# Short Questions Use this as the quick self-test bank while moving through the notes. The answer blocks are easy to hide while self-testing and easy to reuse when making flashcards. ## Foundations - Question 1. MCQ: Which statement best describes a quantum amplitude? A. A nonnegative frequency B. A complex number whose squared magnitude gives probability C. A hidden classical bit D. A hardware noise parameter ``` B. A complex number whose squared magnitude gives probability. ``` - Question 2. MCQ: Which feature lets quantum algorithms make bad computational paths cancel? A. Decoherence B. Interference C. Thermalization D. Classical majority vote ``` B. Interference. ``` - Question 3. What is the Born rule for a normalized state $|\psi\rangle=\sum_x \alpha_x|x\rangle$? ``` The probability of measuring outcome x is |alpha_x|^2. ``` - Question 4. MCQ: What physical effect does multiplying an entire state by a global phase have? A. It changes all measurement probabilities B. It changes only X-basis probabilities C. It has no observable effect D. It creates entanglement ``` C. It has no observable effect. ``` - Question 5. What is the difference between global phase and relative phase? ``` Global phase multiplies the whole state and leaves all probabilities unchanged. Relative phase changes phase between components and can affect later interference. ``` - Question 6. MCQ: Which matrix is the Hadamard gate? A. $\begin{bmatrix}0&1\\1&0\end{bmatrix}$ B. $\frac{1}{\sqrt2}\begin{bmatrix}1&1\\1&-1\end{bmatrix}$ C. $\begin{bmatrix}1&0\\0&i\end{bmatrix}$ D. $\begin{bmatrix}1&0\\0&-1\end{bmatrix}$ ``` B. (1/sqrt(2)) [[1, 1], [1, -1]]. ``` - Question 7. What does it mean for a gate to be unitary? ``` Its inverse is its conjugate transpose, so U dagger U = I. It preserves inner products and state normalization. ``` - Question 8. MCQ: Which operation is not generally unitary? A. X gate B. Hadamard gate C. Measurement D. Phase gate ``` C. Measurement. ``` - Question 9. What does the tensor product represent in quantum mechanics? ``` It combines subsystem Hilbert spaces into the joint state space of a composite system. ``` - Question 10. MCQ: How many complex amplitudes describe a general pure state of n qubits before normalization and global phase conventions? A. n B. n^2 C. 2n D. 2^n ``` D. 2^n. ``` ## Density Matrices And Entanglement - Question 11. MCQ: Which condition identifies a pure density matrix in finite dimension? A. Tr(rho)=0 B. Tr(rho^2)=1 C. det(rho)=1 always D. all off-diagonal entries vanish ``` B. Tr(rho^2)=1. ``` - Question 12. What is the density matrix of a pure state $|\psi\rangle$? ``` rho = |psi><psi|. ``` - Question 13. MCQ: What distinguishes $|+\rangle\langle+|$ from the 50-50 mixture of $|0\rangle$ and $|1\rangle$? A. Their Z-basis probabilities B. Their normalization C. Their off-diagonal coherence terms D. Their dimension ``` C. Their off-diagonal coherence terms. ``` - Question 14. What does the partial trace compute? ``` It computes the reduced density matrix of a subsystem by tracing out the subsystem being ignored. ``` - Question 15. MCQ: What is the reduced state of one qubit of a Bell pair? A. $|0\rangle\langle0|$ B. $|1\rangle\langle1|$ C. $I/2$ D. The original Bell state ``` C. I/2. ``` - Question 16. What is the main conceptual warning about mixed states? ``` Mixedness can represent classical preparation uncertainty, but it can also arise from tracing out part of a larger entangled pure state. ``` - Question 17. MCQ: For a pure two-qubit state with coefficient matrix $M$, what rank indicates separability? A. rank 0 B. rank 1 C. rank 2 D. rank 4 ``` B. rank 1. ``` - Question 18. What determinant condition tests separability for $a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$? ``` The state is separable exactly when ad - bc = 0. ``` - Question 19. MCQ: Which state is classically correlated but separable? A. $(|00\rangle+|11\rangle)/\sqrt2$ B. $\frac12|00\rangle\langle00|+\frac12|11\rangle\langle11|$ C. $(|01\rangle-|10\rangle)/\sqrt2$ D. A GHZ state ``` B. The 50-50 mixture of |00><00| and |11><11|. ``` - Question 20. What is Schmidt rank? ``` The number of nonzero Schmidt coefficients in a pure bipartite state's Schmidt decomposition. ``` - Question 21. MCQ: A pure bipartite state is separable if and only if its Schmidt rank is: A. 0 B. 1 C. 2 D. maximal ``` B. 1. ``` - Question 22. What do Schmidt coefficients determine? ``` They determine the nonzero eigenvalues of the reduced density matrices and therefore the pure-state entanglement entropy. ``` - Question 23. MCQ: Entanglement entropy is a clean entanglement measure for which case? A. Any mixed state B. Any noisy hardware sample C. Pure bipartite states D. Classical probability distributions only ``` C. Pure bipartite states. ``` - Question 24. What is one ebit? ``` The entanglement entropy of a maximally entangled two-qubit Bell pair, equal to 1 bit. ``` ## Protocols, Bell, And Cryptography - Question 25. MCQ: What does the no-cloning theorem forbid? A. Copying known classical bits B. Preparing two identical known states C. Perfectly copying an arbitrary unknown quantum state D. Measuring in the computational basis ``` C. Perfectly copying an arbitrary unknown quantum state. ``` - Question 26. What resource does teleportation consume? ``` One shared Bell pair and two classical bits of communication. ``` - Question 27. MCQ: Does teleportation copy the unknown input state? A. Yes B. No C. Only if the state is entangled D. Only on a simulator ``` B. No. ``` - Question 28. What resource does superdense coding consume? ``` Shared entanglement, usually one Bell pair. ``` - Question 29. MCQ: In superdense coding, one transmitted qubit plus one shared Bell pair can communicate how many classical bits? A. 1 B. 2 C. 3 D. 4 ``` B. 2. ``` - Question 30. What is the basic BB84 eavesdropping signal under intercept-resend? ``` An elevated error rate in the sifted key, typically 25 percent for the simplest intercept-resend attack. ``` - Question 31. MCQ: What does Bell violation rule out under standard assumptions? A. Relativity B. All randomness C. Local hidden-variable explanations D. Measurement itself ``` C. Local hidden-variable explanations. ``` - Question 32. What is the classical CHSH winning bound? ``` 3/4. ``` - Question 33. MCQ: What is the optimal quantum CHSH winning probability? A. 1/2 B. 3/4 C. $\cos^2(\pi/8)$ D. 1 ``` C. cos^2(pi/8). ``` - Question 34. Why does Bell violation not imply faster-than-light signalling? ``` The local marginal distribution for either party does not depend on the other party's input; the stronger-than-classical structure appears only in the compared correlations. ``` - Question 35. MCQ: What does Tsirelson's bound show? A. Quantum correlations are unlimited B. Quantum correlations beat classical local bounds but still have a finite ceiling C. Classical strategies can always match quantum strategies D. Bell pairs are unnecessary ``` B. Quantum correlations beat classical local bounds but still have a finite ceiling. ``` - Question 36. What makes nonlocal games useful pedagogically? ``` They convert correlation claims into operational input-output tasks with measurable winning probabilities. ``` ## Algorithms And Complexity - Question 37. MCQ: What does query complexity count? A. Number of qubits only B. Number of oracle calls C. Number of classical bits in the answer D. Number of lab notebooks ``` B. Number of oracle calls. ``` - Question 38. What is the main promise in Deutsch-Jozsa? ``` The Boolean function is promised to be either constant or balanced. ``` - Question 39. MCQ: Bernstein-Vazirani finds what hidden object? A. A hidden period in modular arithmetic B. A hidden string in a linear Boolean function C. A marked item in an unstructured database D. A stabilizer generator ``` B. A hidden string in a linear Boolean function. ``` - Question 40. What is Simon's algorithm trying to recover? ``` A hidden xor mask s such that f(x)=f(y) exactly when x xor y is either 0 or s. ``` - Question 41. MCQ: Grover search gives which generic black-box speedup? A. No speedup B. Quadratic C. Exponential for all NP-complete problems D. Infinite ``` B. Quadratic. ``` - Question 42. What is the BBBV lesson about Grover search? ``` The quadratic speedup for unstructured black-box search is essentially optimal; Grover does not make arbitrary search exponentially easy. ``` - Question 43. MCQ: What is the real quantum target in Shor's factoring algorithm? A. Sorting B. Period or order finding C. Linear regression D. Bell-state preparation ``` B. Period or order finding. ``` - Question 44. What does the QFT help reveal? ``` It makes periodic or phase structure visible in computational-basis samples. ``` - Question 45. MCQ: Which claim is correct about BQP? A. BQP is known to contain all NP-complete problems B. BQP is bounded-error polynomial-time quantum computation C. BQP means any exponential state can be read out D. BQP is the same as P ``` B. BQP is bounded-error polynomial-time quantum computation. ``` - Question 46. Why is "quantum states have exponentially many amplitudes" not enough to prove exponential classical information output? ``` Measurement gives limited classical information, and quantum algorithms must arrange interference so the desired information becomes observable. ``` ## Hamiltonians, Noise, And QEC - Question 47. MCQ: What equation describes closed-system Hamiltonian time evolution? A. $\rho'=A\rho+B$ B. $|\psi(t)\rangle=e^{-iHt}|\psi(0)\rangle$ C. $p=x+y$ D. $U^\dagger U=0$ ``` B. |psi(t)> = exp(-i H t) |psi(0)>. ``` - Question 48. What does a spectral gap mean in adiabatic algorithms? ``` It is the energy difference between the ground state and the nearest excited state; small gaps make adiabatic evolution harder. ``` - Question 49. MCQ: Which condition must a valid density matrix satisfy? A. Trace one and positive semidefinite B. Trace zero and negative eigenvalues C. Nonunitary always D. Only diagonal entries allowed ``` A. Trace one and positive semidefinite. ``` - Question 50. What is a Kraus channel required to preserve? ``` Trace, with Kraus operators satisfying sum_k E_k dagger E_k = I. ``` - Question 51. MCQ: What does a syndrome measurement reveal? A. The protected logical amplitudes B. The error subspace or error pattern information C. The global phase D. The answer to an NP-complete problem ``` B. The error subspace or error pattern information. ``` - Question 52. Why does no-cloning not prevent quantum error correction? ``` Quantum codes do not clone an unknown state. They encode logical information into a larger entangled subspace and measure syndromes that reveal errors without revealing the logical amplitudes. ``` - Question 53. MCQ: Which stabilizers check the three-qubit bit-flip code? A. XX and YY B. ZZI and IZZ C. H and T D. CNOT only ``` B. ZZI and IZZ. ``` - Question 54. What is a stabilizer of a state? ``` An operator S such that S|psi> = |psi>. ``` - Question 55. MCQ: What does the Gottesman-Knill theorem say at a high level? A. All quantum circuits are classically easy B. Clifford/stabilizer circuits can be simulated efficiently classically C. Shor's algorithm is impossible D. Measurement is unnecessary ``` B. Clifford/stabilizer circuits can be simulated efficiently classically. ``` - Question 56. Why does Gottesman-Knill not make quantum computing classically easy? ``` It applies to a restricted Clifford/stabilizer fragment; universal quantum computation needs non-Clifford resources outside that efficiently simulable regime. ``` ## Tooling And Practical Course Checks - Question 57. MCQ: Which tool is the primary circuit-lab path in this course? A. Qiskit B. Excel C. LaTeX only D. Git alone ``` A. Qiskit. ``` - Question 58. When is QuTiP especially useful? ``` For density matrices, Hamiltonians, open systems, and noise dynamics. ``` - Question 59. MCQ: What should every serious notebook rely on instead of hiding all logic inline? A. Untested handwritten cells only B. Tested helpers in qcourse/ C. Screenshots only D. Hardware queues only ``` B. Tested helpers in qcourse/. ``` - Question 60. What is the practical loop for each concept note? ``` Read the note, run the per-concept notebook, inspect the qcourse helper, and check the corresponding test. ```