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Quantum Computing

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Introduction

Meet Alice, Bob, and Cardy What’s the Big Deal About Quantum Computing?A Brief Preview How to Use This Course: Encouragement and Coaching

Traditional Computing

Traditional Computing vs. Quantum Computing Binary Digits Logic Gates and Truth Tables Quiz: Traditional Computing Summary: Traditional Computing

Traditional Bits in New Clothing

Introduction State Symbols State Vectors Gates as Matrices Quiz: Traditional Bits in New Clothing Summary: Traditional Bits in New Clothing

Qubits and Quantum States

What Is a Qubit?What Is a Quantum State?Quantum States and Quantum Computers Quantum States and State Space Polarized Light Photons Ordinary Vectors Back to Polarized Light Orthogonality of Two-State Vectors Quiz: Qubits and Quantum States Summary: Qubits and Quantum States

Quantum Measurements

Quantum Measurements: What Are They?Light Intensity from a Two-Polarizer Setup State Amplitudes and Probabilities More on State Amplitudes and Probabilities State Preparation Quiz: Quantum Measurements Summary: Quantum Measurements

Quantum Gates and Quantum Circuits

What Is a Quantum Gate?Single-Qubit Quantum Gates Successive Quantum Gates Mathematical Interlude Successive Application of the Same Quantum Gate Measurement Devices are not Reversible Quiz: Quantum Gates and Quantum Circuits Summary: Quantum Gates and Quantum Circuits

Putting a Spin on Spin

Quantum Spin Systems Representations of Spin-1/2 Systems Quantum Gates and Spin-1/2 Systems Quiz: Putting a Spin on Spin Summary: Putting a Spin on Spin

Your Basis, My Basis

Changing Basis States

Changing Quantum Basis States

Three Basis Sets: 0°, 60°, and −60°

Dot and Inner Products of State Vectors

Spin-1/2 States

General Basis State Transformations

The Three-Polarizing-Sheets Experiment

A “Minus 1” Mystery

Bennett–Brassard-84 Encryption Protocol

Quiz: Your Basis, My Basis

Summary: Your Basis, My Basis

Multi-Qubit Systems, Entanglement, and Quantum Weirdness

Two-Qubit Quantum States General Two-Qubit States Two-Qubit Basis Vectors are Orthogonal Single Qubits and Product States Quantum Entanglement Generalized Born Rule and Entanglement Entanglement and the States of Just One of the Qubits More on Entangled States Changing Basis States for Two-Qubit States Two Qubits and Three Measurement Base Quiz: Multi-Qubit Systems, Entanglement, and Quantum Weirdness Summary: Multi-Qubit Systems, Entanglement, and Quantum Weirdness

Quantum Circuits and Multi-Qubit Applications

Two Qubit Gates CNOT Gates Quantum Circuits (Quantum Gate Arrays)The Bell Circuit The Reverse Bell Circuit Entanglement Swapping No-Cloning Theorem Superdense Coding Quantum State Teleportation SWAP Gates Quiz: Quantum Circuits and Multi-Qubit Applications Summary: Quantum Circuits and Multi-Qubit Applications

Quantum Computing Algorithms

Welcome to the Quantum Algorithm Zoo The Deutsch Algorithm Implementing Function Evaluation with Simple Quantum Gates The Simon Search Algorithm Multi-Qubit Hadamard Gates Quiz: Quantum Computing Algorithms Summary: Quantum Computing Algorithms

Mini Project

Implement the Deutsch Algorithm with Qiskit

More Quantum Algorithms

Introduction Grover Search Algorithm Geometric Approach Quantum Operator Approach Amplitude Amplification Approach Error Correction Quantum Computational Chemistry Quiz: More Quantum Algorithms Summary: More Quantum Algorithms

Factoring Algorithm

RSA Encryption and Factoring Number Theory Warm-Up The RSA Algorithm The Shor Factoring Algorithm Connecting Periodic Functions and Factoring Period Finding and Fourier Analysis Quantum Fourier Analysis Modular Exponentiation Greatest Common Divisor Quiz: Factoring Algorithm Summary: Factoring Algorithm

Mini Project

Implement Shor’s Factoring Algorithm Using Qiskit

Fundamental Quantum Issues

Introduction Bell’s Theorem Exploring Quantum Mysteries The Measurement Problem Measurement and Decoherence The Relationship Between Classical and Quantum Computing Entanglement and Quantum Measurements Entanglement and Correlations of Measurement Outcomes Another Look at Averages in Quantum Mechanics Where Is the Planck’s Constant?Quiz: Fundamental Quantum Issues Summary: Fundamental Quantum Issues

Complexifying Quantum States

Complex Numbers and Variables Complex Exponentials Quantum Fourier Transform Relative Phases in Quantum States The Bloch Sphere Quiz: Complexifying Quantum States Summary: Complexifying Quantum States

Present and Future QIS and QC

QC and QIS Overview Programming Quantum Computers Physics of Qubits Quantum Information Science and the Quantum Internet Quantum Computing and Machine Learning Alternative Quantum Computer Architectures Ethical Issues Preparing for Your Quantum Career Summary: Present and Future QIS and QC Final Remarks

Spin-1/2 States

Apply the basis transformation of spin-1/2 states and learn how the state representations change in real space and state space.

Now let’s apply these ideas to a spin- $½$ system with two different measurement bases rotated by $90\degree$ in real space ( $45\degree$ in state space). See the following figure. This is a combination we will see frequently in QIS and QC.

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