Quantum Cellular Automata
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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. -
Elementary Gates for Quantum Computation
Elementary gates for quantum computation Adriano Barenco Charles H. Bennett Richard Cleve y z Oxford University IBM Research University of Calgary David P. DiVincenzo Norman Margolus Peter Shor x { y IBM Research MIT AT&T Bell Labs Tycho Sleator John Smolin Harald Weinfurter yy k UCLA Univ. of Innsbruck New York Univ. submitted to Physical Review A, March 22, 1995 (AC5710) Abstract We show that a set of gates that consists of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (that maps Bo olean values (x; y )to(x; x y )) is universal in the sense that all unitary op erations on arbitrarily many bits n n (U(2 )) can b e expressed as comp ositions of these gates. Weinvestigate the numb er of the ab ove gates required to implement other gates, such as gener- alized Deutsch-To oli gates, that apply a sp eci c U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satis ed. These gates play a central role in many prop osed constructions of quantum com- putational networks. We derive upp er and lower b ounds on the exact number Clarendon Lab oratory, Oxford OX1 3PU, UK; [email protected]. y Yorktown Heights, New York, NY 10598, USA; b ennetc/[email protected]. z Department of Computer Science, Calgary, Alb erta, Canada T2N 1N4; [email protected]. x Lab oratory for Computer Science, Cambridge MA 02139 USA; [email protected]. -
Hybrid Qubits Solve Key Hurdle to Quantum Computing 28 December 2018
Hybrid qubits solve key hurdle to quantum computing 28 December 2018 In 1998, Daniel Loss, one of the authors of the current study, came up with a proposal, along with David DiVincenzo of IBM, to build a quantum computer by using the spins of electrons embedded in a quantum dot—a small particle that behaves like an atom, but that can be manipulated, so that they are sometimes called "artificial atoms." In the time since then, Loss and his team have endeavored to build practical devices. There are a number of barriers to developing practical devices in terms of speed. First, the device must be able to be initialized quickly. Initialization is the process of putting a qubit into a certain state, and if that cannot be done rapidly it slows down the device. Second, it must maintain coherence for a time long enough to make a measurement. Coherence refers to the Schematic of the device. Credit: RIKEN entanglement between two quantum states, and ultimately this is used to make the measurement, so if qubits become decoherent due to environmental noise, for example, the device Spin-based quantum computers have the potential becomes worthless. And finally, the ultimate state to tackle difficult mathematical problems that of the qubit must be able to be quickly read out. cannot be solved using ordinary computers, but many problems remain in making these machines While a number of methods have been proposed scalable. Now, an international group of for building a quantum computer, the one proposed researchers led by the RIKEN Center for Emergent by Loss and DiVincenzo remains one of the most Matter Science have crafted a new architecture for practically feasible, as it is based on quantum computing. -
The Quantum Times) in Producing the First in Their List of Guidelines Nor Was It Reported As Operating Laser, Are All Interesting and Enlightening
TThhee QQuuaannttuumm TTiimmeess APS Topical Group on Quantum Information, Concepts, and Computation Summer 2007 Volume 2, Number 2 Advances & Challenges in Quantum Key Distribution Quantum Key Distribution (QKD) is the flagship success of quantum communication. Since Bennett and Brassard’s 1984 publication [2] it has given a new direction to the field. (For a review see e.g. [3].) Up to then in quantum communication, the properties of quantum mechanics were used, for example, to improve on the through-put of optical channels. With the arrival of QKD an application has been created that achieves what Inside… cannot be achieved with classical communication alone: … you will find a little bit of provably secure protocols with no assumption about the everything. We begin with an computational power of an adversary. (This scenario is referred article from Norbert Lütkenhaus on to as unconditional security, a technical term in cryptography. quantum key distribution (QKD) Still, the devices of sender and receiver are assumed to be that grew out of a news article from perfect and out of bound of the adversary, an assumption without our last issue. Norbert speaks from which no cryptography is possible at all.) It is sufficient to the standpoint of someone at the prepare non-orthogonal states as signals and to measure them at forefront of quantum cryptography the receiver’s end. Then, using an authenticated public channel, and his article should be read by both parties can distill a secret key from the data. The procedure anyone interested in the current needs to be kick-started with some secret key in order to state and future direction of QKD. -
Collision-Based Computing Springer-Verlag London Ltd
Collision-Based Computing Springer-Verlag London Ltd. Andrew Adamatzky (Ed.) Collision-Based Computing , Springer Andrew Adamatzky Facu1ty of Computing, Engineering and Mathematical Sciences, University of the West of England, Bristol, BS16 lQY British Library Cataloguing in Publication Data Collision-based computing l.Cellular automata 1. Adamatzky, Andrew 511.3 ISBN 978-1-85233-540-3 Library of Congress Cataloging -in -Publication Data A catalog record for this book is available from the Library of Congress. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. ISBN 978-1-85233-540-3 ISBN 978-1-4471-0129-1 (eBook) DOI 10.1007/978-1-4471-0129-1 http://www.springer.co. uk © Springer-Verlag London 2002 Originally published by Springer-Verlag London Berlin Heidelberg in 2002 The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. -
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. -
Issues in Physics & Astronomy
Issues in Physics & Astronomy Board on Physics and Astronomy • The National Academies • Washington, D.C. • 202-334-3520 • national-academies.org/bpa • Summer 2008 Understanding the Impact of Selling the Helium Reserve Michael H. Moloney, BPA Staff nder the sponsorship of the monatomic element, it passes easily containing 0.3 percent helium is consid- Bureau of Land Management at through tiny orifices and is therefore used ered economically viable. A few gas Uthe Department of the Interior, for leak detection in many scientific and deposits contain as much as 8 percent the BPA, in cooperation with the Na- technical applications. Its density is only helium. By comparison, the atmosphere tional Materials Advisory Board, has 15 percent that of air, making it useful as a contains only about 0.0005 percent. He- initiated a study to understand the lifting gas for aerostats and other devices. lium from wells that produce impact on the scientific community of Its high heat capacity, along with its uneconomically low concentrations, or the continuing sale of the U.S. helium inertness, makes it the preferred quench- from wells that produce higher concentra- reserve and recent developments in the ing medium for many applications in tions but do not flow through an extrac- helium market. materials processing, such as the produc- tion plant, is often vented to the atmo- The element helium has unique tion of high-quality superalloy powders. It sphere when the natural gas is burned. A properties. Liquefying near absolute is the preferred carrier gas for gas chroma- relatively minor amount of helium also is zero, it is the only option for many tography, a widely applied technique for vented at extraction plants that have no cryogenic applications, such as cooling chemical separations. -
Topological Phases and Applications to Quantum Information Processing" ______List of Organizers
Proposed title: "Topological Phases and Applications to Quantum Information Processing" __________________________________________________________ List of organizers: Nicholas E. Bonesteel (Florida State University and NHMFL) Phone: (850) 644-7805, E-mail: [email protected] James P. Eisenstein (Caltech) Phone: (626) 395-4649, E-mail: [email protected] Michael H. Freedman (Microsoft Research) Phone: (805) 893-6313, E-mail: [email protected] Kirill Shtengel (UC Riverside) Phone: (951) 827-1058, E-mail: [email protected] Steven H. Simon (Lucent Technologies, Bell Labs) Phone: (908) 582-6006, E-mail: [email protected] __________________________________________________________ Proposed length: 4 weeks, preferably July 2 through July 29 or June 25 through July 22. However, anytime between June 18 and Sept 2, 2007 is acceptable. Pushing it toward earlier or later dates will result in conflicts with teaching for many of the potential attendees. __________________________________________________________ Abstract: Quantum computers, if realized in practice, promise exponential speed-up of some of the computational tasks that conventional classical algorithms are intrinsically incapable of handling efficiently. Perhaps even more intriguing possibility arises from being able to use quantum computers to simulate the behavior of other physical systems -- an exciting idea dating back to Feynman. Unfortunately, realizing a quantum computer in practice proves to be very difficult, chiefly due to the debilitating effects of decoherence plaguing all possible schemes which use microscopic degrees of freedom (such as nuclear or electronic spins) as basic building blocks. >From this perspective, topological quantum computing offers an attractive alternative by encoding quantum information in nonlocal topological degrees of freedom that are intrinsically protected from decoherence due to local noise. -
Quantum Computation with Spins in Quantum Dots
Quantum Computation With Spins In Quantum Dots Fikeraddis Damtie June 10, 2013 1 Introduction Physical implementation of quantum computating has been an active area of research since the last couple of decades. In classical computing, the transmission and manipulation of classical information is carried out by physical machines (computer hardwares, etc.). In theses machines the manipulation and transmission of information can be described using the laws of classical physics. Since Newtonian mechanics is a special limit of quantum mechanics, computers making use of the laws of quantum mechanics have greater computational power than classical computers. This need to create a powerful computing machine is the driving motor for research in the field of quantum computing. Until today, there are a few different schemes for implementing a quantum computer based on the David Divincenzo chriterias. Among these are: Spectral hole burning, Trapped ion, e-Helium, Gated qubits, Nuclear Magnetic Resonance, Optics, Quantum dots, Neutral atom, superconductors and doped silicon. In this project only the quantum dot scheme will be discussed. In the year 1997 Daniel Loss and David P.DiVincenzo proposed a spin-qubit quantum computer also called The Loss-DiVincenzo quantum computer. This proposal is now considered to be one of the most promising candidates for quantum computation in the solid state. The main idea of the proposal was to use the intrinsic spin-1=2 degrees of freedom of individual elecrons confined in semiconductor quantum dots. The proposal was made in a way to satisfy the five requirements for quantum computing by David diVincenzo which will be described in sec. -
Local and Distributed Quantum Computation
Local and Distributed Quantum Computation Rodney Van Meter and Simon J. Devitt May 24, 2016 Abstract Experimental groups are now fabricating quantum processors powerful enough to execute small instances of quantum algorithms and definitively demonstrate quantum error correction that extends the lifetime of quantum data, adding ur- gency to architectural investigations. Although other options continue to be ex- plored, effort is coalescing around topological coding models as the most prac- tical implementation option for error correction on realizable microarchitectures. Scalability concerns have also motivated architects to propose distributed memory multicomputer architectures, with experimental efforts demonstrating some of the basic building blocks to make such designs possible. We compile the latest re- sults from a variety of different systems aiming at the construction of a scalable quantum computer. arXiv:1605.06951v1 [quant-ph] 23 May 2016 1 1 Introduction Quantum computers and networks look like increasingly inevitable extensions to our already astonishing classical computing and communication capabilities [1, 2]. How do they work, and once built, what capabilities will they bring? Quantum computation and communication can be understood through seven key con- cepts (see sidebar). Each concept is simple, but collectively they imply that our clas- sical notion of computation is incomplete, and that quantum effects can be used to efficiently solve some previously intractable problems. In the 1980s and 90s, a handful of algorithms were developed and the foundations of quantum computational complexity were laid, but the full range of capabilities and the process of creating new algorithms were poorly understood [1, 3, 4, 5, 6]. Over the last fifteen years, a deeper understanding of this process has developed, and the num- ber of proposed algorithms has exploded 1.