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A Brief Course of Quantum Theory

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A Brief Course of Quantum Theory A Brief Course of Quantum Theory DONG-SHENG WANG October 20, 2019 Contents 1 Qubits9 1.1 Bit, pbit................................9 1.2 Qubit................................. 10 1.3 Unitary evolution........................... 11 1.3.1 Rabi oscillation........................ 12 1.3.2 Periodic Hamiltonian..................... 12 1.3.3 Parameter-dependent Hamiltonian.............. 13 1.3.4 Quantum control....................... 16 1.4 Non-unitary evolution......................... 18 1.4.1 Quantum channels...................... 18 1.4.2 Thermodynamics....................... 21 1.5 Observable and Measurement..................... 22 1.5.1 Projective measurement................... 23 1.5.2 Non-commuting set of observable.............. 24 1.5.3 Tomography, Estimation, Discrimination.......... 25 1.6 The power of qubit.......................... 27 1.6.1 Black-box encoding..................... 27 1.6.2 Quantum key distribution................... 28 1.6.3 Measure classical values................... 29 1.6.4 Leggett–Garg inequalities.................. 30 1.6.5 Contextuality......................... 31 1.7 Zoo of qubits............................. 32 2 Basic Formalism 35 2.1 Hilbert space............................. 35 2.1.1 States............................. 36 2.1.2 Norm............................. 40 2.1.3 Dynamics........................... 41 2.1.4 Measurement......................... 43 2.2 Quantum notions........................... 46 2.2.1 Information and entropy................... 46 2.2.2 Entanglement......................... 48 i ii CONTENTS 2.2.3 Uncertainty.......................... 49 2.3 Geometric phases........................... 51 2.3.1 Aharonov-Anandan phase.................. 51 2.3.2 Applications......................... 52 2.3.3 Generalizations........................ 53 2.4 Quantum channels.......................... 54 2.4.1 Representations........................ 54 2.4.2 Lindblad equation...................... 59 2.4.3 Markovianity......................... 60 2.4.4 Beyond............................ 63 2.5 Matrix product states......................... 64 2.5.1 MPS circuit.......................... 66 2.5.2 Avoid the final projection................... 67 2.5.3 Composition......................... 69 2.5.4 Extract the edge state..................... 70 2.5.5 Area law........................... 70 3 Advanced Formalism 73 3.1 Phase space and quantization..................... 73 3.1.1 Hamiltonization....................... 74 3.1.2 Wigner function....................... 76 3.1.3 Quantization......................... 78 3.2 Relativistic subtheory......................... 81 3.2.1 Special Relativity....................... 81 3.2.2 Relativistic equations..................... 84 3.3 Semi-classical subtheory....................... 88 3.3.1 Bohmian mechanics..................... 89 3.3.2 Path integral......................... 91 3.3.3 Stabilizer formalism..................... 92 3.4 Quantum field theory......................... 93 3.4.1 Bosonic and fermionic fields................. 93 3.4.2 Topological fields....................... 97 3.4.3 Conformal fields....................... 99 4 Quantum Computation 107 4.1 Universal computing models..................... 107 4.1.1 Turing machine........................ 108 4.1.2 Circuit model......................... 110 4.1.3 Turing machine vs. circuit model.............. 111 4.1.4 Simulation.......................... 113 4.2 Quantum gate operations....................... 116 4.2.1 Teleportation......................... 116 CONTENTS 1 4.2.2 Anyon braiding........................ 118 4.2.3 Other methods........................ 119 4.3 Universal fault-tolerant QC...................... 119 4.3.1 Quantum codes........................ 119 4.3.2 Universal vs. Fault-tolerant gate set............. 121 4.4 Examples of universal fault-tolerant QC............... 124 4.4.1 Topological QC via nanyons................. 125 4.4.2 3D gauge color codes..................... 127 4.4.3 Triorthogonal codes..................... 128 4.4.4 Concatenated codes...................... 128 4.5 Quantum algorithms......................... 129 4.5.1 Computational complexity.................. 132 4.5.2 Examples of quantum algorithms.............. 133 5 Condensed Matter Physics 137 5.1 Symmetry............................... 137 5.2 Ising model.............................. 140 5.3 Ising world.............................. 142 5.3.1 Fermionization........................ 142 5.3.2 Bosonization......................... 143 5.3.3 Gauging............................ 144 5.3.4 Defects............................ 145 5.3.5 XXZ model.......................... 146 5.3.6 Dimer models......................... 148 5.4 Topological phases.......................... 150 5.4.1 Toric code........................... 152 5.4.2 Fractional quantum Hall effect................ 155 5.4.3 Topological insulator..................... 158 2 CONTENTS Introduction Question 1. What is ‘quanta’? Quanta, or ‘Quantum’ is by no means a good name; however, there is no better one. Max Planck discovered the constant h = 6:62606957(29) × 10−34J · s, which is treated as the element, or ‘quanta’, of action. What does ‘quantum’ mean? This is a primary but difficult question. Intuitively, a new constant will lead to new principle of nature, which, unfortunately, is not so clear yet, but implicitly sets the foundation of quantum theory. Here I spell my answer first, then I will explain more later. Quantum theory is a theory that describes the structure of motion of objects. The structure of motion is the notion that differs from motion, object, and structure of object. Traditionally in physics and our common sense, motion is the movement in space and time, object is a ‘thing’ with mass or energy, e.g., a ball, an electron, or the earth, and structure of object is the details of the parts of object, could be geometrical, chemical, etc. From classical mechanics we know phase space. We can view quantum theory as a generalization of it. It is a more powerful, complete version of phase space paradigm. Phase space is based on conjugate variables, such as position r and mo- mentum p, and their product is nothing but the action. With a Hamiltonian H(x; p), the Hamilton’s equation is ¶H ¶H = −p˙; = x˙: (1) ¶x ¶ p This is, furthermore, equivalent to Lagrangian version by L := xp˙ − H and the prin- ciple of least action. In quantum theory, conjugate variables are not numbers anymore. Instead, they are collections of numbers, i.e., they are matrices, or, operators. The consequence is that, conjugate operators, also called observable, cannot be measured at the same time since they do not commute. This leads to the complementarity principle of Niels Bohr, or the duality property. Why a matrix is needed to describe a ‘thing’? Well, the short answer is, the thing is complicated. It cannot be well described by a single number or function. Let’s see a most trivial thing: people wear clothes. He/she can wear different ones, he/she can wear the same one in different situations, he/she can choose one depending on 3 4 CONTENTS many factors, such as pocket, mood, etc. This complicated thing of wearing clothes, especially for females, has to be described by an exotic thing: an operator. Question 2. What is the primary quantum equation? In quantum physics, structure of the motion of one object, e.g., an electron, are described as a complex vector jyi in Hilbert space, and the evolution is represented by one operator acting on the state changing its magnitude and phase. Suppose the dynamics of the particle is driven by a Hamiltonian H, then the quantum equation is ih¯jy˙ i = Hjyi; (2) with h¯ = h=2p which appears much more unique than h itself. H is an operator, for which the eigenvalues are the possible energies. If the state jyi initially is expressed in the basis of the eigenstates of H, then the evolution of the state will include: the evolution of the coefficients of the eigenstates, which changes the relative populations (i.e. probabilities) and relative dynamical phases of the eigenstates, and the rotation of the eigenstates themselves which generates geometric phases. Let us name the above equation as state equation, since it provides the possible states of the particle, the probabilities for the particle to be in the states, and also the relative phases among the states. The state equation is originally discovered by Erwin Schrodinger¨ for a special case of particles without internal degree of freedoms (d.o.f). Briefly, the state equation provides the structure of motion of an object. As the result, quantum theory is a new kind of description of motion, which is different from other theories, including classical mechanics, wave mechanics etc. Question 3. Except jyi and H, why there are h¯ and i? The obvious reason for i is that it transfers the effects of Hamiltonian into a phase, and the constant h¯ servers as the action element. In the early age of quantum physics, h¯ is more important. However, it is clear now the quantum state jyi (and also H) is more important. There is a limit by h¯ ! 0, which shall implies there is no quantum effect anymore. This is the correspondence principle of Bohr. The disappearance of h¯ means the ignorance of the structure of motion, e.g., averaging the detailed structure of motion, thus leading to classical results. Question 4. What is the magic of quantum coherence? By extending from numbers to operators, quantum theory brings us the most important concept: quantum coherence. Please do not mix it with other notions of coherence that you know.
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