Restricted Versions of the Two-Local Hamiltonian Problem
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Quantum Computer: Quantum Model and Reality
Quantum Computer: Quantum Model and Reality Vasil Penchev, [email protected] Bulgarian Academy of Sciences: Institute of Philosophy and Sociology: Dept. of Logical Systems and Models Abstract. Any computer can create a model of reality. The hypothesis that quantum computer can generate such a model designated as quantum, which coincides with the modeled reality, is discussed. Its reasons are the theorems about the absence of “hidden variables” in quantum mechanics. The quantum modeling requires the axiom of choice. The following conclusions are deduced from the hypothesis. A quantum model unlike a classical model can coincide with reality. Reality can be interpreted as a quantum computer. The physical processes represent computations of the quantum computer. Quantum information is the real fundament of the world. The conception of quantum computer unifies physics and mathematics and thus the material and the ideal world. Quantum computer is a non-Turing machine in principle. Any quantum computing can be interpreted as an infinite classical computational process of a Turing machine. Quantum computer introduces the notion of “actually infinite computational process”. The discussed hypothesis is consistent with all quantum mechanics. The conclusions address a form of neo-Pythagoreanism: Unifying the mathematical and physical, quantum computer is situated in an intermediate domain of their mutual transformations. Key words: model, quantum computer, reality, Turing machine Eight questions: There are a few most essential questions about the philosophical interpretation of quantum computer. They refer to the fundamental problems in ontology and epistemology rather than philosophy of science or that of information and computation only. The contemporary development of quantum mechanics and the theory of quantum information generate them. -
The Complexity Zoo
The Complexity Zoo Scott Aaronson www.ScottAaronson.com LATEX Translation by Chris Bourke [email protected] 417 classes and counting 1 Contents 1 About This Document 3 2 Introductory Essay 4 2.1 Recommended Further Reading ......................... 4 2.2 Other Theory Compendia ............................ 5 2.3 Errors? ....................................... 5 3 Pronunciation Guide 6 4 Complexity Classes 10 5 Special Zoo Exhibit: Classes of Quantum States and Probability Distribu- tions 110 6 Acknowledgements 116 7 Bibliography 117 2 1 About This Document What is this? Well its a PDF version of the website www.ComplexityZoo.com typeset in LATEX using the complexity package. Well, what’s that? The original Complexity Zoo is a website created by Scott Aaronson which contains a (more or less) comprehensive list of Complexity Classes studied in the area of theoretical computer science known as Computa- tional Complexity. I took on the (mostly painless, thank god for regular expressions) task of translating the Zoo’s HTML code to LATEX for two reasons. First, as a regular Zoo patron, I thought, “what better way to honor such an endeavor than to spruce up the cages a bit and typeset them all in beautiful LATEX.” Second, I thought it would be a perfect project to develop complexity, a LATEX pack- age I’ve created that defines commands to typeset (almost) all of the complexity classes you’ll find here (along with some handy options that allow you to conveniently change the fonts with a single option parameters). To get the package, visit my own home page at http://www.cse.unl.edu/~cbourke/. -
Arxiv:1506.08857V1 [Quant-Ph] 29 Jun 2015
Absolutely Maximally Entangled states, combinatorial designs and multi-unitary matrices Dardo Goyeneche National Quantum Information Center of Gda´nsk,81-824 Sopot, Poland and Faculty of Applied Physics and Mathematics, Technical University of Gda´nsk,80-233 Gda´nsk,Poland Daniel Alsina Dept. Estructura i Constituents de la Mat`eria,Universitat de Barcelona, Spain. Jos´e I. Latorre Dept. Estructura i Constituents de la Mat`eria,Universitat de Barcelona, Spain. and Center for Theoretical Physics, MIT, USA Arnau Riera ICFO-Institut de Ciencies Fotoniques, Castelldefels (Barcelona), Spain Karol Zyczkowski_ Institute of Physics, Jagiellonian University, Krak´ow,Poland and Center for Theoretical Physics, Polish Academy of Sciences, Warsaw, Poland (Dated: June 29, 2015) Absolutely Maximally Entangled (AME) states are those multipartite quantum states that carry absolute maximum entanglement in all possible partitions. AME states are known to play a relevant role in multipartite teleportation, in quantum secret sharing and they provide the basis novel tensor networks related to holography. We present alternative constructions of AME states and show their link with combinatorial designs. We also analyze a key property of AME, namely their relation to tensors that can be understood as unitary transformations in every of its bi-partitions. We call this property multi-unitarity. I. INTRODUCTION entropy S(ρ) = −Tr(ρ log ρ) ; (1) A complete characterization, classification and it is possible to show [1, 2] that the average entropy quantification of entanglement for quantum states re- of the reduced state σ = TrN=2j ih j to N=2 qubits mains an unfinished long-term goal in Quantum Infor- reads: N mation theory. -
Non-Unitary Quantum Computation in the Ground Space of Local Hamiltonians
Non-Unitary Quantum Computation in the Ground Space of Local Hamiltonians Na¨ıri Usher,1, ∗ Matty J. Hoban,2, 3 and Dan E. Browne1 1Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom. 2University of Oxford, Department of Computer Science, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom. 3School of Informatics, University of Edinburgh, 10 Crichton Street, Edinburgh EH8 9AB, UK A central result in the study of Quantum Hamiltonian Complexity is that the k-local hamilto- nian problem is QMA-complete [1]. In that problem, we must decide if the lowest eigenvalue of a Hamiltonian is bounded below some value, or above another, promised one of these is true. Given the ground state of the Hamiltonian, a quantum computer can determine this question, even if the ground state itself may not be efficiently quantum preparable. Kitaev's proof of QMA-completeness encodes a unitary quantum circuit in QMA into the ground space of a Hamiltonian. However, we now have quantum computing models based on measurement instead of unitary evolution, further- more we can use post-selected measurement as an additional computational tool. In this work, we generalise Kitaev's construction to allow for non-unitary evolution including post-selection. Fur- thermore, we consider a type of post-selection under which the construction is consistent, which we call tame post-selection. We consider the computational complexity consequences of this construc- tion and then consider how the probability of an event upon which we are post-selecting affects the gap between the ground state energy and the energy of the first excited state of its corresponding Hamiltonian. -
CS286.2 Lectures 5-6: Introduction to Hamiltonian Complexity, QMA-Completeness of the Local Hamiltonian Problem
CS286.2 Lectures 5-6: Introduction to Hamiltonian Complexity, QMA-completeness of the Local Hamiltonian problem Scribe: Jenish C. Mehta The Complexity Class BQP The complexity class BQP is the quantum analog of the class BPP. It consists of all languages that can be decided in quantum polynomial time. More formally, Definition 1. A language L 2 BQP if there exists a classical polynomial time algorithm A that ∗ maps inputs x 2 f0, 1g to quantum circuits Cx on n = poly(jxj) qubits, where the circuit is considered a sequence of unitary operators each on 2 qubits, i.e Cx = UTUT−1...U1 where each 2 2 Ui 2 L C ⊗ C , such that: 2 i. Completeness: x 2 L ) Pr(Cx accepts j0ni) ≥ 3 1 ii. Soundness: x 62 L ) Pr(Cx accepts j0ni) ≤ 3 We say that the circuit “Cx accepts jyi” if the first output qubit measured in Cxjyi is 0. More j0i specifically, letting P1 = j0ih0j1 be the projection of the first qubit on state j0i, j0i 2 Pr(Cx accepts jyi) =k (P1 ⊗ In−1)Cxjyi k2 The Complexity Class QMA The complexity class QMA (or BQNP, as Kitaev originally named it) is the quantum analog of the class NP. More formally, Definition 2. A language L 2 QMA if there exists a classical polynomial time algorithm A that ∗ maps inputs x 2 f0, 1g to quantum circuits Cx on n + q = poly(jxj) qubits, such that: 2q i. Completeness: x 2 L ) 9jyi 2 C , kjyik2 = 1, such that Pr(Cx accepts j0ni ⊗ 2 jyi) ≥ 3 2q 1 ii. -
STRONGLY UNIVERSAL QUANTUM TURING MACHINES and INVARIANCE of KOLMOGOROV COMPLEXITY (AUGUST 11, 2007) 2 Such That the Aforementioned Halting Conditions Are Satisfied
STRONGLY UNIVERSAL QUANTUM TURING MACHINES AND INVARIANCE OF KOLMOGOROV COMPLEXITY (AUGUST 11, 2007) 1 Strongly Universal Quantum Turing Machines and Invariance of Kolmogorov Complexity Markus M¨uller Abstract—We show that there exists a universal quantum Tur- For a compact presentation of the results by Bernstein ing machine (UQTM) that can simulate every other QTM until and Vazirani, see the book by Gruska [4]. Additional rele- the other QTM has halted and then halt itself with probability one. vant literature includes Ozawa and Nishimura [5], who gave This extends work by Bernstein and Vazirani who have shown that there is a UQTM that can simulate every other QTM for necessary and sufficient conditions that a QTM’s transition an arbitrary, but preassigned number of time steps. function results in unitary time evolution. Benioff [6] has As a corollary to this result, we give a rigorous proof that worked out a slightly different definition which is based on quantum Kolmogorov complexity as defined by Berthiaume et a local Hamiltonian instead of a local transition amplitude. al. is invariant, i.e. depends on the choice of the UQTM only up to an additive constant. Our proof is based on a new mathematical framework for A. Quantum Turing Machines and their Halting Conditions QTMs, including a thorough analysis of their halting behaviour. Our discussion will rely on the definition by Bernstein and We introduce the notion of mutually orthogonal halting spaces and show that the information encoded in an input qubit string Vazirani. We describe their model in detail in Subsection II-B. -
6.845 Quantum Complexity Theory, Lecture 16
6.896 Quantum Complexity Theory Oct. 28, 2008 Lecture 16 Lecturer: Scott Aaronson 1 Recap Last time we introduced the complexity class QMA (quantum Merlin-Arthur), which is a quantum version for NP. In particular, we have seen Watrous’s Group Non-Membership (GNM) protocol which enables a quantum Merlin to prove to a quantum Arthur that some element x of a group G is not a member of H, a subgroup of G. Notice that this is a problem that people do not know how to solve in classical world. To understand the limit of QMA, we have proved that QMA ⊆ P P (and also QMA ⊆ P SP ACE). We have also talked about QMA-complete problems, which are the quantum version of NP complete problem. In particular we have introduced the Local Hamiltonians problem, which is the quantum version of the SAT problem. We also have a quantum version of the Cook-Levin theorem, due to Kitaev, saying that the Local Hamiltonians problem is QMA complete. We will prove Kitaev’s theorem in this lecture. 2 Local Hamiltonians is QMA-complete De¯nition 1 (Local Hamiltonians Problem) Given m measurements E1; : : : ; Em each of which acts on at most k (out of n) qubits where k is a constant, the Local Hamiltonians problem is to decide which of the following two statements is true, promised that one is true: Pm 1. 9 an n-qubit state j'i such that i=1 Pr[Ei accepts j'i] ¸ b; or Pm 2. 8 n-qubit state j'i, i=1 Pr[Ei accepts j'i] · a. -
Arxiv:Quant-Ph/0612113V1 14 Dec 2006 † ∗ Nalo Hs Rosi Euto Ewe Unu Icisadtim and Circuits Quantum C Between Reduction QMA a Is Is Hamiltonian Complete
A new construction for a QMA complete 3-local Hamiltonian. 1, , Daniel Nagaj ∗ and Shay Mozes † 1Center for Theoretical Physics, MIT, Cambridge, MA 02139 (Dated: December 12, 2006) We present a new way of encoding a quantum computation into a 3-local Hamiltonian. Our construction is novel in that it does not include any terms that induce legal-illegal clock transitions. Therefore, the weights of the terms in the Hamiltonian do not scale with the size of the problem as in previous constructions. This improves the construction by Kempe and Regev [2], who were the first to prove that 3-local Hamiltonian is complete for the complexity class QMA, the quantum analogue of NP. Quantum k-SAT, a restricted version of the local Hamiltonian problem using only projector terms, was introduced by Bravyi [1] as an analogue of the classical k-SAT problem. Bravyi proved that quantum 4-SAT is complete for the class QMA with one-sided error (QMA1) and that quantum 2-SAT is in P. We give an encoding of a quantum circuit into a quantum 4-SAT Hamiltonian using only 3-local terms. As an intermediate step to this 3-local construction, we show that quantum 3-SAT for particles with dimensions 3 × 2 × 2 (a qutrit and two qubits) is QMA1 complete. The complexity of quantum 3-SAT with qubits remains an open question. I. INTRODUCTION In recent years, quantum complexity classes have been defined and studied in an attempt to understand the capacity and limitations of quantum computers and quantum algorithms and their relation to classical complexity classes. -
On the Simulation of Quantum Turing Machines
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 304 (2003) 103–128 www.elsevier.com/locate/tcs On the simulation ofquantum turing machines Marco Carpentieri Universita di Salerno, 84081 Baronissi (SA), Italy Received 20 March 2002; received in revised form 11 December 2002; accepted 12 December 2002 Communicated by G. Ausiello Abstract In this article we shall review several basic deÿnitions and results regarding quantum compu- tation. In particular, after deÿning Quantum Turing Machines and networks the paper contains an exposition on continued fractions and on errors in quantum networks. The topic of simulation ofQuantum Turing Machines by means ofobvious computation is introduced. We give a full discussion ofthe simulation ofmultitape Quantum Turing Machines in a slight generalization of the class introduced by Bernstein and Vazirani. As main result we show that the Fisher-Pippenger technique can be used to give an O(tlogt) simulation ofa multi-tape Quantum Turing Machine by another belonging to the extended Bernstein and Vazirani class. This result, even ifregarding a slightly restricted class ofQuantum Turing Machines improves the simulation results currently known in the literature. c 2003 Published by Elsevier B.V. 1. Introduction Even ifthe present technology does consent realizing only very simple devices based on the principles ofquantum mechanics, many authors considered to be worth it ask- ing whether a theoretical model ofquantum computation could o:er any substantial beneÿts over the correspondent theoretical model based on the assumptions ofclas- sical physics. Recently, this question has received considerable attention because of the growing beliefthat quantum mechanical processes might be able to performcom- putation that traditional computing machines can only perform ine;ciently. -
Adiabatic Quantum Computation Is Equivalent To
SIAM REVIEW c 2008 Society for Industrial and Applied Mathematics Vol. 50, No. 4, pp. 755–787 Adiabatic Quantum Computation Is Equivalent to ∗ Standard Quantum Computation Dorit Aharonov† Wim van Dam‡ Julia Kempe§ Zeph Landau¶ Seth Lloyd Oded Regev§ Abstract. The model of adiabatic quantum computation is a relatively recent model of quantum com- putation that has attracted attention in the physics and computer science communities. We describe an efficient adiabatic simulation of any given quantum circuit. This implies that the adiabatic computation model and the standard circuit-based quantum compu- tation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a two-dimensional grid with nearest neighbor inter- actions. The equivalence between the models allows one to state the main open problems in quantum computation using well-studied mathematical objects such as eigenvectors and spectral gaps of Hamiltonians. Key words. quantum computation,adiabatic computation,nearest neighbor interactions AMS subject classifications. 81P68,68Q05 DOI. 10.1137/080734479 1. Introduction. The model of quantum computation has been thoroughly in- vestigated in the last two decades and is by now a well-established one [47]. In this ∗ Published electronically November 5,2008. This paper originally appeared in SIAM Journal on Computing,Volume 37,Number 1,2007,pages 166–194. This work was performed by an em- ployee of the U.S. Government or under U.S. Government contract. The U.S. Government retains a nonexclusive,royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so,for U.S. -
A Quantum Computer Foundation for the Standard Model and Superstring Theories*
A Quantum Computer Foundation for the Standard Model and SuperString Theories* By Stephen Blaha** * Excerpted from the book Cosmos and Consciousness by Stephen Blaha (1stBooks Library, Bloomington, IN, 2000) available from amazon.com and bn.com. ** Associate of the Harvard University Physics Department. ABSTRACT 1. SuperString Theory can naturally be based on a Quantum Computer foundation. This provides a totally new view of SuperString Theory. 2. The Standard Model of elementary particles can be viewed as defining a Quantum Computer Grammar and language. 3. A Quantum Computer can be represented in part as a second-quantized Fermi field. 4. A Quantum Computer in a certain limit naturally forms a Superspace upon which Supersymmetry rotations can be defined – a Continuum Quantum Computer. 5. A representation of Quantum Computers exists that is similar to Turing Machines - a Quantum Turing Machine. As part of this development we define various types of Quantum Grammars. 6. High level Quantum Computer languages are described for the first time. New linguistic views of the most fundamental theories of Physics, the Standard Model and SuperString Theory are described. In these new linguistic representations particles become literally symbols or letters, and particle interactions become grammar rules. This view is NOT the same as the often-expressed view that Mathematics is the language of Physics. The linguistic representation is a specific new mathematical construct. We show how to create a SuperString Quantum Computer that naturally provides a framework for SuperStrings in general and heterotic SuperStrings in particular. There are also a number of new developments relating to Quantum Computers and Quantum Turing Machines that are of interest to Computer Science. -
Quantum Computing : a Gentle Introduction / Eleanor Rieffel and Wolfgang Polak
QUANTUM COMPUTING A Gentle Introduction Eleanor Rieffel and Wolfgang Polak The MIT Press Cambridge, Massachusetts London, England ©2011 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. For information about special quantity discounts, please email [email protected] This book was set in Syntax and Times Roman by Westchester Book Group. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Rieffel, Eleanor, 1965– Quantum computing : a gentle introduction / Eleanor Rieffel and Wolfgang Polak. p. cm.—(Scientific and engineering computation) Includes bibliographical references and index. ISBN 978-0-262-01506-6 (hardcover : alk. paper) 1. Quantum computers. 2. Quantum theory. I. Polak, Wolfgang, 1950– II. Title. QA76.889.R54 2011 004.1—dc22 2010022682 10987654321 Contents Preface xi 1 Introduction 1 I QUANTUM BUILDING BLOCKS 7 2 Single-Qubit Quantum Systems 9 2.1 The Quantum Mechanics of Photon Polarization 9 2.1.1 A Simple Experiment 10 2.1.2 A Quantum Explanation 11 2.2 Single Quantum Bits 13 2.3 Single-Qubit Measurement 16 2.4 A Quantum Key Distribution Protocol 18 2.5 The State Space of a Single-Qubit System 21 2.5.1 Relative Phases versus Global Phases 21 2.5.2 Geometric Views of the State Space of a Single Qubit 23 2.5.3 Comments on General Quantum State Spaces