The SIC Question: History and State of Play
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Quantum Information
Quantum Information J. A. Jones Michaelmas Term 2010 Contents 1 Dirac Notation 3 1.1 Hilbert Space . 3 1.2 Dirac notation . 4 1.3 Operators . 5 1.4 Vectors and matrices . 6 1.5 Eigenvalues and eigenvectors . 8 1.6 Hermitian operators . 9 1.7 Commutators . 10 1.8 Unitary operators . 11 1.9 Operator exponentials . 11 1.10 Physical systems . 12 1.11 Time-dependent Hamiltonians . 13 1.12 Global phases . 13 2 Quantum bits and quantum gates 15 2.1 The Bloch sphere . 16 2.2 Density matrices . 16 2.3 Propagators and Pauli matrices . 18 2.4 Quantum logic gates . 18 2.5 Gate notation . 21 2.6 Quantum networks . 21 2.7 Initialization and measurement . 23 2.8 Experimental methods . 24 3 An atom in a laser field 25 3.1 Time-dependent systems . 25 3.2 Sudden jumps . 26 3.3 Oscillating fields . 27 3.4 Time-dependent perturbation theory . 29 3.5 Rabi flopping and Fermi's Golden Rule . 30 3.6 Raman transitions . 32 3.7 Rabi flopping as a quantum gate . 32 3.8 Ramsey fringes . 33 3.9 Measurement and initialisation . 34 1 CONTENTS 2 4 Spins in magnetic fields 35 4.1 The nuclear spin Hamiltonian . 35 4.2 The rotating frame . 36 4.3 On-resonance excitation . 38 4.4 Excitation phases . 38 4.5 Off-resonance excitation . 39 4.6 Practicalities . 40 4.7 The vector model . 40 4.8 Spin echoes . 41 4.9 Measurement and initialisation . 42 5 Photon techniques 43 5.1 Spatial encoding . -
Lecture Notes: Qubit Representations and Rotations
Phys 711 Topics in Particles & Fields | Spring 2013 | Lecture 1 | v0.3 Lecture notes: Qubit representations and rotations Jeffrey Yepez Department of Physics and Astronomy University of Hawai`i at Manoa Watanabe Hall, 2505 Correa Road Honolulu, Hawai`i 96822 E-mail: [email protected] www.phys.hawaii.edu/∼yepez (Dated: January 9, 2013) Contents mathematical object (an abstraction of a two-state quan- tum object) with a \one" state and a \zero" state: I. What is a qubit? 1 1 0 II. Time-dependent qubits states 2 jqi = αj0i + βj1i = α + β ; (1) 0 1 III. Qubit representations 2 A. Hilbert space representation 2 where α and β are complex numbers. These complex B. SU(2) and O(3) representations 2 numbers are called amplitudes. The basis states are or- IV. Rotation by similarity transformation 3 thonormal V. Rotation transformation in exponential form 5 h0j0i = h1j1i = 1 (2a) VI. Composition of qubit rotations 7 h0j1i = h1j0i = 0: (2b) A. Special case of equal angles 7 In general, the qubit jqi in (1) is said to be in a superpo- VII. Example composite rotation 7 sition state of the two logical basis states j0i and j1i. If References 9 α and β are complex, it would seem that a qubit should have four free real-valued parameters (two magnitudes and two phases): I. WHAT IS A QUBIT? iθ0 α φ0 e jqi = = iθ1 : (3) Let us begin by introducing some notation: β φ1 e 1 state (called \minus" on the Bloch sphere) Yet, for a qubit to contain only one classical bit of infor- 0 mation, the qubit need only be unimodular (normalized j1i = the alternate symbol is |−i 1 to unity) α∗α + β∗β = 1: (4) 0 state (called \plus" on the Bloch sphere) 1 Hence it lives on the complex unit circle, depicted on the j0i = the alternate symbol is j+i: 0 top of Figure 1. -
Introduction to Quantum Information and Computation
Introduction to Quantum Information and Computation Steven M. Girvin ⃝c 2019, 2020 [Compiled: May 2, 2020] Contents 1 Introduction 1 1.1 Two-Slit Experiment, Interference and Measurements . 2 1.2 Bits and Qubits . 3 1.3 Stern-Gerlach experiment: the first qubit . 8 2 Introduction to Hilbert Space 16 2.1 Linear Operators on Hilbert Space . 18 2.2 Dirac Notation for Operators . 24 2.3 Orthonormal bases for electron spin states . 25 2.4 Rotations in Hilbert Space . 29 2.5 Hilbert Space and Operators for Multiple Spins . 38 3 Two-Qubit Gates and Entanglement 43 3.1 Introduction . 43 3.2 The CNOT Gate . 44 3.3 Bell Inequalities . 51 3.4 Quantum Dense Coding . 55 3.5 No-Cloning Theorem Revisited . 59 3.6 Quantum Teleportation . 60 3.7 YET TO DO: . 61 4 Quantum Error Correction 63 4.1 An advanced topic for the experts . 68 5 Yet To Do 71 i Chapter 1 Introduction By 1895 there seemed to be nothing left to do in physics except fill out a few details. Maxwell had unified electriciy, magnetism and optics with his theory of electromagnetic waves. Thermodynamics was hugely successful well before there was any deep understanding about the properties of atoms (or even certainty about their existence!) could make accurate predictions about the efficiency of the steam engines powering the industrial revolution. By 1895, statistical mechanics was well on its way to providing a microscopic basis in terms of the random motions of atoms to explain the macroscopic predictions of thermodynamics. However over the next decade, a few careful observers (e.g. -
Studies in the Geometry of Quantum Measurements
Irina Dumitru Studies in the Geometry of Quantum Measurements Studies in the Geometry of Quantum Measurements Studies in Irina Dumitru Irina Dumitru is a PhD student at the department of Physics at Stockholm University. She has carried out research in the field of quantum information, focusing on the geometry of Hilbert spaces. ISBN 978-91-7911-218-9 Department of Physics Doctoral Thesis in Theoretical Physics at Stockholm University, Sweden 2020 Studies in the Geometry of Quantum Measurements Irina Dumitru Academic dissertation for the Degree of Doctor of Philosophy in Theoretical Physics at Stockholm University to be publicly defended on Thursday 10 September 2020 at 13.00 in sal C5:1007, AlbaNova universitetscentrum, Roslagstullsbacken 21, and digitally via video conference (Zoom). Public link will be made available at www.fysik.su.se in connection with the nailing of the thesis. Abstract Quantum information studies quantum systems from the perspective of information theory: how much information can be stored in them, how much the information can be compressed, how it can be transmitted. Symmetric informationally- Complete POVMs are measurements that are well-suited for reading out the information in a system; they can be used to reconstruct the state of a quantum system without ambiguity and with minimum redundancy. It is not known whether such measurements can be constructed for systems of any finite dimension. Here, dimension refers to the dimension of the Hilbert space where the state of the system belongs. This thesis introduces the notion of alignment, a relation between a symmetric informationally-complete POVM in dimension d and one in dimension d(d-2), thus contributing towards the search for these measurements. -
Quantifying Entanglement in a 68-Billion-Dimensional Quantum State Space
ARTICLE https://doi.org/10.1038/s41467-019-10810-z OPEN Quantifying entanglement in a 68-billion- dimensional quantum state space James Schneeloch 1,5, Christopher C. Tison1,2,3, Michael L. Fanto1,4, Paul M. Alsing1 & Gregory A. Howland 1,4,5 Entanglement is the powerful and enigmatic resource central to quantum information pro- cessing, which promises capabilities in computing, simulation, secure communication, and 1234567890():,; metrology beyond what is possible for classical devices. Exactly quantifying the entanglement of an unknown system requires completely determining its quantum state, a task which demands an intractable number of measurements even for modestly-sized systems. Here we demonstrate a method for rigorously quantifying high-dimensional entanglement from extremely limited data. We improve an entropic, quantitative entanglement witness to operate directly on compressed experimental data acquired via an adaptive, multilevel sampling procedure. Only 6,456 measurements are needed to certify an entanglement-of- formation of 7.11 ± .04 ebits shared by two spatially-entangled photons. With a Hilbert space exceeding 68 billion dimensions, we need 20-million-times fewer measurements than the uncompressed approach and 1018-times fewer measurements than tomography. Our tech- nique offers a universal method for quantifying entanglement in any large quantum system shared by two parties. 1 Air Force Research Laboratory, Information Directorate, Rome, NY 13441, USA. 2 Department of Physics, Florida Atlantic University, Boca Raton, FL 33431, USA. 3 Quanterion Solutions Incorporated, Utica, NY 13502, USA. 4 Rochester Institute of Technology, Rochester, NY 14623, USA. 5These authors contributed equally: James Schneeloch, Gregory A. Howland. Correspondence and requests for materials should be addressed to G.A.H. -
The Bloch Equations a Quick Review of Spin- 1 Conventions and Notation
Ph195a lecture notes, 11/21/01 The Bloch Equations 1 A quick review of spin- 2 conventions and notation 1 The quantum state of a spin- 2 particle is represented by a vector in a two-dimensional complex Hilbert space H2. Let | +z 〉 and | −z 〉 be eigenstates of the operator corresponding to component of spin along the z coordinate axis, ℏ S | + 〉 =+ | + 〉, z z 2 z ℏ S | − 〉 = − | − 〉. 1 z z 2 z In this basis, the operators corresponding to spin components projected along the z,y,x coordinate axes may be represented by the following matrices: ℏ ℏ 10 Sz = σz = , 2 2 0 −1 ℏ ℏ 01 Sx = σx = , 2 2 10 ℏ ℏ 0 −i Sy = σy = . 2 2 2 i 0 We commonly use | ±x 〉 to denote eigenstates of Sx, and similarly for Sy. The dimensionless matrices σx,σy,σz are known as the Pauli matrices, and satisfy the following commutation relations: σx,σy = 2iσz, σy,σz = 2iσx, σz,σx = 2iσy. 3 In addition, Trσiσj = 2δij, 4 and 2 2 2 σx = σy = σz = 1. 5 The Pauli matrices are both Hermitian and unitary. 1 An arbitrary state for an isolated spin- 2 particle may be written θ ϕ θ ϕ |Ψ 〉 = cos exp −i | + 〉 + sin exp i | − 〉, 6 2 2 z 2 2 z where θ and ϕ are real parameters that may be chosen in the ranges 0 ≤ θ ≤ π and 0 ≤ ϕ ≤ 2π. Note that there should really be four real degrees of freedom for a vector in a 1 two-dimensional complex Hilbert space, but one is removed by normalization and another because we don’t care about the overall phase of the state of an isolated quantum system. -
Spectral Quantum Tomography
www.nature.com/npjqi ARTICLE OPEN Spectral quantum tomography Jonas Helsen 1, Francesco Battistel1 and Barbara M. Terhal1,2 We introduce spectral quantum tomography, a simple method to extract the eigenvalues of a noisy few-qubit gate, represented by a trace-preserving superoperator, in a SPAM-resistant fashion, using low resources in terms of gate sequence length. The eigenvalues provide detailed gate information, supplementary to known gate-quality measures such as the gate fidelity, and can be used as a gate diagnostic tool. We apply our method to one- and two-qubit gates on two different superconducting systems available in the cloud, namely the QuTech Quantum Infinity and the IBM Quantum Experience. We discuss how cross-talk, leakage and non-Markovian errors affect the eigenvalue data. npj Quantum Information (2019) 5:74 ; https://doi.org/10.1038/s41534-019-0189-0 INTRODUCTION of the form λ = exp(−γ)exp(iϕ), contain information about the A central challenge on the path towards large-scale quantum quality of the implemented gate. Intuitively, the parameter γ computing is the engineering of high-quality quantum gates. To captures how much the noisy gate deviates from unitarity due to achieve this goal, many methods that accurately and reliably entanglement with an environment, while the angle ϕ can be characterize quantum gates have been developed. Some of these compared to the rotation angles of the targeted gate U. Hence ϕ methods are scalable, meaning that they require an effort which gives information about how much one over- or under-rotates. scales polynomially in the number of qubits on which the gates The spectrum of S can also be related to familiar gate-quality 1–8 act. -
Topological Complexity for Quantum Information
Topological Complexity for Quantum Information Zhengwei Liu Tsinghua University Joint with Arthur Jaffe, Xun Gao, Yunxiang Ren and Shengtao Wang Seminar at Dublin IAS, Oct 14, 2020 Z. Liu (Tsinghua University) Topological Complexity for QI Oct 14, 2020 1 / 31 Topological Complexity: The complexity of computing matrix products or contractions of tensors may grows exponentially using the state sum over the basis. Using topological ideas, such as knot-theoretical isotopy, one could compute the tensors in a more efficient way. Fractionalization: We open the \black box" in tensor network and explore \internal pictorial relations" using the 3D quon language, a fractionalization of tensor network. (Euler's formula, Yang-Baxter equation/relation, star-triangle equation, Kramer-Wannier duality, Jordan-Wigner transformation etc.) Applications: We show that two well-known efficiently classically simulable families, Clifford gates and matchgates, correspond to two kinds of topological complexities. Besides the two simulable families, we introduce a new method to design efficiently classically simulable tensor networks and new families of exactly solvable models. Z. Liu (Tsinghua University) Topological Complexity for QI Oct 14, 2020 2 / 31 Qubits and Gates 2 A qubit is a vector state in C . 2 n An n-qubit is a vector state jφi in (C ) . 2 n A n-qubit gate is a unitary on (C ) . Z. Liu (Tsinghua University) Topological Complexity for QI Oct 14, 2020 3 / 31 Quantum Simulation Feynman and Manin both proposed Quantum Simulation in 1980. Simulate a quantum process by local interactions on qubits. Present an n-qubit gate as a composition of 1-qubit gates and adjacent 2-qubits gates. -
Quantum Tomography of an Electron T Jullien, P Roulleau, B Roche, a Cavanna, Y Jin, Christian Glattli
Quantum tomography of an electron T Jullien, P Roulleau, B Roche, A Cavanna, Y Jin, Christian Glattli To cite this version: T Jullien, P Roulleau, B Roche, A Cavanna, Y Jin, et al.. Quantum tomography of an electron. Nature, Nature Publishing Group, 2014, 514, pp.603 - 607. 10.1038/nature13821. cea-01409215 HAL Id: cea-01409215 https://hal-cea.archives-ouvertes.fr/cea-01409215 Submitted on 5 Dec 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LETTER doi:10.1038/nature13821 Quantum tomography of an electron T. Jullien1*, P. Roulleau1*, B. Roche1, A. Cavanna2, Y. Jin2 & D. C. Glattli1 The complete knowledge of a quantum state allows the prediction of pbyffiffiffiffiffiffiffiffi mixing with a coherent field (local oscillator) with amplitude the probability of all possible measurement outcomes, a crucial step NLOE0utðÞ{x=c , where NLO is the mean photon number. Then a 1 in quantum mechanics. It can be provided by tomographic methods classical measurement of u can be made.p becauseffiffiffiffiffiffiffiffi the fundamentalp quan-ffiffiffiffiffiffiffiffi 2,3 4 5,6 which havebeenappliedtoatomic ,molecular , spin and photonic tum measurement uncertainty , 1 NLO vanishes for large NLO. -
QUANTUM STATE TOMOGRAPHY of SINGLE QUBIT with LASER IR 808Nm USING DENSITY MATRIX
QUANTUM STATE TOMOGRAPHY OF SINGLE QUBIT WITH LASER IR 808nm USING DENSITY MATRIX A BACHELOR THESIS Submitted in Partial Fulfillment the Requirements for Gaining the Bachelor Degree in Physics Education Department Author: SYAFI’I FAHMI BASTIAN 15690044 PHYSICS EDUCATION DEPARTMENT FACULTY OF SCIENCE AND TECHNOLOGY UNIVERSITAS ISLAM NEGERI SUNAN KALIJAGA YOGYAKARTA 2019 ii iii iv MOTTO Never give up! v DEDICATION Dedicated to: My beloved parents All of my families My almamater UIN Sunan Kalijaga Yogyakarta vi ACKNOWLEDGEMENT Alhamdulillahirabbil’alamin and greatest thanks. Allah SWT has given blessing to the writer by giving guidance, health, knowledge. The writer’s prayers are always given to Prophet Muhammad SAW as the messenger of Allah SWT. In This paper has been finished because all supports from everyone who help the writer. In this nice occasion, the writer would like to express the special thanks of gratitude to: 1. Dr. Murtono, M.Sc as Dean of faculty of science and technology, UIN Sunan Kalijaga Yogyakarta and as academic advisor. 2. Drs. Nur Untoro, M. Si, as head of physics education department, faculty of science and technology, UIN Sunan Kalijaga Yogyakarta. 3. Joko Purwanto, M.Sc as project advisor 1 who always gave a new knowledge and suggestions to the writer. 4. Dr. Pruet Kalasuwan as project advisor 2 who introduces and teaches a quantum physics especially about the nanophotonics when the writer enrolled the PSU Student Mobility Program. 5. Pi Mai as the writer’s lab partner who was so patient in giving the knowledge. 6. Norma Sidik Risdianto, M.Sc as a lecturer who giving many motivations to learn physics more and more. -
Appendix a Linear Algebra Basics
Appendix A Linear algebra basics A.1 Linear spaces Linear spaces consist of elements called vectors. Vectors are abstract mathematical objects, but, as the name suggests, they can be visualized as geometric vectors. Like regular numbers, vectors can be added together and subtracted from each other to form new vectors; they can also be multiplied by numbers. However, vectors cannot be multiplied or divided by one another as numbers can. One important peculiarity of the linear algebra used in quantum mechanics is the so-called Dirac notation for vectors. To denote vectors, instead of writing, for example, ~a, we write jai. We shall see later how convenient this notation turns out to be. Definition A.1. A linear (vector) space V over a field1 F is a set in which the following operations are defined: 1. Addition: for any two vectors jai;jbi 2 V, there exists a unique vector in V called their sum, denoted by jai + jbi. 2. Multiplication by a number (“scalar”): For any vector jai 2 V and any number l 2 F, there exists a unique vector in V called their product, denoted by l jai ≡ jail. These operations obey the following axioms. 1. Commutativity of addition: jai + jbi = jbi + jai. 2. Associativity of addition: (jai + jbi) + jci = jai + (jbi + jci). 3. Existence of zero: there exists an element of V called jzeroi such that, for any vector jai, jai + jzeroi= ja i.2 A solutions manual for this appendix is available for download at https://www.springer.com/gp/book/9783662565827 1 Field is a term from algebra which means a complete set of numbers. -
Quantum State Tomography with a Single Observable
Quantum State Tomography with a Single Observable Dikla Oren1, Maor Mutzafi1, Yonina C. Eldar2, Mordechai Segev1 1Physics Department and Solid State Institute, Technion, 32000 Haifa, Israel 2 Electrical Engineering Department, Technion, 32000 Haifa, Israel Quantum information has been drawing a wealth of research in recent years, shedding light on questions at the heart of quantum mechanics1–5, as well as advancing fields such as complexity theory6–10, cryptography6, key distribution11, and chemistry12. These fundamental and applied aspects of quantum information rely on a crucial issue: the ability to characterize a quantum state from measurements, through a process called Quantum State Tomography (QST). However, QST requires a large number of measurements, each derived from a different physical observable corresponding to a different experimental setup. Unfortunately, changing the setup results in unwanted changes to the data, prolongs the measurement and impairs the assumptions that are always made about the stationarity of the noise. Here, we propose to overcome these drawbacks by performing QST with a single observable. A single observable can often be realized by a single setup, thus considerably reducing the experimental effort. In general, measurements of a single observable do not hold enough information to recover the quantum state. We overcome this lack of information by relying on concepts inspired by Compressed Sensing (CS)13,14, exploiting the fact that the sought state – in many applications of quantum information - is close to a pure state (and thus has low rank). Additionally, we increase the system dimension by adding an ancilla that couples to information evolving in the system, thereby providing more measurements, enabling the recovery of the original quantum state from a single-observable measurements.