Use this page for course review. Each section points back to the concept notes where the full explanation, lab, and homework live. For source readings, use [[Source Reading Guide]]. For software, hardware, articles, and lecture links, use [[Resources]]. ## Foundations Related notes: [[concepts/Extended Church-Turing Thesis]], [[concepts/Probability Theory and Quantum Mechanics]], [[concepts/Basic Rules of Quantum Mechanics]], [[concepts/Quantum Gates and Circuits]] > [!question] What is the difference between probability and amplitude? >> [!answer] Probabilities are nonnegative numbers that can be read as outcome chances. Amplitudes are complex numbers that combine before measurement; probabilities appear only after applying the Born rule. > [!question] Why is interference the first real quantum-computing idea? >> [!answer] Quantum algorithms work by arranging amplitudes so bad alternatives cancel and useful alternatives reinforce. That is a computational use of interference, not just a physics demonstration. > [!question] Why does global phase not matter? >> [!answer] Multiplying a whole state by one phase leaves all measurement probabilities unchanged. Only relative phase between components can affect later interference. ## States, Mixtures, And Entanglement Related notes: [[concepts/Pure vs. Mixed States]], [[concepts/Density Matrices and Partial Trace]], [[concepts/Separable vs. Entangled States]], [[concepts/Schmidt Decomposition]], [[concepts/Entanglement Entropy]] > [!question] What is the shortest way to distinguish a superposition from a mixture? >> [!answer] Look for coherence. A superposition can have off-diagonal density-matrix terms that later gates can turn into observable interference. A mixture lacks those coherence terms. > [!question] Why does the partial trace matter? >> [!answer] It gives the state available to an observer who can access only part of a larger system. It explains why subsystems of a pure entangled state can look mixed. > [!question] When is entropy a clean entanglement measure? >> [!answer] For a pure bipartite state, the entropy of either reduced state measures entanglement across that bipartition. For mixed states, entropy can also reflect classical uncertainty or environmental coupling. ## Communication And Cryptography Related notes: [[concepts/Bloch Sphere and No-Cloning Theorem]], [[concepts/Quantum Money and Quantum Key Distribution]], [[concepts/Superdense Coding]], [[concepts/Quantum Teleportation]], [[concepts/GHZ States, Entanglement Swapping, and Monogamy]] > [!question] Why does no-cloning matter operationally? >> [!answer] It prevents perfect copying of arbitrary unknown quantum states. That restriction supports quantum money, QKD security intuition, and the logic of teleportation. > [!question] What resource does superdense coding spend? >> [!answer] It spends shared entanglement. With one shared Bell pair, sending one qubit can communicate two classical bits. > [!question] What resource does teleportation spend? >> [!answer] It spends shared entanglement plus two classical bits of communication. The original unknown state is not copied; its information is transferred through the protocol. ## Bell, Nonlocality, And Randomness Related notes: [[concepts/Bell's Inequality and CHSH]], [[concepts/Nonlocal Games]], [[concepts/Einstein-Certified Randomness]], [[concepts/Interpretations of Quantum Mechanics]] > [!question] What does Bell violation rule out? >> [!answer] It rules out local hidden-variable explanations for the observed correlations under the test assumptions. It does not allow faster-than-light signalling. > [!question] Why are nonlocal games useful? >> [!answer] They turn correlation questions into input-output tasks with measurable winning probabilities, making classical and quantum resources easier to compare. > [!question] Why can Bell violation certify randomness? >> [!answer] A strong enough Bell violation means outputs cannot be fully explained as predetermined local data, so the observed correlations bound predictability under the model assumptions. ## Query Algorithms Related notes: [[concepts/Quantum Query Complexity and Deutsch-Jozsa]], [[concepts/Bernstein-Vazirani and Simon's Algorithm]], [[concepts/Grover's Algorithm]] > [!question] What is query complexity counting? >> [!answer] It counts oracle calls. The point is to compare how much black-box access classical and quantum algorithms need, without hiding work inside the oracle. > [!question] Why is Simon's algorithm a bridge to Shor? >> [!answer] Simon recovers a hidden XOR period, while Shor uses periodicity in modular arithmetic. Both use quantum sampling to expose hidden structure and classical post-processing to finish. > [!question] What does Grover search prove and not prove? >> [!answer] It proves a quadratic speedup for unstructured search. BBBV-style lower bounds show that this is essentially optimal in the black-box model, so it does not make arbitrary exponential search easy. ## Shor, QFT, And Complexity Related notes: [[concepts/RSA, Period Finding, and Shor's Algorithm]], [[concepts/Quantum Fourier Transform]], [[concepts/Quantum Complexity Theory]] > [!question] What is the real target of Shor's quantum subroutine? >> [!answer] Period or order information. The factoring consequence comes from classical reductions and post-processing around that quantum subroutine. > [!question] Why is the QFT not just a circuit trick? >> [!answer] It changes representation so periodic or phase structure becomes visible in computational-basis samples. > [!question] Why is BQP not "NP-complete problems are easy"? >> [!answer] BQP captures bounded-error polynomial-time quantum computation. Known quantum speedups exploit specific structure or oracle settings, and Grover-like lower bounds warn against overgeneralizing. ## Hamiltonians, Noise, And Hardware Related notes: [[concepts/Hamiltonians]], [[concepts/The Adiabatic Algorithm]], [[concepts/Quantum Error Correction]], [[concepts/Stabilizer Formalism]] > [!question] Why learn Hamiltonians if the course uses circuits? >> [!answer] Real systems evolve under Hamiltonians. Circuits are one control model, but Hamiltonian language is needed for simulation, adiabatic algorithms, noise, and hardware. > [!question] What makes the adiabatic algorithm sensitive to gaps? >> [!answer] The runtime depends on staying near the ground state as the Hamiltonian changes. Small spectral gaps make transitions to excited states harder to avoid. > [!question] Why is fault tolerance the answer to the "analog amplitudes are fragile" objection? >> [!answer] Fault tolerance uses encoding, syndrome extraction, and threshold behavior to prevent small physical errors from accumulating directly into logical failure, provided the hardware error model is controlled enough. ## Tool Choice Checks > [!question] When should I use Qiskit or IBM Quantum? >> [!answer] Use them for the main circuit labs, textbook protocols, shot-count experiments, and hardware-facing workflows. > [!question] When should I use QuTiP? >> [!answer] Use it for density matrices, partial traces, Hamiltonian time evolution, open systems, and noise models. > [!question] When should I use CUDA-Q or tensor-network material? >> [!answer] Use them when simulation scale, acceleration, hybrid workflows, or structured many-body simulation becomes the point of the exercise. ## Concept Study Map Use these as the direct review path for the added concept-matched material. The detailed prompts live inside the linked notes. **Mixed-state and entanglement checks.** Use [[concepts/Pure vs. Mixed States]], [[concepts/Density Matrices and Partial Trace]], [[concepts/Separable vs. Entangled States]], and [[concepts/Entanglement Entropy]]. These cover density-matrix decompositions, partial traces, separability, mixtures, superpositions, and entropy calculations. **Nonlocality and no-communication.** Use [[concepts/Bell's Inequality and CHSH]] and [[concepts/Nonlocal Games]]. These cover CHSH bounds, shared randomness, Tsirelson's limit, and why entanglement does not permit signalling. **Algorithmic speedups and limits.** Use [[concepts/Bernstein-Vazirani and Simon's Algorithm]], [[concepts/Grover's Algorithm]], [[concepts/RSA, Period Finding, and Shor's Algorithm]], and [[concepts/Quantum Complexity Theory]]. These cover Simon samples, Grover timing, Shor/order finding, black-box lower bounds, and BQP caveats. **Cryptography, encryption, and error correction.** Use [[concepts/Quantum Money and Quantum Key Distribution]] and [[concepts/Quantum Error Correction]]. These cover Wiesner money, quantum one-time pad security, one-bit impossibility, Shor-code checks, and syndrome logic.