schr¨odingerevolution on a larger system and tracing out the ancillary system: the cost of such a reduction incurs an efficiency overhead of O(t2/) even before the Hamiltonian evolution simulation begins. Instead, the approach of our algorithm is to use a novel variation of the \linear combinations of unitaries" construction that pertains to channels. 1 Introduction The problem of simulating the evolution of closed systems (captured by the Schr¨odingerequation) was proposed by Feynman [11] in 1982 as a motivation for building quantum computers. Since then, several quantum algorithms have appeared for this problem (see subsection 1.1 for references to these algorithms). However, many quantum systems of interest are not closed but are well-captured by the Lindblad Master equation [20, 12]. -
Arxiv:1611.07995V1 [Quant-Ph] 23 Nov 2016
Factoring using 2n + 2 qubits with Toffoli based modular multiplication Thomas H¨aner∗ Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA and Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland Martin Roettelery and Krysta M. Svorez Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA We describe an implementation of Shor's quantum algorithm to factor n-bit integers using only 2n+2 qubits. In contrast to previous space-optimized implementations, ours features a purely Toffoli based modular multiplication circuit. The circuit depth and the overall gate count are in (n3) and (n3 log n), respectively. We thus achieve the same space and time costs as TakahashiO et al. [1], Owhile using a purely classical modular multiplication circuit. As a consequence, our approach evades most of the cost overheads originating from rotation synthesis and enables testing and localization of faults in both, the logical level circuit and an actual quantum hardware implementation. Our new (in-place) constant-adder, which is used to construct the modular multiplication circuit, uses only dirty ancilla qubits and features a circuit size and depth in (n log n) and (n), respectively. O O I. INTRODUCTION Cuccaro [5] Takahashi [6] Draper [4] Our adder Size Θ(n) Θ(n) Θ(n2) Θ(n log n) Quantum computers offer an exponential speedup over Depth Θ(n) Θ(n) Θ(n) Θ(n) their classical counterparts for solving certain problems, n the most famous of which is Shor's algorithm [2] for fac- Ancillas n+1 (clean) n (clean) 0 2 (dirty) toring a large number N | an algorithm that enables the breaking of many popular encryption schemes including Table I. -
Quantum Leap FINAL Case Study
Case study A Quantum Leap at INI Case study: Mathematics for Quantum Information “The application of quantum technologies to In their original proposal, two years earlier, MQI encryption algorithms threatens to dramatically programme organisers Richard Jozsa (Cambridge), impact the US government’s ability to both protect Noah Linden (Bristol), Peter Shor (MIT) and its communications and eavesdrop on the Andreas Winter (Bristol/CQT, Singapore) had communications of foreign governments,” correctly anticipated the readiness of their according to an internal document leaked by discipline for such breakthroughs predicting whistle-blower Edward Snowden. That the UK Government takes quantum computing seriously “we see this as a moment of substantial scientific is evinced by its commitment to “provide £270 opportunity: we believe we are at the beginning million over 5 years to fund a programme to of a period in the subject where diverse areas of support translation of the UK’s world leading mathematics will play an increasing important quantum research into application and new role. A Newton Institute programme at this time industries – from quantum computation to secure offers the possibility of major influence in the communication”, as announced by Chancellor of speed and direction of the field”. the Exchequer George Osborne in his 2013 Autumn Statement. This statement by the UK In October 2013, Alexander Holevo, a pioneer in Chancellor came during the closing weeks of an the field of Quantum Information Science and INI programme on Mathematics for Quantum winner of the Humbolt Prize, gave the Rothschild Information (MQI) and within days of an Lecture for the MQI programme. -
International Congress of Mathematicians
International Congress of Mathematicians Hyderabad, August 19–27, 2010 Abstracts Plenary Lectures Invited Lectures Panel Discussions Editor Rajendra Bhatia Co-Editors Arup Pal G. Rangarajan V. Srinivas M. Vanninathan Technical Editor Pablo Gastesi Contents Plenary Lectures ................................................... .. 1 Emmy Noether Lecture................................. ................ 17 Abel Lecture........................................ .................... 18 Invited Lectures ................................................... ... 19 Section 1: Logic and Foundations ....................... .............. 21 Section 2: Algebra................................... ................. 23 Section 3: Number Theory.............................. .............. 27 Section 4: Algebraic and Complex Geometry............... ........... 32 Section 5: Geometry.................................. ................ 39 Section 6: Topology.................................. ................. 46 Section 7: Lie Theory and Generalizations............... .............. 52 Section 8: Analysis.................................. .................. 57 Section 9: Functional Analysis and Applications......... .............. 62 Section 10: Dynamical Systems and Ordinary Differential Equations . 66 Section 11: Partial Differential Equations.............. ................. 71 Section 12: Mathematical Physics ...................... ................ 77 Section 13: Probability and Statistics................. .................. 82 Section 14: Combinatorics........................... -
Hamiltonian Simulation with Nearly Optimal Dependence on All Parameters
Hamiltonian simulation with nearly optimal dependence on all parameters Dominic W. Berry∗ Andrew M. Childsy Robin Kothariz We present an algorithm for sparse Hamiltonian simulation that has optimal dependence on all pa- rameters of interest (up to log factors). Previous algorithms had optimal or near-optimal scaling in some parameters at the cost of poor scaling in others. Hamiltonian simulation via quantum walk has optimal dependence on the sparsity d at the expense of poor scaling in the allowed error . In contrast, an ap- proach based on fractional-query simulation provides optimal scaling in at the expense of poor scaling in d. Here we combine the two approaches, achieving the best features of both. By implementing a linear combination of quantum walk steps with coefficients given by Bessel functions, our algorithm achieves near-linear scaling in τ := dkHkmaxt and sublogarithmic scaling in 1/. Our dependence on is optimal, and we prove a new lower bound ruling out sublinear dependence on τ. The problem of simulating the dynamics of quantum systems was the original motivation for quantum com- puters [14] and remains one of their major potential applications. The first explicit quantum simulation algorithm [16] gave a method for simulating Hamiltonians that are sums of local interaction terms. Aharonov and Ta-Shma gave an efficient simulation algorithm for the more general class of sparse Hamiltonians [2], and much subsequent work has given improved simulations [4,5,6,7,8, 12, 17, 18]. Sparse Hamiltonians include most physically realistic Hamiltonians as a special case. In addition, sparse Hamiltonian simulation can also be used to design other quantum algorithms [9, 10, 15]. -
Quantum Computing: Lecture Notes
Quantum Computing: Lecture Notes Ronald de Wolf Preface These lecture notes were formed in small chunks during my “Quantum computing” course at the University of Amsterdam, Feb-May 2011, and compiled into one text thereafter. Each chapter was covered in a lecture of 2 45 minutes, with an additional 45-minute lecture for exercises and × homework. The first half of the course (Chapters 1–7) covers quantum algorithms, the second half covers quantum complexity (Chapters 8–9), stuff involving Alice and Bob (Chapters 10–13), and error-correction (Chapter 14). A 15th lecture about physical implementations and general outlook was more sketchy, and I didn’t write lecture notes for it. These chapters may also be read as a general introduction to the area of quantum computation and information from the perspective of a theoretical computer scientist. While I made an effort to make the text self-contained and consistent, it may still be somewhat rough around the edges; I hope to continue polishing and adding to it. Comments & constructive criticism are very welcome, and can be sent to [email protected] Attribution and acknowledgements Most of the material in Chapters 1–6 comes from the first chapter of my PhD thesis [71], with a number of additions: the lower bound for Simon, the Fourier transform, the geometric explanation of Grover. Chapter 7 is newly written for these notes, inspired by Santha’s survey [62]. Chapters 8 and 9 are largely new as well. Section 3 of Chapter 8, and most of Chapter 10 are taken (with many changes) from my “quantum proofs” survey paper with Andy Drucker [28]. -
Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Codes
Analysis of quantum error-correcting codes: symplectic lattice codes and toric codes Thesis by James William Harrington Advisor John Preskill In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2004 (Defended May 17, 2004) ii c 2004 James William Harrington All rights Reserved iii Acknowledgements I can do all things through Christ, who strengthens me. Phillipians 4:13 (NKJV) I wish to acknowledge first of all my parents, brothers, and grandmother for all of their love, prayers, and support. Thanks to my advisor, John Preskill, for his generous support of my graduate studies, for introducing me to the studies of quantum error correction, and for encouraging me to pursue challenging questions in this fascinating field. Over the years I have benefited greatly from stimulating discussions on the subject of quantum information with Anura Abeyesinge, Charlene Ahn, Dave Ba- con, Dave Beckman, Charlie Bennett, Sergey Bravyi, Carl Caves, Isaac Chenchiah, Keng-Hwee Chiam, Richard Cleve, John Cortese, Sumit Daftuar, Ivan Deutsch, Andrew Doherty, Jon Dowling, Bryan Eastin, Steven van Enk, Chris Fuchs, Sho- hini Ghose, Daniel Gottesman, Ted Harder, Patrick Hayden, Richard Hughes, Deborah Jackson, Alexei Kitaev, Greg Kuperberg, Andrew Landahl, Chris Lee, Debbie Leung, Carlos Mochon, Michael Nielsen, Smith Nielsen, Harold Ollivier, Tobias Osborne, Michael Postol, Philippe Pouliot, Marco Pravia, John Preskill, Eric Rains, Robert Raussendorf, Joe Renes, Deborah Santamore, Yaoyun Shi, Pe- ter Shor, Marcus Silva, Graeme Smith, Jennifer Sokol, Federico Spedalieri, Rene Stock, Francis Su, Jacob Taylor, Ben Toner, Guifre Vidal, and Mas Yamada. Thanks to Chip Kent for running some of my longer numerical simulations on a computer system in the High Performance Computing Environments Group (CCN-8) at Los Alamos National Laboratory. -
International Online Conference One-Parameter Semigroups of Operators
International Online Conference One-Parameter Semigroups of Operators BOOK OF ABSTRACTS stable version from 20:21 (UTC+0), Friday 9th April, 2021 Nizhny Novgorod 2021 International Online Conference “One-Parameter Semigroups of Operators 2021” Preface International online conference One-Parameter Semigroups of Operators (OPSO 2021), 5-9 April 2021, is organized by the International laboratory of dynamical systems and applications, and research group Evolution semigroups and applications, both located in Russia, Nizhny Novgorod city. The Laboratory was created in 2019 at the National research university Higher School of Economics (HSE). Website of the Laboratory: https://nnov.hse.ru/en/bipm/dsa/ HSE is a young university (established in 1992) which rapidly become one of the leading Russian universities according to international ratings. In 2021 HSE is a large university focused on only on economics. There are departments of Economics (including Finance, Statistics etc), Law, Mathematics, Computer Science, Media and Design, Physics, Chemistry, Biotechnology, Geography and Geoinformation Technologies, Foreign Languages and some other. Website of the HSE: https://www.hse.ru/en/ The OPSO 2021 online conference has connected 106 speakers and 26 participants without a talk from all over the world. The conference covered the following topics: 1. One-parameter groups and semigroups of linear operators; 2. Nonlinear flows and semiflows; 3. Interplay between linear infinite-dimensional systems and nonlinear finite-dimensional systems; 4. Quan- tum -
Andrew Childs University of Maryland Arxiv:1711.10980
Toward the first quantum simulation with quantum speedup Andrew Childs University of Maryland Dmitri Maslov Yunseong Nam Neil Julien Ross Yuan Su arXiv:1711.10980 Simulating Hamiltonian dynamics on a small quantum computer Andrew Childs Department of Combinatorics & Optimization and Institute for Quantum Computing University of Waterloo based in part on joint work with Dominic Berry, Richard Cleve, Robin Kothari, and Rolando Somma Workshop on What do we do with a small quantum computer? IBM Watson, 9 December 2013 Toward practical quantum speedup IBM Google/UCSB Delft Maryland Important early goal: demonstrate quantum computational advantage … but can we find a practical application of near-term devices? Challenges • Improve experimental systems • Improve algorithms and their implementation, making the best use of available hardware Our goal: Produce concrete resource estimates for the simplest possible practical application of quantum computers Quantum simulation “… nature isn’t classical, dammit, A classical computer cannot even represent and if you want to make a the state efficiently. simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, A quantum computer cannot produce a because it doesn’t look so easy.” complete description of the state. Richard Feynman (1981) Simulating physics with computers But given succinct descriptions of • the initial state (suitable for a quantum computer to prepare it efficiently) and Quantum simulation problem: Given a • a final measurement (say, measurements description of the Hamiltonian H, an of the individual qubits in some basis), evolution time t, and an initial state (0) , a quantum computer can efficiently answer produce the final state ( t ) (to within| i questions that (apparently) a classical one some error tolerance ²|) i cannot.