Lecture notes for Scott Aaronson's CS378 Quantum Information Science at UT Austin, Spring 2017
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An introduction to Quantum Information Science. A few important concepts are introduced (Probability, Locality, Local Realism, the Church-Turing Thesis and its extended variation) to contextualize how quantum mechanics affects our understanding of physics.
Quantum mechanics challenges Computational Universality.
The Double Slit Experiment introduces Decoherence and Interference. It motivates us to use Amplitudes to measure quantum chance, which are related to probabilities through the Born Rule. Linear Algebra can model classical probabilities using Stochastic Matrices and Tensor Products.
Quantum States and the Qubit warrant using Bra-Ket Notation. Linear Algebra can model quantum states too, but for that we need Unitary and Orthogonal Matrices, as well as Unitary Transformations.
Several examples of Quantum Gates get us working in multiple bases. The compatibility (or lack thereof) between Unitary Transformations and Measurement is explored. Quantum Circuit Notation is introduced, along with phenomena occurring with a single qubit (Quantum Zeno Effect, Watched Pot Effect, Elitzur-Vaidman Bomb).
Our first quantum protocol distinguishes between a fair and biased coin. The distinguishability of quantum states is explored.
Considering the state of two qubits with linear algebra and quantum circuit notation introduces the Partial Measurement Rule, the Controlled NOT, and the Bell Pair. Entanglement comes into play, and we see why it need not require the existence of faster-than-light communication.
Density Matrices are introduced to represent Mixed States. We see the properties of density matrixes including Trace and Rank, as well as processes we may want to do with them like applying unitary transformations, performing Eigendecomposition, and Tracing Out.
The Bloch Sphere is introduced as a useful representation of possible states of a qubit.  The No Communication Theorem and the No Cloning Theorem limit what can be done with quantum information. These limits allow for the creation of Quantum Money schemes, such as Wiesner’s Scheme.
Attacks on Wiesner’s Scheme are explored, including an Interactive Attack, and an Attack Based on the Elitzur Vaidman Bomb.
BB84 is a Quantum Key Distribution scheme allowing two parties to generate a shared secret.
Using entanglement as a resource allows for Superdense Coding, transmitting two classical bits via one qubit, and Quantum Teleportation, transmitting a qubit via classical communication.
Quantum Teleportation is further explored and extended to arbitrary quantum states. Quantifying entanglement leads us to Entanglement Swapping, the GHZ State and the Monogamy of Entanglement, as well as Schmidt Decomposition.
Measuring entropy of a quantum state with Von Neumann Entropy and Entanglement Entropy. The Measurement Problem leads us into interpretation of quantum mechanics, the Copenhagen Interpretation and S.U.A.C., as well as useful though experiments, Schrödinger’s Cat and Wigner’s Friend.
Dynamic Collapse theories such as GRW and the Penrose Interpretation suggest that we’re still missing part of the puzzle. Everett’s Many Worlds Interpretation suggests that the universe branches every time a measurement happens.
A further discussion of Many Worlds tackles the practicality of an unfalsifiable interpretation and the Prefered Basis Problem.
Hidden Variable Theories such as Bohmian Mechanics lead to a search for a local hidden variable theory which proves to be impossible given the Bell Inequality, leading us to the CHSH Game.
The optimality of our strategy for the CHSH Game is discussed and proven through Tsirelson’s Inequality. The implications of experimentally Testing the Bell Inequality lead us to Superdeterminism and modern skepticism of quantum mechanics.
Other non-local games (The Odd Cycle Game and The Magic Square Game) are covered. 2 
The CHSH Game can be applied to Generating Guaranteed Random Numbers, and many other tasks, which brings us to Quantum Computing. We discuss the intellectual origins of the field and a few conceptual points.
The roles of interference and entanglement in quantum computing lead us to cover the construction of both classical and quantum Universal Gate Sets. We discuss Quantum Complexity and see our first quantum algorithm, Deutsch’s Algorithm.
We finish our discussion of Universal Gate Sets and the usage of black-box functions, which leads us to Uncomputing. Revisiting Deutsch’s Algorithm, we see it’s generalization the Deutsch-Jozsa Algorithm, as well as an introduction to the Bernstein-Vazirani Problem.
Further discussion of the reliability of qubits leads us to Stabilizer Sets and their compact representations through Generator Sets of Pauli Matrices. The Gottesman-Knill Theorem explains why stabilizer sets aren’t universal, and leads to the use of Tableau Representation.
Quantum error correction codes with Transversality are prefered.
Practical implementations of of quantum computing are discussed, including the important speedups it could provide, leading to a discussion of the HHL Theorem. DiVincenzo Criteria could be satisfied with Trapped Ions or Superconducting Qubits, as well as Photonics (bringing us to the KLM Theorem and Boson Sampling) or Non-abelian Anyons.