Realization of a Quantum Integer-Spin Chain with Controllable Interactions
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Dynamical Control of Matter Waves in Optical Lattices Sune Schøtt
Aarhus University Faculty of Science Department of Physics and Astronomy PhD Thesis Dynamical Control of Matter Waves in Optical Lattices by Sune Schøtt Mai November 2010 This thesis has been submitted to the Faculty of Science at Aarhus Uni- versity in order to fulfill the requirements for obtaining a PhD degree in physics. The work has been carried out under the supervision of profes- sor Klaus Mølmer and associate professor Jan Arlt at the Department of Physics and Astronomy. Contents Contents i 1 Introduction 3 1.1 Thesis Outline .......................... 3 1.2 Bose-Einstein Condensation .................. 4 2 Experimental Setup and Methods 9 2.1 Overview ............................. 9 2.2 Magneto-Optic Trap and Optical Pumping .......... 9 2.3 Transport with Movable Quadrupole Traps . 13 2.4 Evaporative Cooling ...................... 14 2.5 Absorption Imaging ....................... 15 3 Optical Lattices 17 3.1 Introduction ........................... 17 3.2 AC Stark-shift Induced Potentials . 18 3.2.1 Classical and Semi-Classical Approaches . 18 3.2.2 Dressed State Picture . 20 3.3 Lattice Band Structure ..................... 22 3.3.1 Reciprocal Space Bloch Theorem . 23 3.3.2 The 1D Lattice Band-Structure . 23 4 Lattice Calibration Techniques 27 4.1 Introduction ........................... 27 4.2 Kapitza-Dirac Scattering .................... 28 4.3 Bloch Oscillation and LZ-Tunneling . 30 4.3.1 Bloch Oscillation .................... 31 4.3.2 Landau-Zener Theory . 31 i ii CONTENTS 4.3.3 Experimental Verification of the Landau-Zener Model 32 4.4 Lattice Modulation ....................... 35 4.5 Summary ............................ 35 5 Dynamically Controlled Lattices 39 5.1 Introduction ........................... 39 5.2 Overview ............................. 39 5.3 Controlled Matter Wave Beam Splitter . -
Simulating Quantum Field Theory with a Quantum Computer
Simulating quantum field theory with a quantum computer John Preskill Lattice 2018 28 July 2018 This talk has two parts (1) Near-term prospects for quantum computing. (2) Opportunities in quantum simulation of quantum field theory. Exascale digital computers will advance our knowledge of QCD, but some challenges will remain, especially concerning real-time evolution and properties of nuclear matter and quark-gluon plasma at nonzero temperature and chemical potential. Digital computers may never be able to address these (and other) problems; quantum computers will solve them eventually, though I’m not sure when. The physics payoff may still be far away, but today’s research can hasten the arrival of a new era in which quantum simulation fuels progress in fundamental physics. Frontiers of Physics short distance long distance complexity Higgs boson Large scale structure “More is different” Neutrino masses Cosmic microwave Many-body entanglement background Supersymmetry Phases of quantum Dark matter matter Quantum gravity Dark energy Quantum computing String theory Gravitational waves Quantum spacetime particle collision molecular chemistry entangled electrons A quantum computer can simulate efficiently any physical process that occurs in Nature. (Maybe. We don’t actually know for sure.) superconductor black hole early universe Two fundamental ideas (1) Quantum complexity Why we think quantum computing is powerful. (2) Quantum error correction Why we think quantum computing is scalable. A complete description of a typical quantum state of just 300 qubits requires more bits than the number of atoms in the visible universe. Why we think quantum computing is powerful We know examples of problems that can be solved efficiently by a quantum computer, where we believe the problems are hard for classical computers. -
Scalability and High-Efficiency of an $(N+ 1) $-Qubit Toffoli Gate Sphere
Scalability and high-efficiency of an (n + 1)-qubit Toffoli gate sphere via blockaded Rydberg atoms Dongmin Yu1, Yichun Gao1, Weiping Zhang2,3, Jinming Liu1 and Jing Qian1,3,† 1State Key Laboratory of Precision Spectroscopy, Department of Physics, School of Physics and Electronic Science, East China Normal University, Shanghai 200062, China 2Department of Physics and Astronomy, Shanghai Jiaotong University and Tsung-Dao Lee Institute, Shanghai 200240, China and 3Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China∗ The Toffoli gate serving as a basic building block for reversible quantum computation, has man- ifested its great potentials in improving the error-tolerant rate in quantum communication. While current route to the creation of Toffoli gate requires implementing sequential single- and two-qubit gates, limited by longer operation time and lower average fidelity. We develop a new theoretical protocol to construct a universal (n + 1)-qubit Toffoli gate sphere based on the Rydberg blockade mechanism, by constraining the behavior of one central target atom with n surrounding control atoms. Its merit lies in the use of only five π pulses independent of the control atom number n which leads to the overall gate time as fast as 125ns and the average fidelity closing to 0.999. The maximal filling number of control atoms can be∼ up to n = 46, determined by the spherical diame- ter which is equal to the blockade radius, as well as by the nearest neighbor spacing between two trapped-atom lattices. Taking n = 2, 3, 4 as examples we comparably show the gate performance with experimentally accessible parameters, and confirm that the gate errors mainly attribute to the imperfect blockade strength, the spontaneous atomic loss and the imperfect ground-state prepa- ration. -
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. -
A Scanning Transmon Qubit for Strong Coupling Circuit Quantum Electrodynamics
ARTICLE Received 8 Mar 2013 | Accepted 10 May 2013 | Published 7 Jun 2013 DOI: 10.1038/ncomms2991 A scanning transmon qubit for strong coupling circuit quantum electrodynamics W. E. Shanks1, D. L. Underwood1 & A. A. Houck1 Like a quantum computer designed for a particular class of problems, a quantum simulator enables quantitative modelling of quantum systems that is computationally intractable with a classical computer. Superconducting circuits have recently been investigated as an alternative system in which microwave photons confined to a lattice of coupled resonators act as the particles under study, with qubits coupled to the resonators producing effective photon–photon interactions. Such a system promises insight into the non-equilibrium physics of interacting bosons, but new tools are needed to understand this complex behaviour. Here we demonstrate the operation of a scanning transmon qubit and propose its use as a local probe of photon number within a superconducting resonator lattice. We map the coupling strength of the qubit to a resonator on a separate chip and show that the system reaches the strong coupling regime over a wide scanning area. 1 Department of Electrical Engineering, Princeton University, Olden Street, Princeton 08550, New Jersey, USA. Correspondence and requests for materials should be addressed to W.E.S. (email: [email protected]). NATURE COMMUNICATIONS | 4:1991 | DOI: 10.1038/ncomms2991 | www.nature.com/naturecommunications 1 & 2013 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms2991 ver the past decade, the study of quantum physics using In this work, we describe a scanning superconducting superconducting circuits has seen rapid advances in qubit and demonstrate its coupling to a superconducting CPWR Osample design and measurement techniques1–3. -
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. -
Chapter 5. Atoms in Optical Lattices
Strongly correlated systems in atomic and condensed matter physics Lecture notes for Physics 284 by Eugene Demler Harvard University September 18, 2014 2 Chapter 5 Atoms in optical lattices Optical lattices provide a powerful tool for creating strongly correlated many- body systems of ultracold atoms. By choosing different lattice geometries one can obtain very different single particle dispersions. The ratio of the interaction and kinetic energies can be controlled by tuning the depth of the lattice. 5.1 Noninteracting particles in optical lattices The simplest possible periodic optical potential is formed by overlapping two counter-propagating beams. Electric field in the resulting standing wave is E(z) = E0 sin(kz + θ) cos !t (5.1) Here k = 2π/λ is the wavevector of the laser light. Following the general recipe for AC Stark effects, we calculate electric dipolar moments induced by this field in the atoms, calculate interaction between dipolar moments and the electric field, and average over fast optical oscillations (see chapter ??). The result is the potential 2 V (z) = −V0 sin (kz + θ) (5.2) 2 where V0 = α(!)E0 =2, with α(!) being polarizability. It is common to express 2 2 V0 in units of the recoil energy Er = ~ k =2m. In real experiments one also needs to take into account the transverse profile of the beam. Hence V (r?; z) = exp{−2r2=w2(z)g×V (z). In most experiments the main effect of the transverse profile is only to renormalize the parabolic confining potentia. Combining three perpendicular sets of standing waves we get a simple cubic lattice V (r) = −Vx 0 cosqxx − Vy 0 cosqyy − Vz 0 cosqzz (5.3) 3 4 CHAPTER 5. -
Veselago Lensing with Ultracold Atoms in an Optical Lattice
ARTICLE Received 9 Dec 2013 | Accepted 27 Jan 2014 | Published 14 Feb 2014 DOI: 10.1038/ncomms4327 Veselago lensing with ultracold atoms in an optical lattice Martin Leder1, Christopher Grossert1 & Martin Weitz1 Veselago pointed out that electromagnetic wave theory allows for materials with a negative index of refraction, in which most known optical phenomena would be reversed. A slab of such a material can focus light by negative refraction, an imaging technique strikingly different from conventional positive refractive index optics, where curved surfaces bend the rays to form an image of an object. Here we demonstrate Veselago lensing for matter waves, using ultracold atoms in an optical lattice. A relativistic, that is, photon-like, dispersion relation for rubidium atoms is realized with a bichromatic optical lattice potential. We rely on a Raman p-pulse technique to transfer atoms between two different branches of the dispersion relation, resulting in a focusing that is completely analogous to the effect described by Veselago for light waves. Future prospects of the demonstrated effects include novel sub-de Broglie wavelength imaging applications. 1 Institut fu¨r Angewandte Physik der Universita¨t Bonn, Wegelerstrae 8, 53115 Bonn, Germany. Correspondence and requests for materials should be addressed to M.W. (email: [email protected]). NATURE COMMUNICATIONS | 5:3327 | DOI: 10.1038/ncomms4327 | www.nature.com/naturecommunications 1 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4327 eselago lensing is a concept based on negative of spatial periodicity l/4, generated by the dispersion of multi- refraction1–3, where a spatially diverging pencil of rays photon Raman transitions17. -
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. -
Restricted Versions of the Two-Local Hamiltonian Problem
The Pennsylvania State University The Graduate School Eberly College of Science RESTRICTED VERSIONS OF THE TWO-LOCAL HAMILTONIAN PROBLEM A Dissertation in Physics by Sandeep Narayanaswami c 2013 Sandeep Narayanaswami Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2013 The dissertation of Sandeep Narayanaswami was reviewed and approved* by the following: Sean Hallgren Associate Professor of Computer Science and Engineering Dissertation Adviser, Co-Chair of Committee Nitin Samarth Professor of Physics Head of the Department of Physics Co-Chair of Committee David S Weiss Professor of Physics Jason Morton Assistant Professor of Mathematics *Signatures are on file in the Graduate School. Abstract The Hamiltonian of a physical system is its energy operator and determines its dynamics. Un- derstanding the properties of the ground state is crucial to understanding the system. The Local Hamiltonian problem, being an extension of the classical Satisfiability problem, is thus a very well-motivated and natural problem, from both physics and computer science perspectives. In this dissertation, we seek to understand special cases of the Local Hamiltonian problem in terms of algorithms and computational complexity. iii Contents List of Tables vii List of Tables vii Acknowledgments ix 1 Introduction 1 2 Background 6 2.1 Classical Complexity . .6 2.2 Quantum Computation . .9 2.3 Generalizations of SAT . 11 2.3.1 The Ising model . 13 2.3.2 QMA-complete Local Hamiltonians . 13 2.3.3 Projection Hamiltonians, or Quantum k-SAT . 14 2.3.4 Commuting Local Hamiltonians . 14 2.3.5 Other special cases . 15 2.3.6 Approximation Algorithms and Heuristics . -
Ultracold Atoms in a Disordered Optical Lattice
ULTRACOLD ATOMS IN A DISORDERED OPTICAL LATTICE BY MATTHEW ROBERT WHITE B.S., University of California, Santa Barbara, 2003 DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 2009 Urbana, Illinois Doctoral Committee: Professor Paul Kwiat, Chair Assistant Professor Brian DeMarco, Director of Research Assistant Professor Raffi Budakian Professor David Ceperley Acknowledgments This work would not have been possible without the support of my advisor Brian DeMarco and coworkers Matt Pasienski, David McKay, Hong Gao, Stanimir Kondov, David Chen, William McGehee, Matt Brinkley, Lauren Aycock, Cecilia Borries, Soheil Baharian, Sarah Gossett, Minsu Kim, and Yutaka Miyagawa. Generous funding was provided by the Uni- versity of Illinois, NSF, ARO, and ONR. ii Table of Contents List of Figures..................................... v Chapter 1 Introduction .............................. 1 1.1 Strongly-interacting Boson Systems.......................1 1.2 Bose-Hubbard Model...............................2 1.3 Disordered Bose-Hubbard Model with Cold Atoms..............3 1.4 Observations...................................4 1.5 Future Work on the DBH Model........................5 1.6 Quantum Simulation...............................5 1.7 Outline......................................6 Chapter 2 BEC Apparatus............................ 8 2.1 Introduction....................................8 2.2 Atomic Properties................................8