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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. Planck and especially Einstein) began to realize that the very foundations were crum- bling. Maxwell correctly taught us that accelerating charges radiate away energy in the form of electromagnetic waves. Why then do electrons orbit- ing the positively charged nuclei in atoms not radiate away all their energy and fall into the nucleus? This doesn't happen, and in fact, physicists soon discovered that the lone electron in the hydrogen atom seems to have only a discrete set of allowed orbits corresponding to certain well-defined ener- gies. Electrons could in fact emit electromagnetic waves, but only in discrete lumps of energy (`quanta' known as `photons') that matched the change in energy as the electron spontaneously jumped from one allowed energy level to a lower allowed energy level. Once the electron reached the lowest allowed energy level (the `ground state'), it could fall no further despite the fact that it was still accelerating while orbiting the nucleus). Maxwell also taught us that the modes of the electromagnetic field are harmonic oscillators. Why then does electromagnetic radiation not obey the 1 CHAPTER 1. INTRODUCTION 2 laws of statistical mechanics developed for harmonic oscillators? Accord- ing to classical statistical mechanics, the average energy stored in the kinetic and potential energy of a harmonic oscillator is kBT , where kB is Boltzmann's constant and T is the absolute temperature. Since electromagnetic radiation comes in a wide range of frequencies running all the way up to infinity, clas- sical statistical mechanics predicts an `ultraviolet catastrophe,' in which the energy density of blackbody radiation (i.e. radiation in thermal equilibrium) should be kBT × 1. Even venerable thermodynamics, which makes predictions of a general nature independent of microscopic details, gave some hints of problems. The thermodynamic entropy (a measure of our ignorance of the microscopic de- tails of the state of the system; or equivalent a measure (the logarithm) of the number of different microscopic states that are consisten with the observed macroscopic properties) of a gas obeying the ideal gas law turns negative at low temperatures. In statistical physics, Josiah Willard Gibbs pointed out the now famous `Gibbs paradox.' He noted that if a partition is removed from a box allowing the gas atoms on each side to mix, the entropy should irreversibly increase, and yet nothing macroscopically observable actually happens. The classical way of counting up the number of different configu- rations of a bunch of atoms was somehow wrong. The Gibbs paradox could be resolved only if exchanging the positions of two particles (say particle 17 and particle 1,230) did not result in a new state{as if the atoms were not merely the same species but truly indistinguishable. 1.1 Two-Slit Experiment, Interference and Measurements TO BE WRITTEN: Wave-like interference and Superposition Principle. Particle-like detection. Which path information and state collapse upon mea- surement. CHAPTER 1. INTRODUCTION 3 1.2 Bits and Qubits A bit is the smallest unit of information. The word `bit' is short for binary digit and can be represented by 0 or 1 in the binary numbering system (base 2 numbers). It is the amount of information contained in the answer to a yes/no or true/false question (assuming you have no prior knowledge of the answer and the two answers are equally likely as far as you know). If Alice gives Bob either a 0 or a 1 drawn randomly with equal probability, then Bob receives one bit of information.1 A register of N classical bits can be in one of 2N distinct states. For example 3 classical bits can be in 8 states: (000), (001), (010), (011), (100), (101), (110), (111), corresponding to the 8 binary numbers whose (base 10) values run from 0 to 7. This is a remarkably efficient encoding{with an `alphabet' of only two symbols (0 and 1) Alice can send Bob one of 232 = 4; 294; 967; 296 possible messages at a cost of only having to transmit 32 bits (or 4 `bytes'). In fact, if all the messages are equally likely to be chosen, this is provably the most efficient possible encoding. Information is physical and is stored in the state of a physical system. In an ordinary computer, bits are stored in the states of switches (typically these days transistors) which can be open/closed or on/off. Information can also be stored in small domains of magnetism in disk drives. Each magnetic domain acts like a small bar magnet whose magnetic moment (a vector quantity pointing in the direction running from the south pole to the north pole of the bar magnet) can pnly point up or down. Ordinarily a bar magnet can point in any direction in space, but the material properties of the disk drive are intentionally chosen to be anisotropic so that only two directions are possible. A quantum bit (`qubit') is information stored in a quantum system that can be in one of two possible states. Using a quantum state notation devel- oped by Paul A.M. Dirac, state j0i might denote an atom in its lowest energy state (the ground state) and state j1i might denote the first excited state of 1Information is `news'. If Bob knew ahead of time that Alice would chose 0 with probability p0 = 0:999, and 1 with probability p1 = 1 − p0, then he is not surprised when he receives a 0 because he already was pretty sure he `knew the answer to the question' ahead of time. The mathematical theory of information quantifies the information that Bob receives and shows that it is much less than one bit of information. Claude Shannon demonstrated that in this situation Bob receives information H = − [p0 ln2 p0 + p1 ln2 p1], 1 where ln2 means log base 2. H has a maximum value of 1 at p0 = p1 = 2 and approaches zero when either of the probabilities approaches unity. CHAPTER 1. INTRODUCTION 4 the atom (or some other selected excited state). However, quantum systems behave strangely{particles can act like waves and waves sometimes act like particles. Two-state systems have wave-like properties in the sense that can be in superposition states represented mathematically by j i = αj0i + βj1i (1.1) where α and β are (complex) probability amplitudes. If one measures whether the qubit has value 0 or 1, the answer is random. The measure- ment yields 0 with probability2 jαj2 and 1 with probability jβj2. Thus we have the important constraint that the probabilities must add up to unity jαj2 + jβj2 = 1: (1.2) We say that the state must be `normalized.' Furthermore, we see that an overall `global phase' of the form eiχj i = eiχαj0i + eiχβj1i (1.3) (where χ is an arbitrary real number) can have no measurable effect because the probabilities for different measurement results depend only on the magni- tude square of the state coefficients. Hence, without loss of generality we can choose the phase angle χ to make α real. The complex number β = a + ib is parametrized by two real numbers a and b. However the normalization constraint removes one of these degrees of freedom. In the end we are left with two real parameters to specify the state of a single qubit. A standard parametrization in terms of two angles θ and ' is the following θ θ j i = cos j1i + ei' sin j0i: (1.4) 2 2 The two angles θ and ϕ can be visualized as the polar and azimuthal angles of a vector pointing from the origin to the surface of the unit sphere as illus- tragted in Fig. 1.1. We will later describe the meaning of the unit vector on the Bloch sphere in relation to the properties of the corresponding quantum state and will also explain why half angles appear in the parametrization of the state in Eq. (1.4). We simply note here that the two states of a classical bit are represented by the Bloch sphere vectors corresponding to the north pole (0; 0; 1) and the south pole (0; 0; −1).
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