# Quantum Algorithms in NMR Experiments. 25 th May 2012 Ling LIN & Michael Loretz

Save this PDF as:

Size: px
Start display at page:

Download "Quantum Algorithms in NMR Experiments. 25 th May 2012 Ling LIN & Michael Loretz"

## Transcription

1 Quantum Algorithms in NMR Experiments 25 th May 2012 Ling LIN & Michael Loretz

2 Contents 1. Introduction 2. Shor s algorithm 3. NMR quantum computer Nuclear spin qubits in a molecule NMR principles 4. Implementing the Shor algorithm Factoring Conclusion and Outlook Quantum Systems for Information Technology FS

3 Introduction to Shor s algorithm Given an integer N, find its prime factors. Classically, prime factorization takes O(exp[1.9(log N) 1/3 (log log N) 2/3 ]) operations. On a quantum computer, it runs in polynomial time, taking only O((log N) 3 ) operations. Much faster! With a sufficient number of qubits, Shor s algorithm can be used to break public-key cryptography schemes such as the widely used RSA scheme. Shor s algorithm consists of two parts: Classical part: A reduction of the factoring problem to the order-finding problem, which can be done on a classical computer. Quantum part: A quantum algorithm is used to solve the order-finding problem. Quantum Systems for Information Technology FS

4 Shor s algorithm (classical part) Task: Given an integer N, find its prime factors Calculate a x mod N = 1; find period r (order of a) of fn,a(x)=a x mod N Procedure: a 2 mod N = 1 <=> (a + 1)(a - 1) = 0 mod N If neither (a + 1) nor (a - 1) multiple of N, than at least one factor of N is in (a + 1) and also in (a - 1) Algorithm: Choose a ϵ {2,..., N - 1} y := gcd(a, N), check y = 1 r := ord(a), check r = 2k z := max{gcd(ar/2-1, N), gcd(ar/2 + 1, N)} Quantum Systems for Information Technology FS

5 Shor s algorithm (classical part) Task: Given an integer N, find its prime factors Calculate a x mod N = 1; find period r (order of a) of fn,a(x)=a x mod N Procedure: a 2 mod N = 1 <=> (a + 1)(a - 1) = 0 mod N If neither (a + 1) nor (a - 1) multiple of N, than at least one factor of N is in (a + 1) and also in (a - 1) Algorithm: Choose a ϵ {2,..., N - 1} y := gcd(a, N), check y = 1 r := ord(a), check r = 2k z := max{gcd(a r/2-1, N), gcd(a r/2 + 1, N)} Example: to factorize 21: Choose a=13 gcd(13,21)=1 ok. ord(13): 13 2 mod 21 =1 => r=2 ok. gcd(12,21)=3, gcd(14,21)=7 => 21=3*7 Quantum Systems for Information Technology FS

6 Shor s algorithm (quantum part) 1.Initialization Q1 Q 1/2 x 0 x0 2.Construction Q1 Q 1/2 x f (x) x0 3.Transformation Q 1 xy y f (x) x y x:f (x) f (x 0 ) xy b (x 0r b )y x 0y b rby Quantum Systems for Information Technology FS

7 Shor s algorithm (quantum part) 1.Initialization Q1 Q 1/2 x 0 x0 2.Construction Q1 Q 1/2 x f (x) x0 3.Transformation Q 1 xy y f (x) x y x:f (x) f (x 0 ) xy b (x 0r b )y a superposition of Q states f(x) as a quantum function and apply it to the above state apply the quantum Fourier transform, and leads to the final state x 0y b rby Quantum Systems for Information Technology FS

8 Shor s algorithm (quantum part) 4.Measurement Q 1 x:f (x) f (x 0 ) xy 2 Q 2 (x 0r b )y b 2 Q 2 b rby 2 Quantum Systems for Information Technology FS

9 Shor s algorithm (quantum part) 4.Measurement Q 1 x:f (x) f (x 0 ) xy 2 Q 2 (x 0r b )y b 2 Q 2 b rby 2 -Perform Continued Fraction Expansion on y/q to make an approximation, and produce some c/r by it that satisfies two conditions: A: r < N B: ly/q - c/r < 1/2Q r would be the appropriate period r with high probability. -Check if f(x) = f(x + r ) <=> a r = 1 (mod N); if so, done. -Otherwise, obtain more candidates for r near y. Quantum Systems for Information Technology FS

10 Qubit system: nuclear spins in a molecule Perfluorobutadienyl molecule with 7 nuclear spins 19 F and 13 C are spin half nuclei 2 level system due to Zeeman splitting in static magnetic field H = n i h ω 0i I iz ω 0i = gμ i B 0 h ω 0i Transition frequency between 0 & T C ω 0i = 125 MHz 19 F ω 0i = 470 MHz Quantum Systems for Information Technology FS

11 Spin properties Chemical shifts in molecule causes inhomogenities in magnetic field, well separated frequencies for each qubit ω 0i = 1 σ i gμ i B 0 h Longitudinal and transvers coherence times T 1 and T 2 in order of seconds Pairwise J coupling between spins H J = i<j 2πhJ ij I iz I jz Quantum Systems for Information Technology FS

12 Experimental realization RF coil in xy plane for applying RF pulses to manipulate spins Spin state is detected by rotating about y-axis by nuclei produce measurable RF field in the coil (ensemble measurement) Qubit system of the molecule fullfils the DiVincenzo criteria For proper initialization temporal averaging can be used Long enough coherence times Pairwise J coupling of spins allows to implement gates Readout of spin states possible Quantum Systems for Information Technology FS

13 Shor s algorithm applied to 15 Quantum Systems for Information Technology FS

14 Quantum circuit Quantum Systems for Information Technology FS

15 Quantum circuit II 3) Perform invers QFT on first register Ψ 3 = 7 y=0 7 x=0 e 2πixy/8 8 y a x modn Sum over y reduces due to periodicity of f(x) to terms with y = 2n c r, with c a constant and r the period of f(x) 4) Measuring the spin states of the first register with the pick up coil Initialization, manipulation and the measurement processes require about 300 RF pulses within 750 ms. Quantum Systems for Information Technology FS

16 Experimental result a=11 (simple case) Spin 1 Spin 2 Spin 3 Theoretical simulation Measured signal Simulation with decoherence Frequency vs. Phase plot for each spin Positive peaks correspond to 0 and negative peaks to 1 Simple case because just one qubit in a mixted state Multiple peaks in the spectra arise due to crosstalk of the spins Quantum Systems for Information Technology FS

17 Experimental result a=11 (simple case) Spin 1 Spin 2 Spin 3 Theoretical simulation Measured signal Simulation with decoherence Mixture of states 000 and 100 or in decimal notation 0 and 4 Period of y is 4 r = = 2 gcd ± 1,15 = 3 & 5 Quantum Systems for Information Technology FS

18 Experimental result a=7 (difficult case) Spin 1 Spin 2 Spin 3 Theoretical simulation Measured signal Simulation with decoherence Mixture of states 000, 010, 100 and 110 or in decimal notation 0, 2, 4 and 6 Period of y is 2 r = = 4 gcd ± 1,15 = 3 & 5 Quantum Systems for Information Technology FS

19 References Shor Pieter W. Polynomial Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer Vandersypen, LMK: et al Experimental realization of Shor s quantum factoring algorithm using nuclear magnetic resonance Quantum Systems for Information Technology FS

20 Conclusion Shor algorithm provides a solution for factoring number in polynominal time using a quantum computer Successful implementation of Shor s factoring algorithm in an NMR quantum computer Good agreement between measured and simulated spectra, discrepancies can be attributed to decoherence First quantum computation experiment for which decoherence is the dominant source of errors Quantum Systems for Information Technology FS

21 Outlook The experiment shows the limits of NMR quantum computers For N>15 a molecule with more spins is needed Problems of single spin accessibility, spin interactions and shorter coherence times Hard to find error correction schemes for quantum computers to overcome imprecision and decoherence Proof for powerful every day applications of quantum computers is missing Task of the on going research to find other qubit systems and new quantum computation algorithms Quantum Systems for Information Technology FS

### Quantum Computing Lecture 7. Quantum Factoring. Anuj Dawar

Quantum Computing Lecture 7 Quantum Factoring Anuj Dawar Quantum Factoring A polynomial time quantum algorithm for factoring numbers was published by Peter Shor in 1994. polynomial time here means that

### Quantum Computing with NMR

Quantum Computing with NMR Sabine Keiber, Martin Krauÿ June 3, 2009 Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, 2009 1 / 46 1 A Short Introduction to NMR 2 Di Vincenzo's Requirements

### Factoring by Quantum Computers

Factoring by Quantum Computers Ragesh Jaiswal University of California, San Diego A Quantum computer is a device that uses uantum phenomenon to perform a computation. A classical system follows a single

### Modular arithmetic. x ymodn if x = y +mn for some integer m. p. 1/??

p. 1/?? Modular arithmetic Much of modern number theory, and many practical problems (including problems in cryptography and computer science), are concerned with modular arithmetic. While this is probably

### Lecture 8: Applications of Quantum Fourier transform

Department of Physical Sciences, University of Helsinki http://theory.physics.helsinki.fi/ quantumgas/ p. 1/25 Quantum information and computing Lecture 8: Applications of Quantum Fourier transform Jani-Petri

### Bits Superposition Quantum Parallelism

7-Qubit Quantum Computer Typical Ion Oscillations in a Trap Bits Qubits vs Each qubit can represent both a or at the same time! This phenomenon is known as Superposition. It leads to Quantum Parallelism

### 0.1 Phase Estimation Technique

Phase Estimation In this lecture we will describe Kitaev s phase estimation algorithm, and use it to obtain an alternate derivation of a quantum factoring algorithm We will also use this technique to design

### Lecture 13: Factoring Integers

CS 880: Quantum Information Processing 0/4/0 Lecture 3: Factoring Integers Instructor: Dieter van Melkebeek Scribe: Mark Wellons In this lecture, we review order finding and use this to develop a method

### QUANTUM COMPUTERS AND CRYPTOGRAPHY. Mark Zhandry Stanford University

QUANTUM COMPUTERS AND CRYPTOGRAPHY Mark Zhandry Stanford University Classical Encryption pk m c = E(pk,m) sk m = D(sk,c) m??? Quantum Computing Attack pk m aka Post-quantum Crypto c = E(pk,m) sk m = D(sk,c)

### Nuclear Magnetic Resonance

Nuclear Magnetic Resonance Introduction Atomic magnetism Nuclear magnetic resonance refers to the behaviour of atomic nuclei in the presence of a magnetic field. The first principle required to understand

### S hor s prime factoring algorithm1 reduces the factorization of a product N 5 pp9 of distinct odd primes p and

OPEN SUBJECT AREAS: QUANTUM INFORMATION QUBITS Received 21 August 2013 Accepted 3 October 2013 Published 28 October 2013 Correspondence and requests for materials should be addressed to M.R.G. (mgeller@uga.

### Quantum Systems for Information Technology

Lecture Quantum Systems for Information Technology fall term (HS) 2010 Lecturer: e Andreas Wallraff office: HPF D 8/9, ETH Hoenggerberg email: qsit-lecture@phys.ethz.ch What is this lecture about? Quantum

### Quantum Computing Architectures

Quantum Computing Architectures 1:-2: Fred Chong (UCD) - Intro, quantum algorithms, and error correction 2:-2:3 Break and discussion 2:3-3:3 Ike Chuang (MIT) - Device technology and implementation issues

### Quantum implementation of elementary arithmetic operations

Quantum implementation of elementary arithmetic operations G. Florio and D. Picca INFN (Sezione di Bari and Dipartimento di Fisica, Università di Bari, Via Orabona 4, 706, Bari, Italy Abstract Quantum

### Instrumental Lab. Nuclear Magnetic Resonance. Dr Alex J. Roche

Instrumental Lab Nuclear Magnetic Resonance Dr Alex J. Roche 1 Nuclear Magnetic Resonance (NMR) Spectroscopy NMR is the most powerful analytical tool currently available to an organic chemist. NMR allows

### Nuclear Magnetic Resonance

Nuclear Magnetic Resonance Author: James Dragan Lab Partner: Stefan Evans Physics Department, Stony Brook University, Stony Brook, NY 794. (Dated: December 5, 23) We study the principles behind Nuclear

### What Has Quantum Mechanics to Do With Factoring? Things I wish they had told me about Peter Shor s algorithm

What Has Quantum Mechanics to Do With Factoring? Things I wish they had told me about Peter Shor s algorithm 1 Question: What has quantum mechanics to do with factoring? Answer: Nothing! 2 Question: What

### Basic Pulse Sequences I

Basic Pulse Sequences I Pascale Legault Département de Biochimie Université de Montréal 1 Outline 1) Introduction to Multidimensional NMR 2) The 2D HSQC Experiment: Limitation of Classical Approaches 3)

### Quantum Computing. Robert Sizemore

Quantum Computing Robert Sizemore Outline Introduction: What is quantum computing? What use is quantum computing? Overview of Quantum Systems Dirac notation & wave functions Two level systems Classical

### A Recent Improvements in Quantum Model and Counter Measures in Quantum Computing

A Recent Improvements in Quantum Model and Counter Measures in Quantum Computing J.Senthil Murugan 1, V.Parthasarathy 2, S.Sathya 3, M.Anand 4 Assistant Professor, VelTech HighTech Dr.Rangarajan Dr.Sakunthala

### Shor s algorithm and secret sharing

Shor s algorithm and secret sharing Libor Nentvich: QC 23 April 2007: Shor s algorithm and secret sharing 1/41 Goals: 1 To explain why the factoring is important. 2 To describe the oldest and most successful

### NMR for Physical and Biological Scientists Thomas C. Pochapsky and Susan Sondej Pochapsky Table of Contents

Preface Symbols and fundamental constants 1. What is spectroscopy? A semiclassical description of spectroscopy Damped harmonics Quantum oscillators The spectroscopic experiment Ensembles and coherence

### Study of algorithms for factoring integers and computing discrete logarithms

Study of algorithms for factoring integers and computing discrete logarithms First Indo-French Workshop on Cryptography and Related Topics (IFW 2007) June 11 13, 2007 Paris, France Dr. Abhijit Das Department

### Nuclear Magnetic Resonance Spectroscopy

Nuclear Magnetic Resonance Spectroscopy Nuclear magnetic resonance spectroscopy is a powerful analytical technique used to characterize organic molecules by identifying carbonhydrogen frameworks within

### "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta". h is the Planck constant he called it

1 2 "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta". h is the Planck constant he called it the quantum of action 3 Newton believed in the corpuscular

### Factoring Algorithms Based on NMR Quantum

1295 2002 69-74 69 Factoring Algorithms Based on NMR Quantum Computers (Noboru Kunihiro) (Shigeru Yamashita) NTT NTT Abstract No polynomial time algorithms have been proposed for the factoring and discrete

### Special versions of COSY can differentiate between short range and long range interactions, as illustrated below.

2D NMR Spectroscopy To record a normal FT NMR spectrum we apply a pulse to our spin system and record the free induction decay (FID) following the pulse. The spectrum is obtained by Fourier Transform where

### 230483 - QOT - Quantum Optical Technologies

Coordinating unit: Teaching unit: Academic year: Degree: ECTS credits: 2015 230 - ETSETB - Barcelona School of Telecommunications Engineering 739 - TSC - Department of Signal Theory and Communications

### Structure Determination by NMR

Structure Determination by NMR * Introduction to NMR * 2D NMR, resonance assignments J Correlated Based Experiments * COSY - Correlated Spectroscopy * NOESY - Nuclear Overhauser Effect Spectroscopy * HETCOR

### Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and

Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study

### Quantum Computers vs. Computers Security. @veorq http://aumasson.jp

Quantum Computers vs. Computers Security @veorq http://aumasson.jp Schrodinger equation Entanglement Bell states EPR pairs Wave functions Uncertainty principle Tensor products Unitary matrices Hilbert

### 4. It is possible to excite, or flip the nuclear magnetic vector from the α-state to the β-state by bridging the energy gap between the two. This is a

BASIC PRINCIPLES INTRODUCTION TO NUCLEAR MAGNETIC RESONANCE (NMR) 1. The nuclei of certain atoms with odd atomic number, and/or odd mass behave as spinning charges. The nucleus is the center of positive

### Pulsed nuclear magnetic resonance to explore relaxation times Hannah Saddler, Adam Egbert, and Will Weigand

Pulsed nuclear magnetic resonance to explore relaxation times Hannah Saddler, Adam Egbert, and Will Weigand (Dated: 26 October 2015) The method of pulsed nuclear magnetic resonance was employed in this

### Pulsed Nuclear Magnetic Resonance Analysis of Glycerin and Mineral Oil

Pulsed Nuclear Magnetic Resonance Analysis of Glycerin and Mineral Oil Janet Chao, Dean Henze, Patrick Smith, Kent Lee February 27, 2013 Abstract Pulsed nuclear magnetic resonance was used in order to

### Quantum control of individual electron and nuclear spins in diamond lattice

Quantum control of individual electron and nuclear spins in diamond lattice Mikhail Lukin Physics Department, Harvard University Collaborators: L.Childress, M.Gurudev Dutt, J.Taylor, D.Chang, L.Jiang,A.Zibrov

### arxiv:quant-ph/9907063v2 12 Dec 1999

Nuclear Magnetic Resonance Quantum Computing Using Liquid Crystal Solvents Costantino S. Yannoni, Mark H. Sherwood, Dolores C. Miller, and Isaac L. Chuang arxiv:quant-ph/99763v2 2 Dec 999 IBM Almaden Research

### Introduction to Quantum Computing

Introduction to Quantum Computing Javier Enciso encisomo@in.tum.de Joint Advanced Student School 009 Technische Universität München April, 009 Abstract In this paper, a gentle introduction to Quantum Computing

### 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008

MIT OpenCourseWare http://ocw.mit.edu 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

### NMR SPECTROSCOPY. Basic Principles, Concepts, and Applications in Chemistry. Harald Günther University of Siegen, Siegen, Germany.

NMR SPECTROSCOPY Basic Principles, Concepts, and Applications in Chemistry Harald Günther University of Siegen, Siegen, Germany Second Edition Translated by Harald Günther JOHN WILEY & SONS Chichester

### NMR Nuclear Magnetic Resonance

NMR Nuclear Magnetic Resonance Nuclear magnetic resonance (NMR) is an effect whereby magnetic nuclei in a magnetic field absorb and re-emit electromagnetic (EM) energy. This energy is at a specific resonance

### Quantum computing in practice

Quantum computing in practice & applications to cryptography Renaud Lifchitz OPPIDA NoSuchCon, November 19-21, 2014 Renaud Lifchitz NoSuchCon, November 19-21, 2014 1 / 68 Speaker s bio French senior security

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### Nuclear Magnetic Resonance Lisa M. Larrimore

Lisa M. The fundamentals of nuclear magnetic resonance (NMR) spectroscopy were explored on samples containing large numbers of protons. Mineral oil and dilluted solutions of CuSO 4 were placed in a permanent

### Modern Factoring Algorithms

Modern Factoring Algorithms Kostas Bimpikis and Ragesh Jaiswal University of California, San Diego... both Gauss and lesser mathematicians may be justified in rejoicing that there is one science [number

### Entanglement and its Role in Shor's Algorithm

ntanglement and its Role in Shor's Algorithm Vivien M. Kendon 1, William J. Munro Trusted Systems Laboratory P Laboratories Bristol PL-2005-215 December 5, 2005* entanglement, Shor's algorithm ntanglement

### Signal to Noise Instrumental Excel Assignment

Signal to Noise Instrumental Excel Assignment Instrumental methods, as all techniques involved in physical measurements, are limited by both the precision and accuracy. The precision and accuracy of a

### Nuclear Magnetic Resonance (NMR) Spectroscopy

April 28, 2016 Exam #3: Graded exams on Tuesday! Final Exam Tuesday, May 10 th, 10:30 a.m. Room: Votey 207 (tentative) Review Session: Sunday, May 8 th, 4 pm, Kalkin 325 (tentative) Office Hours Next week:

### Spectroscopy. energy Low λ High ν. UV-visible

Spectroscopy frequency 10 20 10 18 10 16 10 14 10 12 10 8 Gamma rays X-rays UV IR Microwaves Radiowaves High energy Low λ High ν visible Low energy quantization of energy levels X-Ray UV-visible Infrared

### APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

### Basic Principles of Magnetic Resonance

Basic Principles of Magnetic Resonance Contents: Jorge Jovicich jovicich@mit.edu I) Historical Background II) An MR experiment - Overview - Can we scan the subject? - The subject goes into the magnet -

### A Scalable Architecture for Spin-based Quantum Computers

A Scalable Architecture for Spin-based Quantum Computers Why Spins? Simple, well isolated quantum system 1> Long decoherence times (~sec) Gate operations are "easy" to implement +? Scalability 0> A scalable

### I. Introduction. MPRI Cours 2-12-2. Lecture IV: Integer factorization. What is the factorization of a random number? II. Smoothness testing. F.

F. Morain École polytechnique MPRI cours 2-12-2 2013-2014 3/22 F. Morain École polytechnique MPRI cours 2-12-2 2013-2014 4/22 MPRI Cours 2-12-2 I. Introduction Input: an integer N; logox F. Morain logocnrs

### A Quantum Full Adder for a. Scalable Nuclear Spin Quantum Computer

A Quantum Full Adder for a Scalable Nuclear Spin Quantum Computer G.P. Berman 1,G.D.Doolen 1, G.V. López 2, and V.I. Tsifrinovich 3 1 T-13 and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico

### Nuclear Magnetic Resonance and the Measurement of Relaxation Times of Acetone with Gadolinium

Nuclear Magnetic Resonance and the Measurement of Relaxation Times of Acetone with Gadolinium Xia Lee and Albert Tsai June 15, 2006 1 1 Introduction Nuclear magnetic resonance (NMR) is a spectroscopic

### Nuclear Magnetic Resonance and Its Application in Condensed Matter Physics

Nuclear Magnetic Resonance and Its Application in Condensed Matter Physics Kangbo Hao 1. Introduction Nuclear Magnetic Resonance (NMR) is a physics phenomenon first observed by Isidor Rabi in 1938. [1]

### Nuclear Magnetic Resonance (NMR) Wade Textbook

Nuclear Magnetic Resonance (NMR) Wade Textbook Background Is a nondestructive structural analysis technique Has the same theoretical basis as magnetic resonance imaging (MRI) Referring to MRI as nuclear

### Spin-lattice and spin-spin relaxation

Spin-lattice and spin-spin relaation Sequence of events in the NMR eperiment: (i) application of a 90 pulse alters the population ratios, and creates transverse magnetic field components (M () ); (ii)

### Lecture 3: Line broadening mechanisms in NMR

Lecture 3: Line broadening mechanisms in NMR Lecture aims to explain: 1. Fourier transform limit (or energy-time uncertainty principle) 2. Resonance broadening due to interaction with neighbouring nuclei

### TWO DIMENSIONAL CORRELATED NMR SPECTROSCOPY

TWO DIMENSIONAL 2D-NMR TWO DIMENSIONAL CORRELATED NMR SPECTROSCOPY Two-dimensional J-resolved spectrum: Frequencies plotted on one axis, coupling constants on the other. Two-dimensional shift correlated

### NMR is the most powerful structure determination tool available to organic chemists.

Nuclear Magnetic esonance (NM) Spectrometry NM is the most powerful structure determination tool available to organic chemists. An NM spectrum provides information about: 1. The number of atoms of a given

### Quantum computation with phase drift errors.

Quantum computation with phase drift errors. César Miquel 1,,3,JuanPabloPaz 1,,3, and Wojciech Hubert Zurek 1,3 1 Institute for Theoretical Physics, University of California, Santa Barbara, CA 9316 3,

### Two-Dimensional Nuclear Magnetic Resonance: Principle and Applications. Guangjin Hou

Two-Dimensional Nuclear Magnetic Resonance: Principle and Applications Guangjin Hou 0.07.008 Outlines Histories of Multi-Dimensional NMR The general scheme for D NMR The classes of D NMR experiments -D

### Introduction to NMR spectroscopy. Swiss Institute of Bioinformatics I.Phan & J. Kopp

Introduction to NMR spectroscopy Swiss Institute of Bioinformatics I.Phan & J. Kopp NMR: the background Complex technique. Requires knowledge in: Mathematics Physics Chemistry Biology (Medicin) Involves

### April 24, 2015. A Classical Perspective. Exam #3: Solution Key online now! Graded exams by Monday!

April 24, 2015 Exam #3: Solution Key online now! Graded exams by Monday! Final Exam Monday, May 4 th, 10:30 a.m. Room: Perkins 107 1 A Classical Perspective A classical view will help us understand the

### Library (versus Language) Based Parallelism in Factoring: Experiments in MPI. Dr. Michael Alexander Dr. Sonja Sewera.

Library (versus Language) Based Parallelism in Factoring: Experiments in MPI Dr. Michael Alexander Dr. Sonja Sewera Talk 2007-10-19 Slide 1 of 20 Primes Definitions Prime: A whole number n is a prime number

### Assume MAS on powders for all problems, unless stated otherwise (or obvious from the context). Calculation Exercise #1 (Wednesday)

Exercises for Solid-State NMR Spectroscopy in Materials Chemistry Mattias Edén, Department of Materials and Environmental Chemistry, Stockholm University Assume MAS on powders for all problems, unless

### Pulsed Nuclear Magnetic Resonance An Experiment for UCSB s Advanced Laboratory

1 Pulsed Nuclear Magnetic Resonance An Experiment for UCSB s Advanced Laboratory Foreword Pulsed nuclear magnetic resonance is fascinating in its own right, and is also an incredibly important tool for

### Nuclear Magnetic Resonance (NMR) Spectroscopy cont... Recommended Reading:

Applied Spectroscopy Nuclear Magnetic Resonance (NMR) Spectroscopy cont... Recommended Reading: Banwell and McCash Chapter 7 Skoog, Holler Nieman Chapter 19 Atkins, Chapter 18 Relaxation processes We need

### Generation and Detection of NMR Signals

Generation and Detection of NMR Signals Hanudatta S. Atreya NMR Research Centre Indian Institute of Science NMR Spectroscopy Spin (I)=1/2h B 0 Energy 0 = B 0 Classical picture (B 0 ) Quantum Mechanical

### Quantum Computing for Beginners: Building Qubits

Quantum Computing for Beginners: Building Qubits Suzanne Gildert Condensed Matter Physics Research (Quantum Devices Group) University of Birmingham 28/03/2007 Overview of this presentation What is a Qubit?

### Nuclear Magnetic Resonance Spectroscopy

Nuclear Magnetic Resonance Spectroscopy Introduction NMR is the most powerful tool available for organic structure determination. It is used to study a wide variety of nuclei: 1 H 13 C 15 N 19 F 31 P 2

### Step Response of RC Circuits

Step Response of RC Circuits 1. OBJECTIVES...2 2. REFERENCE...2 3. CIRCUITS...2 4. COMPONENTS AND SPECIFICATIONS...3 QUANTITY...3 DESCRIPTION...3 COMMENTS...3 5. DISCUSSION...3 5.1 SOURCE RESISTANCE...3

### NMR - Basic principles

NMR - Basic principles Subatomic particles like electrons, protons and neutrons are associated with spin - a fundamental property like charge or mass. In the case of nuclei with even number of protons

### The Limits of Adiabatic Quantum Computation

The Limits of Adiabatic Quantum Computation Alper Sarikaya June 11, 2009 Presentation of work given on: Thesis and Presentation approved by: Date: Contents Abstract ii 1 Introduction to Quantum Computation

### Introduction to NMR Part 1. Revised 2/19/07 Anne M. Gorham

Introduction to NMR Part 1 Revised 2/19/07 Anne M. Gorham What is an NMR? Niobium-tin-copper clad coil wound like a spool of thread. The current runs through this coil, creating the magnetic field. This

### H NMR (proton NMR): determines number and type of H atoms 13. C NMR (proton NMR): determines number and type of C atoms

14.1 An Introduction to NMR Spectroscopy A. The Basics of Nuclear Magnetic Resonance (NMR) Spectroscopy nuclei with odd atomic number have a S = ½ with two spin states (+1/2 and -1/2) 1 H NMR (proton NMR):

### Prof.M.Perucca CORSO DI APPROFONDIMENTO DI FISICA ATOMICA: (III-INCONTRO) RISONANZA MAGNETICA NUCLEARE

Prof.M.Perucca CORSO DI APPROFONDIMENTO DI FISICA ATOMICA: (III-INCONTRO) RISONANZA MAGNETICA NUCLEARE SUMMARY (I/II) Angular momentum and the spinning gyroscope stationary state equation Magnetic dipole

### Fourier Analysis. Nikki Truss. Abstract:

Fourier Analysis Nikki Truss 09369481 Abstract: The aim of this experiment was to investigate the Fourier transforms of periodic waveforms, and using harmonic analysis of Fourier transforms to gain information

### A New Quantum Algorithm for NP-complete Problems

CDMTCS Research Report Series A New Quantum Algorithm for NP-complete Problems Masanori Ohya Igor V. Volovich Science University of Tokyo Steklov Mathematical Institute CDMTCS-194 September 00 Centre for

### Elements of Applied Cryptography Public key encryption

Network Security Elements of Applied Cryptography Public key encryption Public key cryptosystem RSA and the factorization problem RSA in practice Other asymmetric ciphers Asymmetric Encryption Scheme Let

### 10 Liquid-state NMR Basics of NMR System and interactions

Liquid-state NMR The first implementation of a quantum computer that has been realized is nuclear magnetic resonance (NMR) in liquids. It encodes the quantum information in the nuclear spin degrees of

### Nuclear Magnetic Resonance Spectroscopy

Most spinning nuclei behave like magnets. Nuclear Magnetic Resonance Spectroscopy asics owever, as opposed to the behavior of a classical magnet the nuclear spin magnetic moment does not always align with

### Factoring Algorithms

Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors

### Quantum Computation as a Dynamical Process ABSTRACT

Quantum Computation as a Dynamical Process G.P. Berman 1, G.D. Doolen 1, and V.I. Tsifrinovich 1 Theoretical Division and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 Department of

### Molecular spectroscopy III: Nuclear Magnetic Resonance (NMR)

Molecular spectroscopy III: Nuclear Magnetic Resonance (NMR) Nuclear magnetic resonance (NMR) is a physical phenomenon in which magnetic nuclei in a magnetic field absorb electromagnetic radiation at a

### NMR and IR spectra & vibrational analysis

Lab 5: NMR and IR spectra & vibrational analysis A brief theoretical background 1 Some of the available chemical quantum methods for calculating NMR chemical shifts are based on the Hartree-Fock self-consistent

### In this lecture. Magnetic Field. Magnetic Field. Moving Charges in a B-FieldB. Magnetism (Part II)

In this lecture Magnetism (Part II) Magnetic field Magnetic force on a moving charge Magnetic Field Lines Magnetism for MRI Magnetic Field Just as there is a field associated with electric charge there

### Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University

Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called

### 1 H and 13 C NMR compared: Both give information about the number of chemically nonequivalent nuclei (nonequivalent

1 H and 13 C NMR compared: 13 C NMR Spectroscopy Both give information about the number of chemically nonequivalent nuclei (nonequivalent hydrogens or nonequivalent carbons) Both give information about

### NMR Spectroscopy. Introduction

Introduction NMR Spectroscopy Over the past fifty years nuclear magnetic resonance spectroscopy, commonly referred to as nmr, has become the most important technique for determining the structure of organic

### Quantum Computers. And How Does Nature Compute? Kenneth W. Regan 1 University at Buffalo (SUNY) 21 May, 2015. Quantum Computers

Quantum Computers And How Does Nature Compute? Kenneth W. Regan 1 University at Buffalo (SUNY) 21 May, 2015 1 Includes joint work with Amlan Chakrabarti, U. Calcutta If you were designing Nature, how would

### There are certain things that we have to take into account before and after we take an FID (or the spectrum, the FID is not that useful after all).

Data acquisition There are certain things that we have to take into account before and after we take an FID (or the spectrum, the FID is not that useful after all). Some deal with the detection system.

### NMR Techniques Applied to Mineral Oil, Water, and Ethanol

NMR Techniques Applied to Mineral Oil, Water, and Ethanol L. Bianchini and L. Coffey Physics Department, Brandeis University, MA, 02453 (Dated: February 24, 2010) Using a TeachSpin PS1-A pulsed NMR device,

### arxiv:quant-ph/9508027v2 25 Jan 1996

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer arxiv:quant-ph/9508027v2 25 Jan 1996 Peter W. Shor Abstract A digital computer is generally believed to

### Proton Nuclear Magnetic Resonance Spectroscopy

CHEM 334L Organic Chemistry Laboratory Revision 2.0 Proton Nuclear Magnetic Resonance Spectroscopy In this laboratory exercise we will learn how to use the Chemistry Department's Nuclear Magnetic Resonance

### Introduction to Nuclear Magnetic Resonance Spectroscopy

Introduction to Nuclear Magnetic Resonance Spectroscopy Dr. Dean L. Olson, NMR Lab Director School of Chemical Sciences University of Illinois Called figures, equations, and tables are from Principles

### 2D NMR Spectroscopy. Lecture 3

2D NMR Spectroscopy Lecture 3 hemical shifts The chemical environment affects the magnetic field of nuclei. B eff = B o - B loc B eff = B o ( - σ ) σ is the magnetic shielding of the nucleus. Factors that

### The Experiment Some nuclei have nuclear magnetic moments; just as importantly, some do not

Chemistry 2600 Lecture Notes Chapter 15 Nuclear Magnetic Resonance Spectroscopy Page 1 of 23 Structure Determination in Organic Chemistry: NMR Spectroscopy Three main techniques are used to determine the