[
  {
    "id": "q001",
    "topic": "Quantum Mathematics",
    "difficulty": "Beginner",
    "question": "For $z=3+4i$, what is $|z|$?",
    "options": [
      "$5$",
      "$7$",
      "$1$",
      "$25$"
    ],
    "answer": 0,
    "explanation": "The modulus is $\\sqrt{3^2+4^2}=5$."
  },
  {
    "id": "q002",
    "topic": "Quantum Mathematics",
    "difficulty": "Beginner",
    "question": "What is the inner product $\\langle 0|1\\rangle$?",
    "options": [
      "$0$",
      "$1$",
      "$-1$",
      "$i$"
    ],
    "answer": 0,
    "explanation": "Computational basis vectors are orthogonal."
  },
  {
    "id": "q003",
    "topic": "Quantum Mathematics",
    "difficulty": "Intermediate",
    "question": "Which condition defines a unitary matrix $U$?",
    "options": [
      "$U^\\dagger U=I$",
      "$U^2=I$",
      "$U^T=U$",
      "$\\det U=0$"
    ],
    "answer": 0,
    "explanation": "A unitary operator preserves inner products and norms."
  },
  {
    "id": "q004",
    "topic": "Quantum Mathematics",
    "difficulty": "Beginner",
    "question": "What is the dimension of the Hilbert space of five qubits?",
    "options": [
      "$32$",
      "$10$",
      "$25$",
      "$64$"
    ],
    "answer": 0,
    "explanation": "The dimension is $2^n$, hence $2^5=32$."
  },
  {
    "id": "q005",
    "topic": "Quantum Mathematics",
    "difficulty": "Beginner",
    "question": "What is the trace of the identity on a qubit?",
    "options": [
      "$2$",
      "$1$",
      "$0$",
      "$4$"
    ],
    "answer": 0,
    "explanation": "The $2\\times2$ identity has two diagonal ones."
  },
  {
    "id": "q006",
    "topic": "Quantum Mathematics",
    "difficulty": "Intermediate",
    "question": "If $A$ is Hermitian, its eigenvalues are…",
    "options": [
      "Real",
      "Purely imaginary",
      "Always positive",
      "Always zero"
    ],
    "answer": 0,
    "explanation": "Hermiticity guarantees real eigenvalues, but they may be negative."
  },
  {
    "id": "q007",
    "topic": "Qubits & States",
    "difficulty": "Beginner",
    "question": "Which expression describes a normalized pure qubit?",
    "options": [
      "$\\alpha|0\\rangle+\\beta|1\\rangle$, $|\\alpha|^2+|\\beta|^2=1$",
      "$\\alpha|0\\rangle+\\beta|1\\rangle$, $\\alpha+\\beta=1$",
      "$|0\\rangle+|1\\rangle$",
      "$2|0\\rangle$"
    ],
    "answer": 0,
    "explanation": "Amplitudes must have squared magnitudes summing to one."
  },
  {
    "id": "q008",
    "topic": "Qubits & States",
    "difficulty": "Beginner",
    "question": "What is $|+\\rangle$?",
    "options": [
      "$(|0\\rangle+|1\\rangle)/\\sqrt2$",
      "$(|0\\rangle-|1\\rangle)/\\sqrt2$",
      "$|0\\rangle+|1\\rangle$",
      "$|1\\rangle$"
    ],
    "answer": 0,
    "explanation": "The plus state is the normalized equal superposition."
  },
  {
    "id": "q009",
    "topic": "Qubits & States",
    "difficulty": "Intermediate",
    "question": "What does a global phase $e^{i\\phi}$ change in an isolated pure state?",
    "options": [
      "No measurement probabilities",
      "Every measurement probability",
      "The Hilbert-space dimension",
      "The state norm"
    ],
    "answer": 0,
    "explanation": "Global phase cancels from Born probabilities."
  },
  {
    "id": "q010",
    "topic": "Qubits & States",
    "difficulty": "Beginner",
    "question": "What is the Bloch vector of $|0\\rangle$?",
    "options": [
      "North pole $+z$",
      "South pole $-z$",
      "Equator $+x$",
      "Equator $+y$"
    ],
    "answer": 0,
    "explanation": "$|0\\rangle$ is the $+1$ eigenstate of $Z$."
  },
  {
    "id": "q011",
    "topic": "Qubits & States",
    "difficulty": "Intermediate",
    "question": "Which state is orthogonal to $|+\\rangle$?",
    "options": [
      "$|-\\rangle$",
      "$|0\\rangle$",
      "$|1\\rangle$",
      "$i|+\\rangle$"
    ],
    "answer": 0,
    "explanation": "The inner product $\\langle +|-\\rangle$ vanishes."
  },
  {
    "id": "q012",
    "topic": "Qubits & States",
    "difficulty": "Advanced",
    "question": "How many independent real parameters specify a pure qubit up to global phase?",
    "options": [
      "$2$",
      "$1$",
      "$3$",
      "$4$"
    ],
    "answer": 0,
    "explanation": "The Bloch sphere uses polar and azimuthal angles."
  },
  {
    "id": "q013",
    "topic": "Quantum Gates",
    "difficulty": "Beginner",
    "question": "What is $X|0\\rangle$?",
    "options": [
      "$|1\\rangle$",
      "$|0\\rangle$",
      "$|+\\rangle$",
      "$i|0\\rangle$"
    ],
    "answer": 0,
    "explanation": "The Pauli-X gate flips the computational basis state."
  },
  {
    "id": "q014",
    "topic": "Quantum Gates",
    "difficulty": "Beginner",
    "question": "What is $H|0\\rangle$?",
    "options": [
      "$|+\\rangle$",
      "$|-\\rangle$",
      "$|1\\rangle$",
      "$|0\\rangle$"
    ],
    "answer": 0,
    "explanation": "Hadamard creates an equal superposition from $|0\\rangle$."
  },
  {
    "id": "q015",
    "topic": "Quantum Gates",
    "difficulty": "Intermediate",
    "question": "Which identity holds for the Hadamard gate?",
    "options": [
      "$H^2=I$",
      "$H^2=X$",
      "$H^2=Z$",
      "$H^2=0$"
    ],
    "answer": 0,
    "explanation": "Applying Hadamard twice returns the input."
  },
  {
    "id": "q016",
    "topic": "Quantum Gates",
    "difficulty": "Beginner",
    "question": "What is $Z|1\\rangle$?",
    "options": [
      "$-|1\\rangle$",
      "$|0\\rangle$",
      "$|1\\rangle$",
      "$i|1\\rangle$"
    ],
    "answer": 0,
    "explanation": "Pauli-Z adds a minus sign to the $|1\\rangle$ component."
  },
  {
    "id": "q017",
    "topic": "Quantum Gates",
    "difficulty": "Intermediate",
    "question": "Which gate has diagonal entries $1$ and $i$?",
    "options": [
      "$S$",
      "$X$",
      "$H$",
      "$Y$"
    ],
    "answer": 0,
    "explanation": "The phase gate is $S=\\mathrm{diag}(1,i)$."
  },
  {
    "id": "q018",
    "topic": "Quantum Gates",
    "difficulty": "Beginner",
    "question": "What is the output of CNOT on $|10\\rangle$ (first qubit control)?",
    "options": [
      "$|11\\rangle$",
      "$|10\\rangle$",
      "$|00\\rangle$",
      "$|01\\rangle$"
    ],
    "answer": 0,
    "explanation": "A control value of one flips the target bit."
  },
  {
    "id": "q019",
    "topic": "Measurement",
    "difficulty": "Beginner",
    "question": "Measuring $|+\\rangle$ in the $Z$ basis gives outcome 0 with probability…",
    "options": [
      "$1/2$",
      "$1$",
      "$0$",
      "$1/4$"
    ],
    "answer": 0,
    "explanation": "Born's rule squares the amplitude $1/\\sqrt2$."
  },
  {
    "id": "q020",
    "topic": "Measurement",
    "difficulty": "Beginner",
    "question": "What is the probability of outcome 1 for $\\sqrt{3/4}|0\\rangle+(1/2)|1\\rangle$?",
    "options": [
      "$1/4$",
      "$3/4$",
      "$1/2$",
      "$1$"
    ],
    "answer": 0,
    "explanation": "The $|1\\rangle$ amplitude has squared magnitude $1/4$."
  },
  {
    "id": "q021",
    "topic": "Measurement",
    "difficulty": "Intermediate",
    "question": "Which gate before a $Z$ measurement effectively measures in the $X$ basis?",
    "options": [
      "$H$",
      "$X$",
      "$Z$",
      "$S$"
    ],
    "answer": 0,
    "explanation": "Hadamard maps the $X$ eigenbasis to the computational basis."
  },
  {
    "id": "q022",
    "topic": "Measurement",
    "difficulty": "Beginner",
    "question": "After obtaining outcome 1 in a projective $Z$ measurement of a qubit, the post-measurement state is…",
    "options": [
      "$|1\\rangle$",
      "$|0\\rangle$",
      "$|+\\rangle$",
      "The original state unchanged"
    ],
    "answer": 0,
    "explanation": "The outcome conditions the state on the corresponding projector."
  },
  {
    "id": "q023",
    "topic": "Measurement",
    "difficulty": "Intermediate",
    "question": "For an observable $A$, what is the expectation in $|\\psi\\rangle$?",
    "options": [
      "$\\langle\\psi|A|\\psi\\rangle$",
      "$\\langle\\psi|\\psi\\rangle A$",
      "$A^2$",
      "$\\det A$"
    ],
    "answer": 0,
    "explanation": "The expectation is the quadratic form of the observable."
  },
  {
    "id": "q024",
    "topic": "Measurement",
    "difficulty": "Advanced",
    "question": "Which statement is correct for nonorthogonal pure states?",
    "options": [
      "They cannot be perfectly distinguished in a single shot",
      "They are always perfectly distinguishable",
      "They have identical amplitudes",
      "They cannot be prepared"
    ],
    "answer": 0,
    "explanation": "Nonzero overlap prevents zero-error single-copy discrimination."
  },
  {
    "id": "q025",
    "topic": "Entanglement",
    "difficulty": "Beginner",
    "question": "Which state is a Bell state?",
    "options": [
      "$(|00\\rangle+|11\\rangle)/\\sqrt2$",
      "$|00\\rangle$",
      "$|+\\rangle|+\\rangle$",
      "$|0\\rangle|+\\rangle$"
    ],
    "answer": 0,
    "explanation": "The Bell state cannot be factored into single-qubit states."
  },
  {
    "id": "q026",
    "topic": "Entanglement",
    "difficulty": "Intermediate",
    "question": "What is the reduced state of either qubit in $|\\Phi^+\\rangle$?",
    "options": [
      "$I/2$",
      "$|0\\rangle\\langle0|$",
      "$|+\\rangle\\langle+|$",
      "$I$"
    ],
    "answer": 0,
    "explanation": "Tracing out one Bell-pair qubit leaves a maximally mixed state."
  },
  {
    "id": "q027",
    "topic": "Entanglement",
    "difficulty": "Intermediate",
    "question": "What is the entanglement entropy of a Bell pair's one-qubit subsystem (base 2)?",
    "options": [
      "$1$ bit",
      "$0$ bits",
      "$2$ bits",
      "$1/2$ bit"
    ],
    "answer": 0,
    "explanation": "$S(I/2)=-2(1/2)\\log_2(1/2)=1$."
  },
  {
    "id": "q028",
    "topic": "Entanglement",
    "difficulty": "Beginner",
    "question": "Which circuit prepares $|\\Phi^+\\rangle$ from $|00\\rangle$?",
    "options": [
      "$H$ on qubit 0, then CNOT 0→1",
      "$X$ on qubit 0 only",
      "$Z$ on both qubits",
      "CNOT 0→1 only"
    ],
    "answer": 0,
    "explanation": "The Hadamard creates two branches; CNOT correlates them."
  },
  {
    "id": "q029",
    "topic": "Entanglement",
    "difficulty": "Advanced",
    "question": "Can local unitary operations alone change the entanglement entropy of a bipartite pure state?",
    "options": [
      "No",
      "Yes, always",
      "Only if the gates are Hadamards",
      "Only for two qubits"
    ],
    "answer": 0,
    "explanation": "Local unitaries preserve Schmidt coefficients."
  },
  {
    "id": "q030",
    "topic": "Entanglement",
    "difficulty": "Beginner",
    "question": "How many computational-basis outcomes can a GHZ state $(|000\\rangle+|111\\rangle)/\\sqrt2$ produce ideally?",
    "options": [
      "$2$",
      "$3$",
      "$4$",
      "$8$"
    ],
    "answer": 0,
    "explanation": "Only 000 and 111 have nonzero amplitudes."
  },
  {
    "id": "q031",
    "topic": "Quantum Circuits",
    "difficulty": "Beginner",
    "question": "In a circuit diagram, time conventionally progresses…",
    "options": [
      "Left to right",
      "Right to left",
      "Bottom to top",
      "Top to bottom"
    ],
    "answer": 0,
    "explanation": "Gate order is read along the wires from left to right."
  },
  {
    "id": "q032",
    "topic": "Quantum Circuits",
    "difficulty": "Intermediate",
    "question": "What does a barrier do in a Qiskit circuit?",
    "options": [
      "Marks a boundary for compilation and visualization",
      "Measures all qubits",
      "Resets the register",
      "Creates entanglement"
    ],
    "answer": 0,
    "explanation": "A barrier constrains some circuit reordering; it is not a quantum gate."
  },
  {
    "id": "q033",
    "topic": "Quantum Circuits",
    "difficulty": "Beginner",
    "question": "Which gate is controlled on two qubits and flips a third target?",
    "options": [
      "Toffoli (CCX)",
      "Hadamard",
      "SWAP",
      "Phase S"
    ],
    "answer": 0,
    "explanation": "CCX flips its target when both controls are one."
  },
  {
    "id": "q034",
    "topic": "Quantum Circuits",
    "difficulty": "Beginner",
    "question": "What is the action of SWAP on $|01\\rangle$?",
    "options": [
      "$|10\\rangle$",
      "$|11\\rangle$",
      "$|00\\rangle$",
      "$|01\\rangle$"
    ],
    "answer": 0,
    "explanation": "SWAP exchanges the states of two wires."
  },
  {
    "id": "q035",
    "topic": "Quantum Circuits",
    "difficulty": "Intermediate",
    "question": "What does circuit depth count?",
    "options": [
      "Longest sequence of dependent gate layers",
      "Total number of measured shots",
      "Number of classical bits",
      "Sum of gate angles"
    ],
    "answer": 0,
    "explanation": "Parallel operations can share a layer; depth tracks sequential layers."
  },
  {
    "id": "q036",
    "topic": "Quantum Circuits",
    "difficulty": "Beginner",
    "question": "How many complex amplitudes describe an arbitrary three-qubit state vector?",
    "options": [
      "$8$",
      "$3$",
      "$6$",
      "$16$"
    ],
    "answer": 0,
    "explanation": "There is one amplitude for each of $2^3$ basis strings."
  },
  {
    "id": "q037",
    "topic": "Quantum Teleportation",
    "difficulty": "Beginner",
    "question": "What pre-shared resource does standard qubit teleportation require?",
    "options": [
      "One entangled Bell pair",
      "Two uncorrelated classical bits",
      "A GHZ state of four qubits",
      "A copied unknown state"
    ],
    "answer": 0,
    "explanation": "Teleportation consumes a shared entangled pair."
  },
  {
    "id": "q038",
    "topic": "Quantum Teleportation",
    "difficulty": "Beginner",
    "question": "How many classical bits does standard single-qubit teleportation send?",
    "options": [
      "$2$",
      "$1$",
      "$0$",
      "$3$"
    ],
    "answer": 0,
    "explanation": "Alice sends her two Bell-measurement outcomes to Bob."
  },
  {
    "id": "q039",
    "topic": "Quantum Teleportation",
    "difficulty": "Intermediate",
    "question": "Does teleportation transmit information faster than light?",
    "options": [
      "No",
      "Yes, for Bell pairs",
      "Yes, if qubits are pure",
      "Yes, without measurements"
    ],
    "answer": 0,
    "explanation": "Bob needs classical communication before he can recover the state."
  },
  {
    "id": "q040",
    "topic": "Quantum Teleportation",
    "difficulty": "Intermediate",
    "question": "What correction gates can Bob require in standard teleportation?",
    "options": [
      "$X$ and $Z$",
      "Only $H$",
      "Only $S$",
      "Only CNOT"
    ],
    "answer": 0,
    "explanation": "The two classical bits select Pauli corrections."
  },
  {
    "id": "q041",
    "topic": "Quantum Teleportation",
    "difficulty": "Intermediate",
    "question": "What happens to Alice's original unknown input state during teleportation?",
    "options": [
      "It is destroyed by the measurement",
      "It remains as a second copy",
      "It becomes a classical probability vector only",
      "It is amplified"
    ],
    "answer": 0,
    "explanation": "Teleportation does not violate no-cloning; Alice no longer retains the original."
  },
  {
    "id": "q042",
    "topic": "Superdense Coding",
    "difficulty": "Beginner",
    "question": "How many classical bits can one transmitted qubit communicate with one pre-shared ebit?",
    "options": [
      "$2$",
      "$1$",
      "$3$",
      "$4$"
    ],
    "answer": 0,
    "explanation": "The four Bell states encode two bits when entanglement is pre-shared."
  },
  {
    "id": "q043",
    "topic": "Superdense Coding",
    "difficulty": "Beginner",
    "question": "How many orthogonal Bell states exist for two qubits?",
    "options": [
      "$4$",
      "$2$",
      "$8$",
      "$1$"
    ],
    "answer": 0,
    "explanation": "The Bell basis spans the four-dimensional two-qubit Hilbert space."
  },
  {
    "id": "q044",
    "topic": "Superdense Coding",
    "difficulty": "Intermediate",
    "question": "Which operation can distinguish Bell states in a standard decoder?",
    "options": [
      "CNOT followed by Hadamard on control",
      "Only an $X$ gate",
      "Only a $Z$ gate",
      "Only a phase gate"
    ],
    "answer": 0,
    "explanation": "The decoder maps Bell states onto computational basis states."
  },
  {
    "id": "q045",
    "topic": "Superdense Coding",
    "difficulty": "Intermediate",
    "question": "Is superdense coding possible with one sent qubit and no shared entanglement under the standard noiseless model?",
    "options": [
      "No",
      "Yes, always",
      "Yes, with a Hadamard only",
      "Yes, with a classical coin"
    ],
    "answer": 0,
    "explanation": "The pre-shared ebit is essential to the two-bit-per-qubit protocol."
  },
  {
    "id": "q046",
    "topic": "Superdense Coding",
    "difficulty": "Beginner",
    "question": "Which quantity is consumed by standard superdense coding?",
    "options": [
      "Shared entanglement",
      "An extra transmitted classical bit",
      "A third qubit sent by Bob",
      "A measurement oracle"
    ],
    "answer": 0,
    "explanation": "The Bell pair is used as the communication resource."
  },
  {
    "id": "q047",
    "topic": "Deutsch–Jozsa",
    "difficulty": "Beginner",
    "question": "What promise does Deutsch–Jozsa assume?",
    "options": [
      "The Boolean function is constant or balanced",
      "The function is injective or surjective",
      "The function is periodic or random",
      "The function is linear or quadratic"
    ],
    "answer": 0,
    "explanation": "The algorithm distinguishes constant from balanced promised functions."
  },
  {
    "id": "q048",
    "topic": "Deutsch–Jozsa",
    "difficulty": "Intermediate",
    "question": "For a constant oracle, what is the ideal probability of measuring $0^n$ on the input register?",
    "options": [
      "$1$",
      "$0$",
      "$1/2$",
      "$1/2^n$"
    ],
    "answer": 0,
    "explanation": "All amplitudes interfere constructively at the all-zero string."
  },
  {
    "id": "q049",
    "topic": "Deutsch–Jozsa",
    "difficulty": "Intermediate",
    "question": "What is the purpose of phase kickback in Deutsch–Jozsa?",
    "options": [
      "Encode $f(x)$ as a relative phase",
      "Clone the oracle register",
      "Remove the need for a unitary",
      "Measure the output without disturbance"
    ],
    "answer": 0,
    "explanation": "The phase $(-1)^{f(x)}$ carries the oracle's information."
  },
  {
    "id": "q050",
    "topic": "Deutsch–Jozsa",
    "difficulty": "Intermediate",
    "question": "How many oracle queries does ideal Deutsch–Jozsa need under its promise?",
    "options": [
      "$1$",
      "$2$",
      "$n$",
      "$2^n$"
    ],
    "answer": 0,
    "explanation": "One coherent oracle query suffices in the standard formulation."
  },
  {
    "id": "q051",
    "topic": "Deutsch–Jozsa",
    "difficulty": "Intermediate",
    "question": "What is the standard target-qubit preparation before the oracle?",
    "options": [
      "$|-\\rangle$",
      "$|0\\rangle$",
      "$|+\\rangle$",
      "$|1\\rangle$"
    ],
    "answer": 0,
    "explanation": "The minus state is an eigenstate of bit-flip with eigenvalue −1."
  },
  {
    "id": "q052",
    "topic": "Bernstein–Vazirani",
    "difficulty": "Beginner",
    "question": "What hidden structure does Bernstein–Vazirani recover?",
    "options": [
      "A bit string $s$ in $f(x)=s\\cdot x\\bmod2$",
      "An unknown prime factor",
      "A Hamiltonian ground state",
      "A noisy channel capacity"
    ],
    "answer": 0,
    "explanation": "The oracle hides a parity mask."
  },
  {
    "id": "q053",
    "topic": "Bernstein–Vazirani",
    "difficulty": "Intermediate",
    "question": "How many ideal oracle calls recover an $n$-bit secret string?",
    "options": [
      "$1$",
      "$n$",
      "$2^n$",
      "$n^2$"
    ],
    "answer": 0,
    "explanation": "Phase interference reveals all bits of the parity mask in one query."
  },
  {
    "id": "q054",
    "topic": "Bernstein–Vazirani",
    "difficulty": "Intermediate",
    "question": "What does the final input-register measurement return ideally?",
    "options": [
      "The hidden string $s$",
      "Only its Hamming weight",
      "A random string",
      "The bitwise complement of $s$"
    ],
    "answer": 0,
    "explanation": "The Hadamard transform decodes the phase pattern."
  },
  {
    "id": "q055",
    "topic": "Bernstein–Vazirani",
    "difficulty": "Intermediate",
    "question": "For secret $s=101$, what is $f(110)$ using binary dot product mod 2?",
    "options": [
      "$1$",
      "$0$",
      "$2$",
      "$3$"
    ],
    "answer": 0,
    "explanation": "$1\\cdot1+0\\cdot1+1\\cdot0=1\\bmod2$."
  },
  {
    "id": "q056",
    "topic": "Bernstein–Vazirani",
    "difficulty": "Beginner",
    "question": "Which mathematical transform appears before and after the oracle?",
    "options": [
      "Hadamard transform",
      "Quantum Fourier transform only",
      "Amplitude damping",
      "Partial trace"
    ],
    "answer": 0,
    "explanation": "Hadamards convert parity phases to a computational-basis answer."
  },
  {
    "id": "q057",
    "topic": "Simon’s Algorithm",
    "difficulty": "Intermediate",
    "question": "What is Simon's hidden promise?",
    "options": [
      "$f(x)=f(y)$ iff $x=y$ or $x\\oplus y=s$",
      "$f(x)=x^2$",
      "$f$ is constant or balanced",
      "$f$ is a permutation only"
    ],
    "answer": 0,
    "explanation": "The nonzero mask $s$ defines a two-to-one periodicity."
  },
  {
    "id": "q058",
    "topic": "Simon’s Algorithm",
    "difficulty": "Intermediate",
    "question": "What equation must measured strings $y$ satisfy in Simon's algorithm?",
    "options": [
      "$y\\cdot s=0\\pmod2$",
      "$y+s=1$",
      "$y=s$",
      "$y^2=s$"
    ],
    "answer": 0,
    "explanation": "Interference yields linear constraints orthogonal to the secret."
  },
  {
    "id": "q059",
    "topic": "Simon’s Algorithm",
    "difficulty": "Intermediate",
    "question": "Over which field are Simon's linear equations solved?",
    "options": [
      "$\\mathbb F_2$",
      "$\\mathbb R$",
      "$\\mathbb C$",
      "$\\mathbb Q$"
    ],
    "answer": 0,
    "explanation": "The equations are binary, with arithmetic modulo two."
  },
  {
    "id": "q060",
    "topic": "Simon’s Algorithm",
    "difficulty": "Intermediate",
    "question": "Why are multiple runs needed in Simon's algorithm?",
    "options": [
      "To collect independent linear constraints",
      "To create a longer secret",
      "To avoid using Hadamards",
      "To turn the oracle classical"
    ],
    "answer": 0,
    "explanation": "One sample generally gives only one linear equation."
  },
  {
    "id": "q061",
    "topic": "Simon’s Algorithm",
    "difficulty": "Advanced",
    "question": "If $s=11$ for two bits, which measured $y$ satisfies $y\\cdot s=0\\bmod2$?",
    "options": [
      "$11$",
      "$01$",
      "$10$",
      "$00$ only"
    ],
    "answer": 0,
    "explanation": "$11\\cdot11=1+1=0\\bmod2$; 00 also satisfies, so the nonzero choice is 11."
  },
  {
    "id": "q062",
    "topic": "Quantum Fourier Transform",
    "difficulty": "Beginner",
    "question": "What is the dimension of the QFT matrix on three qubits?",
    "options": [
      "$8\\times8$",
      "$3\\times3$",
      "$6\\times6$",
      "$16\\times16$"
    ],
    "answer": 0,
    "explanation": "Three qubits span eight basis states."
  },
  {
    "id": "q063",
    "topic": "Quantum Fourier Transform",
    "difficulty": "Intermediate",
    "question": "What is the QFT of $|0\\rangle$ on an $N$-dimensional register?",
    "options": [
      "Uniform superposition $\\frac1{\\sqrt N}\\sum_k|k\\rangle$",
      "$|N-1\\rangle$",
      "$|0\\rangle$ only",
      "The zero vector"
    ],
    "answer": 0,
    "explanation": "All phases equal one for input index zero."
  },
  {
    "id": "q064",
    "topic": "Quantum Fourier Transform",
    "difficulty": "Intermediate",
    "question": "Which gates typically implement the QFT?",
    "options": [
      "Hadamards and controlled phase rotations",
      "Only X gates",
      "Only CNOT gates",
      "Only measurements"
    ],
    "answer": 0,
    "explanation": "The QFT uses phase rotations conditioned on other qubits."
  },
  {
    "id": "q065",
    "topic": "Quantum Fourier Transform",
    "difficulty": "Intermediate",
    "question": "Is the QFT unitary?",
    "options": [
      "Yes",
      "No, because it measures",
      "No, because it is irreversible",
      "Only for one qubit"
    ],
    "answer": 0,
    "explanation": "The Fourier matrix's columns are orthonormal."
  },
  {
    "id": "q066",
    "topic": "Quantum Fourier Transform",
    "difficulty": "Intermediate",
    "question": "Why are SWAP gates sometimes placed at the end of a QFT circuit?",
    "options": [
      "To reverse qubit order",
      "To create ancillas",
      "To measure phase",
      "To reset qubits"
    ],
    "answer": 0,
    "explanation": "The standard decomposition produces output wires in reversed significance order."
  },
  {
    "id": "q067",
    "topic": "Phase Estimation",
    "difficulty": "Intermediate",
    "question": "What quantity does quantum phase estimation infer?",
    "options": [
      "$\\theta$ in $U|u\\rangle=e^{2\\pi i\\theta}|u\\rangle$",
      "Only the norm of $U$",
      "The number of CNOT gates",
      "The physical temperature"
    ],
    "answer": 0,
    "explanation": "Eigenphases are encoded through controlled powers of $U$."
  },
  {
    "id": "q068",
    "topic": "Phase Estimation",
    "difficulty": "Intermediate",
    "question": "What is the role of inverse QFT in phase estimation?",
    "options": [
      "Decode phase into a binary measurement distribution",
      "Prepare a Bell pair",
      "Reset the eigenstate",
      "Eliminate controlled unitaries"
    ],
    "answer": 0,
    "explanation": "Inverse QFT transforms phase gradients into computational basis peaks."
  },
  {
    "id": "q069",
    "topic": "Phase Estimation",
    "difficulty": "Advanced",
    "question": "What happens if the target is an exact eigenstate of $U$?",
    "options": [
      "Controlled powers imprint the eigenphase on controls",
      "The state is necessarily destroyed",
      "The target must be measured first",
      "The unitary becomes nonunitary"
    ],
    "answer": 0,
    "explanation": "The eigenstate contributes a scalar phase under each controlled power."
  },
  {
    "id": "q070",
    "topic": "Phase Estimation",
    "difficulty": "Intermediate",
    "question": "If $\\theta=1/4$ and two counting qubits suffice exactly, which binary outcome is expected (MSB first)?",
    "options": [
      "$01$",
      "$10$",
      "$11$",
      "$00$"
    ],
    "answer": 0,
    "explanation": "$1/4=0.01_2$ on two counting bits."
  },
  {
    "id": "q071",
    "topic": "Phase Estimation",
    "difficulty": "Intermediate",
    "question": "What improves the binary precision of ideal phase estimation?",
    "options": [
      "More counting qubits",
      "Fewer controlled operations",
      "Measuring only the target",
      "Replacing QFT with X gates"
    ],
    "answer": 0,
    "explanation": "Additional counting qubits resolve finer binary phase digits."
  },
  {
    "id": "q072",
    "topic": "Grover Search",
    "difficulty": "Beginner",
    "question": "What is Grover's ideal query complexity for one marked item among $N$?",
    "options": [
      "$O(\\sqrt N)$",
      "$O(N)$",
      "$O(\\log N)$",
      "$O(N^2)$"
    ],
    "answer": 0,
    "explanation": "Amplitude amplification achieves quadratic query improvement."
  },
  {
    "id": "q073",
    "topic": "Grover Search",
    "difficulty": "Intermediate",
    "question": "What does the Grover oracle do to a marked basis state?",
    "options": [
      "Flips its phase",
      "Measures it",
      "Copies it",
      "Deletes it"
    ],
    "answer": 0,
    "explanation": "A phase oracle multiplies marked amplitudes by −1."
  },
  {
    "id": "q074",
    "topic": "Grover Search",
    "difficulty": "Intermediate",
    "question": "What does the diffusion operator reflect amplitudes about?",
    "options": [
      "Their mean",
      "Zero only",
      "The largest amplitude",
      "The imaginary axis only"
    ],
    "answer": 0,
    "explanation": "The inversion-about-the-mean step amplifies marked amplitudes."
  },
  {
    "id": "q075",
    "topic": "Grover Search",
    "difficulty": "Advanced",
    "question": "Approximately how many Grover iterations are optimal for one marked item?",
    "options": [
      "$(\\pi/4)\\sqrt N$",
      "$N$",
      "$N/2$",
      "$\\log_2 N$"
    ],
    "answer": 0,
    "explanation": "Rotations in the two-dimensional marked/unmarked subspace determine the optimum."
  },
  {
    "id": "q076",
    "topic": "Grover Search",
    "difficulty": "Intermediate",
    "question": "What can happen if Grover iterations continue far beyond the optimum?",
    "options": [
      "Success probability decreases again",
      "Success becomes exactly one forever",
      "The oracle stops being unitary",
      "The register becomes classical"
    ],
    "answer": 0,
    "explanation": "Grover amplification is an oscillation, not a monotone process."
  },
  {
    "id": "q077",
    "topic": "Shor & Order Finding",
    "difficulty": "Intermediate",
    "question": "What classical subproblem does Shor's factoring method reduce to quantum order finding?",
    "options": [
      "Find $r$ with $a^r\\equiv1\\pmod N$",
      "Find the largest prime below $N$",
      "Compute a matrix determinant",
      "Sort a list of integers"
    ],
    "answer": 0,
    "explanation": "The quantum stage estimates the multiplicative order modulo $N$."
  },
  {
    "id": "q078",
    "topic": "Shor & Order Finding",
    "difficulty": "Intermediate",
    "question": "What condition is required before seeking the order of $a$ modulo $N$?",
    "options": [
      "$\\gcd(a,N)=1$",
      "$a=N$",
      "$a=0$",
      "$N$ must be even"
    ],
    "answer": 0,
    "explanation": "Only coprime elements belong to the multiplicative group modulo $N$."
  },
  {
    "id": "q079",
    "topic": "Shor & Order Finding",
    "difficulty": "Advanced",
    "question": "What classical method recovers a rational approximation to an estimated phase?",
    "options": [
      "Continued fractions",
      "Bubble sort",
      "Gaussian elimination only",
      "Newton's cooling law"
    ],
    "answer": 0,
    "explanation": "Continued fractions help reconstruct the order denominator."
  },
  {
    "id": "q080",
    "topic": "Shor & Order Finding",
    "difficulty": "Advanced",
    "question": "If $a^r\\equiv1\\pmod N$ and $r$ is even, which values can yield nontrivial factors?",
    "options": [
      "$\\gcd(a^{r/2}\\pm1,N)$",
      "$\\gcd(a^r,N)$",
      "$a+r$ and $a-r$",
      "$N/2$ only"
    ],
    "answer": 0,
    "explanation": "Factoring $a^r-1$ as a difference of squares enables gcd extraction."
  },
  {
    "id": "q081",
    "topic": "Shor & Order Finding",
    "difficulty": "Intermediate",
    "question": "Does a toy order-finding circuit automatically constitute a scalable implementation of Shor's algorithm?",
    "options": [
      "No",
      "Yes, always",
      "Only if it has two qubits",
      "Only if it uses a Hadamard"
    ],
    "answer": 0,
    "explanation": "Scalable modular exponentiation and fault-tolerant resources are substantial requirements."
  },
  {
    "id": "q082",
    "topic": "Quantum Noise",
    "difficulty": "Beginner",
    "question": "Which channel models energy relaxation from $|1\\rangle$ toward $|0\\rangle$?",
    "options": [
      "Amplitude damping",
      "Pure dephasing",
      "Ideal unitary rotation",
      "Identity channel"
    ],
    "answer": 0,
    "explanation": "Amplitude damping captures loss of excitation."
  },
  {
    "id": "q083",
    "topic": "Quantum Noise",
    "difficulty": "Intermediate",
    "question": "Which process suppresses off-diagonal density-matrix coherence without necessarily changing populations?",
    "options": [
      "Dephasing",
      "Bit-flip only",
      "SWAP",
      "Teleportation"
    ],
    "answer": 0,
    "explanation": "Phase noise reduces coherence between basis states."
  },
  {
    "id": "q084",
    "topic": "Quantum Noise",
    "difficulty": "Advanced",
    "question": "What is a valid quantum channel required to be?",
    "options": [
      "Completely positive and trace preserving",
      "Only Hermitian",
      "Only unitary",
      "Always invertible"
    ],
    "answer": 0,
    "explanation": "CPTP maps describe deterministic physical quantum channels."
  },
  {
    "id": "q085",
    "topic": "Quantum Noise",
    "difficulty": "Intermediate",
    "question": "What does $T_1$ commonly characterize?",
    "options": [
      "Energy relaxation time",
      "Gate clock frequency",
      "Number of qubits",
      "State-vector dimension"
    ],
    "answer": 0,
    "explanation": "$T_1$ describes longitudinal population relaxation."
  },
  {
    "id": "q086",
    "topic": "Quantum Noise",
    "difficulty": "Intermediate",
    "question": "What does $T_2$ commonly characterize?",
    "options": [
      "Transverse coherence decay time",
      "Number of measurements",
      "Classical bit capacity",
      "Oracle depth"
    ],
    "answer": 0,
    "explanation": "$T_2$ captures decoherence of off-diagonal terms."
  },
  {
    "id": "q087",
    "topic": "Quantum Error Correction",
    "difficulty": "Beginner",
    "question": "What is the basic idea of a logical qubit?",
    "options": [
      "Encode information across multiple physical qubits",
      "Measure the unknown amplitude directly",
      "Copy an arbitrary state perfectly",
      "Remove all gates"
    ],
    "answer": 0,
    "explanation": "Redundant entangled encoding protects logical information against certain errors."
  },
  {
    "id": "q088",
    "topic": "Quantum Error Correction",
    "difficulty": "Intermediate",
    "question": "What does a stabilizer syndrome measurement reveal ideally?",
    "options": [
      "Information about errors without directly revealing logical amplitudes",
      "The complete unknown logical state",
      "A copied logical qubit",
      "Every gate's runtime"
    ],
    "answer": 0,
    "explanation": "Syndrome measurements identify error signatures while preserving encoded information."
  },
  {
    "id": "q089",
    "topic": "Quantum Error Correction",
    "difficulty": "Intermediate",
    "question": "What is the distance of the three-qubit repetition code against bit flips?",
    "options": [
      "$3$",
      "$1$",
      "$2$",
      "$5$"
    ],
    "answer": 0,
    "explanation": "The logical codewords 000 and 111 differ in three positions."
  },
  {
    "id": "q090",
    "topic": "Quantum Error Correction",
    "difficulty": "Intermediate",
    "question": "Which errors does the three-qubit bit-flip repetition code directly correct?",
    "options": [
      "One physical $X$ error",
      "Arbitrary one-qubit errors",
      "Two independent $X$ errors",
      "Any three-qubit unitary"
    ],
    "answer": 0,
    "explanation": "It corrects a single bit flip, not general phase errors."
  },
  {
    "id": "q091",
    "topic": "Quantum Error Correction",
    "difficulty": "Advanced",
    "question": "What is a code of distance $d$ guaranteed to correct under standard assumptions?",
    "options": [
      "Up to $\\lfloor(d-1)/2\\rfloor$ arbitrary physical-qubit errors",
      "Exactly $d$ errors",
      "Any number of errors",
      "Only measurement errors"
    ],
    "answer": 0,
    "explanation": "Distance bounds the number of correctable independent errors."
  },
  {
    "id": "q092",
    "topic": "Quantum Cryptography",
    "difficulty": "Beginner",
    "question": "What physical principle helps detect eavesdropping in BB84?",
    "options": [
      "Disturbance from measuring incompatible bases",
      "Perfect cloning",
      "Superluminal signaling",
      "Classical one-time pads alone"
    ],
    "answer": 0,
    "explanation": "An eavesdropper's wrong-basis measurements introduce detectable errors."
  },
  {
    "id": "q093",
    "topic": "Quantum Cryptography",
    "difficulty": "Beginner",
    "question": "How many polarization or qubit states are used in ideal BB84 encoding?",
    "options": [
      "$4$",
      "$2$",
      "$6$",
      "$8$"
    ],
    "answer": 0,
    "explanation": "BB84 uses two mutually unbiased bases with two states each."
  },
  {
    "id": "q094",
    "topic": "Quantum Cryptography",
    "difficulty": "Intermediate",
    "question": "What happens during basis sifting in BB84?",
    "options": [
      "Alice and Bob retain rounds with matching preparation/measurement bases",
      "They publish the raw secret bits",
      "They amplify all photons",
      "They eliminate all losses"
    ],
    "answer": 0,
    "explanation": "Matching bases are retained before error estimation and privacy amplification."
  },
  {
    "id": "q095",
    "topic": "Quantum Cryptography",
    "difficulty": "Intermediate",
    "question": "Does BB84 security require an authenticated classical channel?",
    "options": [
      "Yes",
      "No",
      "Only if no eavesdropper exists",
      "Only for single photons"
    ],
    "answer": 0,
    "explanation": "Authentication prevents man-in-the-middle substitution of classical messages."
  },
  {
    "id": "q096",
    "topic": "Quantum Cryptography",
    "difficulty": "Advanced",
    "question": "What does privacy amplification aim to reduce?",
    "options": [
      "An adversary's information about the final key",
      "Quantum gate depth",
      "The number of bases",
      "Optical wavelength"
    ],
    "answer": 0,
    "explanation": "Hashing compresses partially secret raw bits into a shorter, more secure key."
  },
  {
    "id": "q097",
    "topic": "Quantum Information Theory",
    "difficulty": "Beginner",
    "question": "What is the von Neumann entropy of a pure quantum state?",
    "options": [
      "$0$",
      "$1$ bit always",
      "$2$ bits always",
      "Infinite"
    ],
    "answer": 0,
    "explanation": "A pure density matrix has eigenvalues 1 and zeros."
  },
  {
    "id": "q098",
    "topic": "Quantum Information Theory",
    "difficulty": "Intermediate",
    "question": "What is the entropy of the maximally mixed single-qubit state in bits?",
    "options": [
      "$1$",
      "$0$",
      "$2$",
      "$1/2$"
    ],
    "answer": 0,
    "explanation": "$S(I/2)=-\\sum_{j=1}^2(1/2)\\log_2(1/2)=1$."
  },
  {
    "id": "q099",
    "topic": "Quantum Information Theory",
    "difficulty": "Intermediate",
    "question": "What is the purity $\\operatorname{Tr}(\\rho^2)$ of a pure state?",
    "options": [
      "$1$",
      "$0$",
      "$1/2$",
      "$2$"
    ],
    "answer": 0,
    "explanation": "Projectors satisfy $\\rho^2=\\rho$ and have unit trace."
  },
  {
    "id": "q100",
    "topic": "Quantum Information Theory",
    "difficulty": "Beginner",
    "question": "What is the trace of any normalized density operator?",
    "options": [
      "$1$",
      "$0$",
      "$2$",
      "It depends on basis"
    ],
    "answer": 0,
    "explanation": "Trace one ensures total probability equals one."
  },
  {
    "id": "q101",
    "topic": "Quantum Information Theory",
    "difficulty": "Intermediate",
    "question": "What does partial trace compute?",
    "options": [
      "A subsystem's reduced density operator",
      "A gate's inverse",
      "A classical parity bit",
      "An eigenphase only"
    ],
    "answer": 0,
    "explanation": "Tracing out the environment gives local statistics."
  },
  {
    "id": "q102",
    "topic": "Variational Algorithms",
    "difficulty": "Beginner",
    "question": "What is optimized in VQE?",
    "options": [
      "A parameterized circuit's energy expectation",
      "The number of classical bits sent",
      "The Hilbert-space dimension",
      "An unknown oracle's exact truth table"
    ],
    "answer": 0,
    "explanation": "VQE minimizes an estimated Hamiltonian expectation."
  },
  {
    "id": "q103",
    "topic": "Variational Algorithms",
    "difficulty": "Intermediate",
    "question": "What mathematical principle makes VQE an upper-bound method for ground-state energy?",
    "options": [
      "Variational principle",
      "No-cloning theorem",
      "Bell inequality",
      "Central limit theorem only"
    ],
    "answer": 0,
    "explanation": "Any normalized trial state's energy is at least the ground-state energy."
  },
  {
    "id": "q104",
    "topic": "Variational Algorithms",
    "difficulty": "Intermediate",
    "question": "What is a classical optimizer's role in VQE?",
    "options": [
      "Update circuit parameters using measured objective values",
      "Execute quantum gates on hardware",
      "Prepare photons directly",
      "Perform quantum measurement collapse"
    ],
    "answer": 0,
    "explanation": "A classical routine chooses parameters between quantum evaluations."
  },
  {
    "id": "q105",
    "topic": "Variational Algorithms",
    "difficulty": "Intermediate",
    "question": "What is the central aim of QAOA?",
    "options": [
      "Optimize a cost function using alternating cost and mixer unitaries",
      "Factor every integer exactly",
      "Teleport arbitrary registers",
      "Measure nonorthogonal states perfectly"
    ],
    "answer": 0,
    "explanation": "QAOA uses variational alternating layers for combinatorial objectives."
  },
  {
    "id": "q106",
    "topic": "Variational Algorithms",
    "difficulty": "Advanced",
    "question": "What is a barren plateau?",
    "options": [
      "A training region with exponentially small gradients in some settings",
      "A noiseless quantum channel",
      "A perfect Bell state",
      "A particular Pauli gate"
    ],
    "answer": 0,
    "explanation": "Vanishing gradients can make variational optimization difficult."
  },
  {
    "id": "q107",
    "topic": "Quantum Hardware",
    "difficulty": "Beginner",
    "question": "Which platform uses superconducting Josephson circuits as qubits?",
    "options": [
      "Superconducting qubits",
      "Trapped ions",
      "Neutral atoms only",
      "Photonic dual-rail qubits"
    ],
    "answer": 0,
    "explanation": "Josephson junction nonlinearity creates addressable circuit levels."
  },
  {
    "id": "q108",
    "topic": "Quantum Hardware",
    "difficulty": "Beginner",
    "question": "Which platform often encodes qubits in internal levels of charged atoms?",
    "options": [
      "Trapped-ion quantum computing",
      "Superconducting transmons",
      "Silicon CMOS logic only",
      "Classical optical fiber"
    ],
    "answer": 0,
    "explanation": "Electromagnetic traps confine ions manipulated with fields or lasers."
  },
  {
    "id": "q109",
    "topic": "Quantum Hardware",
    "difficulty": "Intermediate",
    "question": "What interaction is often used to entangle neutral-atom Rydberg qubits?",
    "options": [
      "Rydberg blockade",
      "Capacitive charging of Josephson junctions",
      "Classical XOR alone",
      "Electron spin resonance in bulk metal"
    ],
    "answer": 0,
    "explanation": "Strong Rydberg interactions enable conditional dynamics."
  },
  {
    "id": "q110",
    "topic": "Quantum Hardware",
    "difficulty": "Intermediate",
    "question": "What is a key practical concern for photonic quantum computing?",
    "options": [
      "Photon loss",
      "Josephson critical current",
      "Ion motional heating only",
      "Nuclear spin diffusion only"
    ],
    "answer": 0,
    "explanation": "Lost photons reduce successful transmission and detection."
  },
  {
    "id": "q111",
    "topic": "Quantum Hardware",
    "difficulty": "Intermediate",
    "question": "What does qubit connectivity constrain on hardware?",
    "options": [
      "Which pairs can directly perform native two-qubit gates",
      "The number of classical variables",
      "The mathematical value of $\\pi$",
      "The meaning of normalization"
    ],
    "answer": 0,
    "explanation": "Limited coupling can require routing and SWAP overhead."
  },
  {
    "id": "q112",
    "topic": "Quantum Sensing & Metrology",
    "difficulty": "Intermediate",
    "question": "What is the standard quantum limit scaling for independent probes in ideal estimation?",
    "options": [
      "$1/\\sqrt N$",
      "$1/N^2$",
      "$N$",
      "$\\sqrt N$"
    ],
    "answer": 0,
    "explanation": "Independent statistical samples typically yield inverse-square-root precision scaling."
  },
  {
    "id": "q113",
    "topic": "Quantum Sensing & Metrology",
    "difficulty": "Advanced",
    "question": "What scaling can ideal entangled probes achieve in suitable noiseless phase-estimation models?",
    "options": [
      "$1/N$",
      "$1/\\sqrt N$",
      "$N$",
      "$N^2$"
    ],
    "answer": 0,
    "explanation": "Heisenberg scaling improves the ideal resource dependence to inverse N."
  },
  {
    "id": "q114",
    "topic": "Quantum Sensing & Metrology",
    "difficulty": "Intermediate",
    "question": "What physical effect is exploited in Ramsey interferometry?",
    "options": [
      "Phase accumulation between coherent superposition components",
      "Permanent copying of unknown states",
      "Classical sorting",
      "Dissipative reset only"
    ],
    "answer": 0,
    "explanation": "A free-evolution interval converts frequency differences to phase."
  },
  {
    "id": "q115",
    "topic": "Quantum Sensing & Metrology",
    "difficulty": "Intermediate",
    "question": "What is a common purpose of squeezed light in precision sensing?",
    "options": [
      "Reduce noise in one quadrature at the expense of the conjugate quadrature",
      "Reduce uncertainty in both conjugate quadratures below their joint bound",
      "Remove photon shot noise in all observables",
      "Make photons massive"
    ],
    "answer": 0,
    "explanation": "Squeezing redistributes quadrature uncertainty while respecting Heisenberg limits."
  },
  {
    "id": "q116",
    "topic": "Quantum Sensing & Metrology",
    "difficulty": "Advanced",
    "question": "What is the quantum Fisher information used for?",
    "options": [
      "Bounding attainable parameter-estimation precision",
      "Counting classical transistors",
      "Factoring integers directly",
      "Measuring temperature without a probe"
    ],
    "answer": 0,
    "explanation": "Quantum Cramér–Rao bounds relate Fisher information to estimation variance."
  },
  {
    "id": "q117",
    "topic": "Quantum Machine Learning",
    "difficulty": "Beginner",
    "question": "What is a quantum feature map?",
    "options": [
      "A procedure encoding classical data into quantum states",
      "A classical encryption key",
      "A measurement that copies states",
      "A hardware cooling cycle"
    ],
    "answer": 0,
    "explanation": "Feature maps embed inputs into a quantum Hilbert space."
  },
  {
    "id": "q118",
    "topic": "Quantum Machine Learning",
    "difficulty": "Intermediate",
    "question": "What is a quantum kernel often based on?",
    "options": [
      "Overlap or fidelity between encoded quantum states",
      "A prime factorization oracle",
      "A classical RAM address",
      "A photon wavelength alone"
    ],
    "answer": 0,
    "explanation": "Quantum kernels quantify similarity through encoded-state overlaps."
  },
  {
    "id": "q119",
    "topic": "Quantum Machine Learning",
    "difficulty": "Beginner",
    "question": "What is a parameterized quantum circuit?",
    "options": [
      "A circuit with adjustable gate parameters",
      "A circuit with no gates",
      "A circuit that never measures",
      "A circuit restricted to Clifford gates"
    ],
    "answer": 0,
    "explanation": "Rotation angles and other gate settings can be trained."
  },
  {
    "id": "q120",
    "topic": "Quantum Machine Learning",
    "difficulty": "Intermediate",
    "question": "Does quantum machine learning universally outperform classical machine learning?",
    "options": [
      "No",
      "Yes, for every dataset",
      "Yes, whenever qubits exceed features",
      "Yes, without data encoding"
    ],
    "answer": 0,
    "explanation": "Performance depends on data, assumptions, hardware, and comparison baselines."
  },
  {
    "id": "q121",
    "topic": "Quantum Machine Learning",
    "difficulty": "Advanced",
    "question": "What is the parameter-shift rule used for in suitable variational circuits?",
    "options": [
      "Estimating gradients from shifted circuit evaluations",
      "Increasing qubit count",
      "Eliminating measurement noise completely",
      "Computing matrix rank only"
    ],
    "answer": 0,
    "explanation": "Certain gate generators permit analytic gradients from shifted expectation values."
  },
  {
    "id": "q122",
    "topic": "Quantum Simulation",
    "difficulty": "Intermediate",
    "question": "What does Hamiltonian simulation approximate?",
    "options": [
      "Time evolution $e^{-iHt/\\hbar}$",
      "The determinant of $H$ only",
      "The transpose of $H$",
      "A classical probability normalization"
    ],
    "answer": 0,
    "explanation": "A Hamiltonian generates unitary dynamics of closed systems."
  },
  {
    "id": "q123",
    "topic": "Quantum Simulation",
    "difficulty": "Intermediate",
    "question": "Which algorithm is commonly used to approximate evolution under a sum of noncommuting terms?",
    "options": [
      "Trotter–Suzuki product formulas",
      "Bubble sort",
      "Deutsch–Jozsa",
      "BB84"
    ],
    "answer": 0,
    "explanation": "Product formulas alternate exponentials of Hamiltonian components."
  },
  {
    "id": "q124",
    "topic": "Quantum Simulation",
    "difficulty": "Intermediate",
    "question": "What is a Pauli-string Hamiltonian?",
    "options": [
      "A weighted sum of tensor products of Pauli operators",
      "A list of classical passwords",
      "A matrix with only positive entries",
      "A circuit with no parameters"
    ],
    "answer": 0,
    "explanation": "Pauli decompositions express spin Hamiltonians as measurable terms."
  },
  {
    "id": "q125",
    "topic": "Quantum Simulation",
    "difficulty": "Advanced",
    "question": "Why is fermion-to-qubit mapping needed?",
    "options": [
      "Fermionic operators must be represented using qubit operators",
      "Qubits cannot encode any occupation numbers",
      "Fermions are always bosons",
      "Measurement is impossible without it"
    ],
    "answer": 0,
    "explanation": "Mappings such as Jordan–Wigner encode anticommutation structure."
  },
  {
    "id": "q126",
    "topic": "Quantum Simulation",
    "difficulty": "Beginner",
    "question": "What is a common purpose of a digital quantum simulator?",
    "options": [
      "Approximate dynamics of a target quantum system using gates",
      "Build a classical transistor",
      "Send classical bits faster than light",
      "Clone arbitrary unknown states"
    ],
    "answer": 0,
    "explanation": "Gate sequences approximate a target model's evolution."
  }
]