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Much as there's no royal road to geometry, getting familiar with quantum computing usually takes a lot of hard work. I would consider anyone who can answer Bertrand's questions in the comments to be pretty knowledgeable, at least having completed the first parts of a good semester of an advanced undergraduate/graduate course or have otherwise engaged in many many hours of self-study.
But there are many good lecture series on-line. YMMV.
Nonetheless knowledge of some topics that may be at least important to have a cocktail-level conversation of quantum computing might start off and include:
- Qubits, two state quantum systems
- The representation of a qubit on the complex plane
- Superposition and the difference between a qubit and a bit
- The Born rule of measurement and post-measurement states, with side topics about interpretations such as Copenhagen and Many-Worlds
- The difference between the computational basis and the Hadamard basis
- The no-cloning theorem
- The Bloch sphere and the representation of a qubit in three dimensions
- Neat applications like the Elitzur-Vaidman bomb tester
- Other applications of qubits in product states, such as quantum money (Wiesner's scheme) and/or quantum cryptography (the BB84 scheme)
Moving on:
- Entanglement and the EPR paper
- The Bell inequality and/or the CHSH game and/or the Mermin-Peres magic square
- Neat applications like teleportation and superdense coding
- Other applications of Bell pairs like the E91 quantum cryptography scheme
- Perhaps the GHZ state and/or the W state
Further on to quantum algorithms you have the greatest hits like:
- The Bernstein-Vazirani algorithm
- The Deutsch-Jozsa algorithm
- The BQP complexity class
- Simon's algorithm
- Hamiltonian simulation
- Quantum error correction
- The crown jewels of Shor's algorithm/Grover's algorithm
etc.