Quantum Algorithms and Mathematical Foundations of Quantum
Computation: MCQ Questions
1. What is the dimension of the state space for n qubits?
a) n
b) 2n
c) 2^n
d) n^2
Answer: c) 2^n
2. In quantum computing, what does the Hadamard gate do?
a) Rotates the qubit by 90 degrees
b) Creates an equal superposition of |0 and |1⟩ ⟩
c) Flips the qubit
d) Measures the qubit
Answer: b) Creates an equal superposition of |0 and |1⟩ ⟩
3. Which of the following is the correct mathematical representation of a qubit?
a) |ψ = α|0 + β|1 , where α^2 + β^2 = 1⟩ ⟩ ⟩
b) |ψ = α|0 + β|1 , where α + β = 1⟩ ⟩ ⟩
c) |ψ = α|0 × β|1 , where α^2 + β^2 = 1⟩ ⟩ ⟩
d) |ψ = α|0 - β|1 , where α^2 - β^2 = 1⟩ ⟩ ⟩
Answer: a) |ψ = α|0 + β|1 , where α^2 + β^2 = 1⟩⟩⟩
4. What is the time complexity of Grover's algorithm for searching an unsorted database?
a) O(N)
b) O(log N)
c) O(√N)
d) O(N log N)
Answer: c) O(√N)
5. Which quantum gate is represented by the matrix [[0, 1], [1, 0]]?
a) Hadamard gate
b) Pauli-X gate
c) Pauli-Z gate
d) Phase gate
Answer: b) Pauli-X gate
6. What is the purpose of quantum error correction?
a) To prevent decoherence
b) To increase the number of qubits
c) To speed up quantum algorithms
d) To simulate classical computers
Answer: a) To prevent decoherence
7. In Shor's algorithm, what mathematical problem is solved efficiently?
a) Matrix multiplication
b) Integer factorization
c) Graph coloring
d) Traveling salesman problem
Answer: b) Integer factorization
8. What is the tensor product of |0 and |1 ?⟩ ⟩
a) |0⟩
b) |1⟩
c) |01⟩
d) |10⟩
Answer: c) |01⟩
9. Which of the following is NOT a universal set of quantum gates?
a) Hadamard, CNOT, and T gates
b) Pauli-X, Pauli-Y, and Pauli-Z gates
c) Hadamard and CNOT gates
d) Toffoli and Hadamard gates
Answer: c) Hadamard and CNOT gates
10. What is the quantum Fourier transform used for in Shor's algorithm?
a) To create entanglement
b) To find the period of a function
c) To measure the qubits
d) To implement error correction
Answer: b) To find the period of a function
11. Which of the following best describes quantum teleportation?
a) Faster-than-light communication
b) Transfer of quantum state using classical communication and entanglement
c) Copying of quantum information
d) Direct transfer of qubits between distant locations
Answer: b) Transfer of quantum state using classical communication and entanglement
12. What is the maximum number of orthogonal states that can be reliably distinguished in a 2-
dimensional Hilbert space?
a) 1
b) 2
c) 3
d) 4
Answer: b) 2
13. In the context of quantum computation, what does NISQ stand for?
a) Non-Interacting Spin Qubits
b) Noisy Intermediate-Scale Quantum
c) Next-Generation Integrated System Quantum
d) Normalized Iterative Schrodinger Quantization
Answer: b) Noisy Intermediate-Scale Quantum
14. What is the time complexity of the classical algorithm for integer factorization?
a) O(N)
b) O(log N)
c) O(exp((log N)^(1/3) * (log log N)^(2/3)))
d) O(N log N)
Answer: c) O(exp((log N)^(1/3) * (log log N)^(2/3)))
15. Which of the following is true about the no-cloning theorem?
a) It allows perfect copying of quantum states
b) It prohibits the creation of entangled states
c) It states that it's impossible to create an exact copy of an arbitrary unknown quantum state
d) It only applies to mixed states
Answer: c) It states that it's impossible to create an exact copy of an arbitrary unknown quantum
state
16. What is the purpose of the phase estimation algorithm?
a) To measure the phase of a qubit
b) To estimate the eigenvalues of a unitary operator
c) To create superposition states
d) To implement quantum error correction
Answer: b) To estimate the eigenvalues of a unitary operator
17. Which of the following is NOT a property of quantum entanglement?
a) Non-locality
b) Faster-than-light communication
c) Einstein-Podolsky-Rosen (EPR) paradox
d) Bell's inequality violation
Answer: b) Faster-than-light communication
18. What is the purpose of amplitude amplification in Grover's algorithm?
a) To increase the number of qubits
b) To amplify the amplitude of the desired state
c) To reduce quantum errors
d) To create entanglement
Answer: b) To amplify the amplitude of the desired state
19. Which of the following quantum algorithms is used for database searching?
a) Shor's algorithm
b) Grover's algorithm
c) Quantum Fourier transform
d) VQE algorithm
Answer: b) Grover's algorithm
20. What is the purpose of the SWAP gate in quantum circuits?
a) To create superposition
b) To measure qubits
c) To exchange the states of two qubits
d) To implement error correction
Answer: c) To exchange the states of two qubits
21. In quantum computing, what does the acronym QRAM stand for?
a) Quantum Random Access Memory
b) Qubit Rotation and Measurement
c) Quantum Recursive Algorithm Method
d) Quantized Resonance Amplitude Modulation
Answer: a) Quantum Random Access Memory
22. What is the time complexity of the HHL algorithm for solving linear systems?
a) O(N)
b) O(log N)
c) O(√N)
d) O(poly(log N))
Answer: d) O(poly(log N))
23. Which of the following is a correct representation of the CNOT gate matrix?
a) [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]]
b) [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]
c) [[0, 1], [1, 0]]
d) [[1, 0], [0, -1]]
Answer: a) [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]]
24. What is the purpose of the Quantum Approximate Optimization Algorithm (QAOA)?
a) To factor large integers
b) To solve combinatorial optimization problems
c) To implement quantum error correction
d) To create maximally entangled states
Answer: b) To solve combinatorial optimization problems
25. Which of the following is NOT a type of quantum error?
a) Bit flip
b) Phase flip
c) Amplitude damping
d) Quantum tunneling
Answer: d) Quantum tunneling
26. What is the main advantage of quantum cryptography over classical cryptography?
a) Faster encryption
b) Larger key sizes
c) Information-theoretic security
d) Easier key distribution
Answer: c) Information-theoretic security
27. In the context of quantum computation, what does VQE stand for?
a) Very Quick Entanglement
b) Variational Quantum Eigensolver
c) Vector Quantization Engine
d) Verified Quantum Encryption
Answer: b) Variational Quantum Eigensolver
28. What is the purpose of the Toffoli gate in quantum circuits?
a) To create superposition
b) To implement a controlled-controlled-NOT operation
c) To measure qubits
d) To create entanglement between two qubits
Answer: b) To implement a controlled-controlled-NOT operation
29. Which of the following is a correct mathematical representation of the Hadamard gate?
a) (1/√2) * [[1, 1], [1, -1]]
b) [[0, 1], [1, 0]]
c) [[1, 0], [0, -1]]
d) [[0, -i], [i, 0]]
Answer: a) (1/√2) * [[1, 1], [1, -1]]
30. What is the primary purpose of quantum annealing?
a) To cool quantum systems
b) To solve optimization problems
c) To implement quantum error correction
d) To create entangled states
Answer: b) To solve optimization problems
31. Which of the following is NOT a postulate of quantum mechanics?
a) Measurement postulate
b) Unitary evolution
c) State space postulate
d) Deterministic outcome postulate
Answer: d) Deterministic outcome postulate
32. What is the purpose of the phase kickback trick in quantum algorithms?
a) To create superposition
b) To implement error correction
c) To transfer phase information from auxiliary qubits to the main register
d) To measure qubits without collapsing the state
Answer: c) To transfer phase information from auxiliary qubits to the main register
33. Which of the following best describes the quantum Zeno effect?
a) The act of observation slows down quantum evolution
b) Rapid measurements speed up quantum computations
c) Entanglement between distant particles
d) Spontaneous emission of photons from excited atoms
Answer: a) The act of observation slows down quantum evolution
34. What is the primary purpose of quantum state tomography?
a) To create superposition states
b) To reconstruct an unknown quantum state through measurements
c) To implement quantum teleportation
d) To perform quantum error correction
Answer: b) To reconstruct an unknown quantum state through measurements
35. In the context of quantum algorithms, what does QSVT stand for?
a) Quantum State Vector Transformation
b) Qubit Stabilization via Teleportation
c) Quantum Signal Variance Threshold
d) Quantum Singular Value Transformation
Answer: d) Quantum Singular Value Transformation
36. What is the purpose of the quantum phase estimation algorithm?
a) To measure the phase of a qubit
b) To estimate the eigenvalues of a unitary operator
c) To create superposition states
d) To implement quantum error correction
Answer: b) To estimate the eigenvalues of a unitary operator
37. Which of the following is a correct mathematical representation of the SWAP gate?
a) [[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]]
b) [[0, 1], [1, 0]]
c) [[1, 0], [0, -1]]
d) [[0, -i], [i, 0]]
Answer: a) [[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]]
38. What is the purpose of the quantum Fourier transform?
a) To create entanglement
b) To transform states from the computational basis to the Fourier basis
c) To measure qubits
d) To implement error correction
Answer: b) To transform states from the computational basis to the Fourier basis
39. Which of the following is NOT a type of quantum gate?
a) Pauli gates
b) Hadamard gate
c) CNOT gate
d) Quantum Entanglement gate
Answer: d) Quantum Entanglement gate
40. What is the time complexity of Deutsch-Jozsa algorithm?
a) O(1)
b) O(log N)
c) O(√N)
d) O(N)
Answer: a) O(1)
41. Which of the following best describes the quantum adiabatic theorem?
a) A system remains in its instantaneous eigenstate if a given perturbation is acted on it
slowly enough
b) Quantum systems always evolve to their ground state
c) The energy of a closed quantum system is always conserved
d) Quantum states can be perfectly copied
Answer: a) A system remains in its instantaneous eigenstate if a given perturbation is acted on it
slowly enough
42. What is the purpose of the Quantum Approximate Optimization Algorithm (QAOA)?
a) To factor large integers
b) To solve combinatorial optimization problems
c) To implement quantum error correction
d) To create maximally entangled states
Answer: b) To solve combinatorial optimization problems
43. Which of the following is a correct mathematical representation of the Pauli-Y gate?
a) [[0, 1], [1, 0]]
b) [[1, 0], [0, -1]]
c) [[0, -i], [i, 0]]
d) [[1, 0], [0, 1]]
Answer: c) [[0, -i], [i, 0]]
44. What is the primary purpose of quantum error correction codes?
a) To increase the number of qubits
b) To protect quantum information from decoherence and other errors
c) To speed up quantum algorithms
d) To create entanglement between qubits
Answer: b) To protect quantum information from decoherence and other errors
45. In the context of quantum computation, what does NISQ stand for?
a) Non-Interacting Spin Qubits
b) Noisy Intermediate-Scale Quantum
c) Next-Generation Integrated System Quantum
d) Normalized Iterative Schrodinger Quantization
Answer: b) Noisy Intermediate-Scale Quantum
46. What is the purpose of the quantum walk algorithm?
a) To simulate classical random walks
b) To solve graph problems faster than classical algorithms
c) To implement quantum error correction
d) To create entangled states
Answer: b) To solve graph problems faster than classical algorithms
47. Which of the following is NOT a type of quantum error correction code?
a) Surface code
b) Shor code
c) Steane code
d) Quantum superposition code
Answer: d) Quantum superposition code
48. What is the primary purpose of the variational quantum eigensolver (VQE) algorithm?
a) To factor large integers
b) To find the ground state energy of molecules
c) To implement quantum teleportation
d) To create maximally entangled states
Answer: b) To find the ground state energy of molecules
49. Which of the following best describes the concept of quantum supremacy?
a) The ability of quantum computers to solve all problems faster than classical computers
b) The point at which quantum computers can solve a problem that classical computers
cannot solve in a reasonable time
c) The creation of a universal quantum computer
d) The achievement of perfect quantum error correction
Answer: b) The point
50. If a quantum system is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |1 ?⟩
a) 1/3
b) 2/3
c) 1/√3
d) √2/√3
Answer: b) 2/3
51. What is the trace of the density matrix ρ = (1/2)|0 0| + (1/2)|1 1|?⟩⟨ ⟩⟨
a) 0
b) 1/2
c) 1
d) 2
Answer: c) 1
52. If the Hadamard gate H is applied twice to a qubit in state |0 , what is the final state?⟩
a) |0⟩
b) |1⟩
c) (1/√2)(|0 + |1 )⟩ ⟩
d) (1/√2)(|0 - |1 )⟩ ⟩
Answer: a) |0⟩
53. What is the tensor product of |+ = (1/√2)(|0 + |1 ) with itself?⟩ ⟩ ⟩
a) (1/2)(|00 + |01 + |10 + |11 )⟩⟩⟩⟩
b) (1/√2)(|00 + |11 )⟩ ⟩
c) (1/2)(|00 - |01 - |10 + |11 )⟩ ⟩ ⟩ ⟩
d) |00⟩
Answer: a) (1/2)(|00 + |01 + |10 + |11 )⟩ ⟩ ⟩ ⟩
54. What is the probability of measuring |0 in the state |ψ = cos(θ/2)|0 + sin(θ/2)|1 ?⟩ ⟩ ⟩ ⟩
a) cos^2(θ/2)
b) sin^2(θ/2)
c) cos(θ/2)
d) sin(θ/2)
Answer: a) cos^2(θ/2)
55. If a qubit is in the state |ψ = α|0 + β|1 , what is the expectation value of the Pauli-Z ⟩ ⟩ ⟩
operator?
a) α^2 - β^2
b) α^2 + β^2
c) 2αβ
d) α - β
Answer: a) α^2 - β^2
56. What is the dimension of the Hilbert space for a system of 3 qubits?
a) 3
b) 6
c) 8
d) 16
Answer: c) 8
57. If the CNOT gate is applied to the state (1/√2)(|00 + |11 ), what is the resulting state?⟩ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: a) (1/√2)(|00 + |11 )⟩ ⟩
58. What is the probability of finding a marked item in a database of N items after one Grover
iteration?
a) 1/N
b) 4/N
c) 1/√N
d) 4/√N
Answer: b) 4/N
59. If |ψ = α|0 + β|1 is a pure state, what is the condition for α and β?⟩ ⟩ ⟩
a) α + β = 1
b) α^2 + β^2 = 1
c) α = β
d) α^2 - β^2 = 1
Answer: b) α^2 + β^2 = 1
60. What is the action of the Pauli-X gate on the state |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) |+⟩
b) |-⟩
c) |0⟩
d) |1⟩
Answer: a) |+⟩
61. If a quantum system is in the state |ψ = (1/√2)|0 + (i/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |0 ?⟩
a) 0
b) 1/4
c) 1/2
d) 1
Answer: c) 1/2
62. What is the trace distance between the states ρ = |0 0| and σ = |+ +|?⟩⟨ ⟩⟨
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
63. If the Hadamard gate H is applied to the state |ψ = α|0 + β|1 , what is the resulting state?⟩ ⟩ ⟩
a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
b) ((α-β)/√2)|0 + ((α+β)/√2)|1⟩ ⟩
c) (α/√2)|0 + (β/√2)|1⟩ ⟩
d) (β/√2)|0 + (α/√2)|1⟩ ⟩
Answer: a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
64. What is the fidelity between the pure states |ψ = (1/√2)(|0 + |1 ) and |φ = (1/√2)(|0 - |1 )?⟩ ⟩ ⟩ ⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: b) 1/2
65. If a qubit is in the state |ψ = cos(θ)|0 + sin(θ)|1 , what is the expectation value of the Pauli-⟩ ⟩ ⟩
X operator?
a) cos(θ)
b) sin(θ)
c) sin(2θ)
d) cos(2θ)
Answer: c) sin(2θ)
66. What is the Schmidt rank of the state (1/√3)(|00 + |11 + |22 )?⟩ ⟩ ⟩
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
67. If the phase gate S is applied twice to a qubit in state |+ , what is the resulting state?⟩
a) |+⟩
b) |-⟩
c) i|+⟩
d) -|+⟩
Answer: b) |-⟩
68. What is the von Neumann entropy of the density matrix ρ = (1/2)|0 0| + (1/2)|1 1|?⟩⟨ ⟩⟨
a) 0
b) log(2)
c) 1/2
d) 1
Answer: b) log(2)
69. If a quantum algorithm requires O(√N) queries to solve a problem, how many queries would
be needed for a problem size of 1,000,000?
a) 10
b) 100
c) 1,000
d) 10,000
Answer: c) 1,000
70. What is the probability of measuring |1 in the state |ψ = (3/5)|0 + (4/5)|1 ?⟩ ⟩ ⟩ ⟩
a) 3/5
b) 4/5
c) 9/25
d) 16/25
Answer: d) 16/25
71. If the CNOT gate is applied to the state (1/√2)(|0 + |1 ) |0 , what is the resulting state?⟩ ⟩ ⊗ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: b) (1/√2)(|00 + |10 )⟩ ⟩
72. What is the quantum circuit depth of a sequential application of n single-qubit gates?
a) 1
b) n
c) log(n)
d) 2^n
Answer: b) n
73. If |ψ = α|0 + β|1 and |φ = γ|0 + δ|1 are two qubit states, what is their inner product ψ|⟩ ⟩ ⟩ ⟩ ⟩ ⟩ ⟨
φ ?⟩
a) αγ + βδ
b) α*γ + β*δ
c) |α|^2 + |β|^2
d) |γ|^2 + |δ|^2
Answer: b) α*γ + β*δ
74. What is the gate complexity of the quantum Fourier transform on n qubits?
a) O(n)
b) O(n log n)
c) O(n^2)
d) O(2^n)
Answer: b) O(n log n)
75. If a qubit is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value of the ⟩ ⟩ ⟩
Pauli-Y operator?
a) 0
b) i√2/3
c) -i√2/3
d) 2/3
Answer: c) -i√2/3
76. What is the tensor product of the Pauli-X and Pauli-Z matrices?
a) [[0, 1], [1, 0]]
b) [[0, -i], [i, 0]]
c) [[0, 1], [-1, 0]]
d) [[1, 0], [0, -1]]
Answer: c) [[0, 1], [-1, 0]]
77. If the Toffoli gate is applied to the state |110 , what is the resulting state?⟩
a) |110⟩
b) |111⟩
c) |100⟩
d) |101⟩
Answer: b) |111⟩
78. What is the trace of the matrix representation of the Hadamard gate?
a) 0
b) 1
c) √2
d) 2
Answer: b) 1
79. If a quantum system is in the state |ψ = (1/√2)|0 + (1/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: d) 1
80. What is the dimension of the Hilbert space for a system of n qudits with d levels each?
a) n
b) d
c) n^d
d) d^n
Answer: d) d^n
81. If the density matrix of a qubit is ρ = (1/2)I + (1/2)σz, where I is the identity matrix and σz is
the Pauli-Z matrix, what is the probability of measuring |0 ?⟩
a) 1/4
b) 1/2
c) 3/4
d) 1
Answer: c) 3/4
82. What is the quantum circuit depth of a CNOT gate followed by a Hadamard gate on the
target qubit?
a) 1
b) 2
c) 3
d) 4
Answer: b) 2
83. If |ψ = α|0 + β|1 is a pure state, what is the purity Tr(ρ^2) of its density matrix ρ?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1
d) 2
Answer: c) 1
84. What is the probability of finding the correct answer in Grover's algorithm after π√N/4
iterations for a database of size N?
a) 1/2
b) 1/√2
c) 1/4
d) 1
Answer: b) 1/√2
85. If a quantum system is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value⟩ ⟩ ⟩
of the observable A = 2|0 0| + 3|1 1|?⟩⟨ ⟩⟨
a) 5/3
b) 8/3
c) 7/3
d) 11/3
Answer: b) 8/3
86. What is the trace distance between the pure states |0 and (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
87. If the Hadamard gate H is applied to each qubit in the state |00 , what is the resulting state?⟩
a) (1/2)(|00 + |01 + |10 + |11 )⟩⟩⟩⟩
b) (1/√2)(|00 + |11 )⟩ ⟩
c) (1/2)(|00 - |01 - |10 + |11 )⟩ ⟩ ⟩ ⟩
d) |++⟩
Answer: a) (1/2)(|00 + |01 + |10 + |11 )⟩ ⟩ ⟩ ⟩
88. What is the quantum circuit depth of a quantum Fourier transform on n qubits using the
standard decomposition?
a) n
b) n^2
c) log(n)
d) 2^n
Answer: b) n^2
What is the probability of measuring |0 in the state |ψ = cos(θ/2)|0 + sin(θ/2)|1 ?⟩ ⟩ ⟩ ⟩
a) cos^2(θ/2)
b) sin^2(θ/2)
c) cos(θ/2)
d) sin(θ/2)
Answer: a) cos^2(θ/2)
55. If a qubit is in the state |ψ = α|0 + β|1 , what is the expectation value of the Pauli-Z ⟩ ⟩ ⟩
operator?
a) α^2 - β^2
b) α^2 + β^2
c) 2αβ
d) α - β
Answer: a) α^2 - β^2
56. What is the dimension of the Hilbert space for a system of 3 qubits?
a) 3
b) 6
c) 8
d) 16
Answer: c) 8
57. If the CNOT gate is applied to the state (1/√2)(|00 + |11 ), what is the resulting state?⟩ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: a) (1/√2)(|00 + |11 )⟩ ⟩
58. What is the probability of finding a marked item in a database of N items after one Grover
iteration?
a) 1/N
b) 4/N
c) 1/√N
d) 4/√N
Answer: b) 4/N
59. If |ψ = α|0 + β|1 is a pure state, what is the condition for α and β?⟩ ⟩ ⟩
a) α + β = 1
b) α^2 + β^2 = 1
c) α = β
d) α^2 - β^2 = 1
Answer: b) α^2 + β^2 = 1
60. What is the action of the Pauli-X gate on the state |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) |+⟩
b) |-⟩
c) |0⟩
d) |1⟩
Answer: a) |+⟩
61. If a quantum system is in the state |ψ = (1/√2)|0 + (i/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |0 ?⟩
a) 0
b) 1/4
c) 1/2
d) 1
Answer: c) 1/2
62. What is the trace distance between the states ρ = |0 0| and σ = |+ +|?⟩⟨ ⟩⟨
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
63. If the Hadamard gate H is applied to the state |ψ = α|0 + β|1 , what is the resulting state?⟩ ⟩ ⟩
a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
b) ((α-β)/√2)|0 + ((α+β)/√2)|1⟩ ⟩
c) (α/√2)|0 + (β/√2)|1⟩ ⟩
d) (β/√2)|0 + (α/√2)|1⟩ ⟩
Answer: a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
64. What is the fidelity between the pure states |ψ = (1/√2)(|0 + |1 ) and |φ = (1/√2)(|0 - |1 )?⟩ ⟩ ⟩ ⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: b) 1/2
65. If a qubit is in the state |ψ = cos(θ)|0 + sin(θ)|1 , what is the expectation value of the Pauli-⟩ ⟩ ⟩
X operator?
a) cos(θ)
b) sin(θ)
c) sin(2θ)
d) cos(2θ)
Answer: c) sin(2θ)
66. What is the Schmidt rank of the state (1/√3)(|00 + |11 + |22 )?⟩ ⟩ ⟩
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
67. If the phase gate S is applied twice to a qubit in state |+ , what is the resulting state?⟩
a) |+⟩
b) |-⟩
c) i|+⟩
d) -|+⟩
Answer: b) |-⟩
68. What is the von Neumann entropy of the density matrix ρ = (1/2)|0 0| + (1/2)|1 1|?⟩⟨ ⟩⟨
a) 0
b) log(2)
c) 1/2
d) 1
Answer: b) log(2)
69. If a quantum algorithm requires O(√N) queries to solve a problem, how many queries would
be needed for a problem size of 1,000,000?
a) 10
b) 100
c) 1,000
d) 10,000
Answer: c) 1,000
70. What is the probability of measuring |1 in the state |ψ = (3/5)|0 + (4/5)|1 ?⟩ ⟩ ⟩ ⟩
a) 3/5
b) 4/5
c) 9/25
d) 16/25
Answer: d) 16/25
71. If the CNOT gate is applied to the state (1/√2)(|0 + |1 ) |0 , what is the resulting state?⟩ ⟩ ⊗ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: b) (1/√2)(|00 + |10 )⟩ ⟩
72. What is the quantum circuit depth of a sequential application of n single-qubit gates?
a) 1
b) n
c) log(n)
d) 2^n
Answer: b) n
73. If |ψ = α|0 + β|1 and |φ = γ|0 + δ|1 are two qubit states, what is their inner product ψ|⟩ ⟩ ⟩ ⟩ ⟩ ⟩ ⟨
φ ?⟩
a) αγ + βδ
b) α*γ + β*δ
c) |α|^2 + |β|^2
d) |γ|^2 + |δ|^2
Answer: b) α*γ + β*δ
74. What is the gate complexity of the quantum Fourier transform on n qubits?
a) O(n)
b) O(n log n)
c) O(n^2)
d) O(2^n)
Answer: b) O(n log n)
75. If a qubit is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value of the ⟩ ⟩ ⟩
Pauli-Y operator?
a) 0
b) i√2/3
c) -i√2/3
d) 2/3
Answer: c) -i√2/3
76. What is the tensor product of the Pauli-X and Pauli-Z matrices?
a) [[0, 1], [1, 0]]
b) [[0, -i], [i, 0]]
c) [[0, 1], [-1, 0]]
d) [[1, 0], [0, -1]]
Answer: c) [[0, 1], [-1, 0]]
77. If the Toffoli gate is applied to the state |110 , what is the resulting state?⟩
a) |110⟩
b) |111⟩
c) |100⟩
d) |101⟩
Answer: b) |111⟩
78. What is the trace of the matrix representation of the Hadamard gate?
a) 0
b) 1
c) √2
d) 2
Answer: b) 1
79. If a quantum system is in the state |ψ = (1/√2)|0 + (1/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: d) 1
80. What is the dimension of the Hilbert space for a system of n qudits with d levels each?
a) n
b) d
c) n^d
d) d^n
Answer: d) d^n
81. If the density matrix of a qubit is ρ = (1/2)I + (1/2)σz, where I is the identity matrix and σz is
the Pauli-Z matrix, what is the probability of measuring |0 ?⟩
a) 1/4
b) 1/2
c) 3/4
d) 1
Answer: c) 3/4
82. What is the quantum circuit depth of a CNOT gate followed by a Hadamard gate on the
target qubit?
a) 1
b) 2
c) 3
d) 4
Answer: b) 2
83. If |ψ = α|0 + β|1 is a pure state, what is the purity Tr(ρ^2) of its density matrix ρ?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1
d) 2
Answer: c) 1
84. What is the probability of finding the correct answer in Grover's algorithm after π√N/4
iterations for a database of size N?
a) 1/2
b) 1/√2
c) 1/4
d) 1
Answer: b) 1/√2
85. If a quantum system is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value⟩ ⟩ ⟩
of the observable A = 2|0 0| + 3|1 1|?⟩⟨ ⟩⟨
a) 5/3
b) 8/3
c) 7/3
d) 11/3
Answer: b) 8/3
86. What is the trace distance between the pure states |0 and (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
87. If the Hadamard gate H is applied to each qubit in the state |00 , what is the resulting state?⟩
a) (1/2)(|00 + |01 + |10 + |11 )⟩⟩⟩⟩
b) (1/√2)(|00 + |11 )⟩ ⟩
c) (1/2)(|00 - |01 - |10 + |11 )⟩ ⟩ ⟩ ⟩
d) |++⟩
Answer: a) (1/2)(|00 + |01 + |10 + |11 )⟩ ⟩ ⟩ ⟩
88. What is the quantum circuit depth of a quantum Fourier transform on n qubits using the
standard decomposition?
a) n
b) n^2
c) log(n)
d) 2^n
Answer: b) n^2
What is the probability of measuring |0 in the state |ψ = cos(θ/2)|0 + sin(θ/2)|1 ?⟩ ⟩ ⟩ ⟩
a) cos^2(θ/2)
b) sin^2(θ/2)
c) cos(θ/2)
d) sin(θ/2)
Answer: a) cos^2(θ/2)
55. If a qubit is in the state |ψ = α|0 + β|1 , what is the expectation value of the Pauli-Z ⟩ ⟩ ⟩
operator?
a) α^2 - β^2
b) α^2 + β^2
c) 2αβ
d) α - β
Answer: a) α^2 - β^2
56. What is the dimension of the Hilbert space for a system of 3 qubits?
a) 3
b) 6
c) 8
d) 16
Answer: c) 8
57. If the CNOT gate is applied to the state (1/√2)(|00 + |11 ), what is the resulting state?⟩ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: a) (1/√2)(|00 + |11 )⟩ ⟩
58. What is the probability of finding a marked item in a database of N items after one Grover
iteration?
a) 1/N
b) 4/N
c) 1/√N
d) 4/√N
Answer: b) 4/N
59. If |ψ = α|0 + β|1 is a pure state, what is the condition for α and β?⟩ ⟩ ⟩
a) α + β = 1
b) α^2 + β^2 = 1
c) α = β
d) α^2 - β^2 = 1
Answer: b) α^2 + β^2 = 1
60. What is the action of the Pauli-X gate on the state |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) |+⟩
b) |-⟩
c) |0⟩
d) |1⟩
Answer: a) |+⟩
61. If a quantum system is in the state |ψ = (1/√2)|0 + (i/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |0 ?⟩
a) 0
b) 1/4
c) 1/2
d) 1
Answer: c) 1/2
62. What is the trace distance between the states ρ = |0 0| and σ = |+ +|?⟩⟨ ⟩⟨
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
63. If the Hadamard gate H is applied to the state |ψ = α|0 + β|1 , what is the resulting state?⟩ ⟩ ⟩
a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
b) ((α-β)/√2)|0 + ((α+β)/√2)|1⟩ ⟩
c) (α/√2)|0 + (β/√2)|1⟩ ⟩
d) (β/√2)|0 + (α/√2)|1⟩ ⟩
Answer: a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
64. What is the fidelity between the pure states |ψ = (1/√2)(|0 + |1 ) and |φ = (1/√2)(|0 - |1 )?⟩ ⟩ ⟩ ⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: b) 1/2
65. If a qubit is in the state |ψ = cos(θ)|0 + sin(θ)|1 , what is the expectation value of the Pauli-⟩ ⟩ ⟩
X operator?
a) cos(θ)
b) sin(θ)
c) sin(2θ)
d) cos(2θ)
Answer: c) sin(2θ)
66. What is the Schmidt rank of the state (1/√3)(|00 + |11 + |22 )?⟩ ⟩ ⟩
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
67. If the phase gate S is applied twice to a qubit in state |+ , what is the resulting state?⟩
a) |+⟩
b) |-⟩
c) i|+⟩
d) -|+⟩
Answer: b) |-⟩
68. What is the von Neumann entropy of the density matrix ρ = (1/2)|0 0| + (1/2)|1 1|?⟩⟨ ⟩⟨
a) 0
b) log(2)
c) 1/2
d) 1
Answer: b) log(2)
69. If a quantum algorithm requires O(√N) queries to solve a problem, how many queries would
be needed for a problem size of 1,000,000?
a) 10
b) 100
c) 1,000
d) 10,000
Answer: c) 1,000
70. What is the probability of measuring |1 in the state |ψ = (3/5)|0 + (4/5)|1 ?⟩ ⟩ ⟩ ⟩
a) 3/5
b) 4/5
c) 9/25
d) 16/25
Answer: d) 16/25
71. If the CNOT gate is applied to the state (1/√2)(|0 + |1 ) |0 , what is the resulting state?⟩ ⟩ ⊗ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: b) (1/√2)(|00 + |10 )⟩ ⟩
72. What is the quantum circuit depth of a sequential application of n single-qubit gates?
a) 1
b) n
c) log(n)
d) 2^n
Answer: b) n
73. If |ψ = α|0 + β|1 and |φ = γ|0 + δ|1 are two qubit states, what is their inner product ψ|⟩ ⟩ ⟩ ⟩ ⟩ ⟩ ⟨
φ ?⟩
a) αγ + βδ
b) α*γ + β*δ
c) |α|^2 + |β|^2
d) |γ|^2 + |δ|^2
Answer: b) α*γ + β*δ
74. What is the gate complexity of the quantum Fourier transform on n qubits?
a) O(n)
b) O(n log n)
c) O(n^2)
d) O(2^n)
Answer: b) O(n log n)
75. If a qubit is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value of the ⟩ ⟩ ⟩
Pauli-Y operator?
a) 0
b) i√2/3
c) -i√2/3
d) 2/3
Answer: c) -i√2/3
76. What is the tensor product of the Pauli-X and Pauli-Z matrices?
a) [[0, 1], [1, 0]]
b) [[0, -i], [i, 0]]
c) [[0, 1], [-1, 0]]
d) [[1, 0], [0, -1]]
Answer: c) [[0, 1], [-1, 0]]
77. If the Toffoli gate is applied to the state |110 , what is the resulting state?⟩
a) |110⟩
b) |111⟩
c) |100⟩
d) |101⟩
Answer: b) |111⟩
78. What is the trace of the matrix representation of the Hadamard gate?
a) 0
b) 1
c) √2
d) 2
Answer: b) 1
79. If a quantum system is in the state |ψ = (1/√2)|0 + (1/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: d) 1
80. What is the dimension of the Hilbert space for a system of n qudits with d levels each?
a) n
b) d
c) n^d
d) d^n
Answer: d) d^n
81. If the density matrix of a qubit is ρ = (1/2)I + (1/2)σz, where I is the identity matrix and σz is
the Pauli-Z matrix, what is the probability of measuring |0 ?⟩
a) 1/4
b) 1/2
c) 3/4
d) 1
Answer: c) 3/4
82. What is the quantum circuit depth of a CNOT gate followed by a Hadamard gate on the
target qubit?
a) 1
b) 2
c) 3
d) 4
Answer: b) 2
83. If |ψ = α|0 + β|1 is a pure state, what is the purity Tr(ρ^2) of its density matrix ρ?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1
d) 2
Answer: c) 1
84. What is the probability of finding the correct answer in Grover's algorithm after π√N/4
iterations for a database of size N?
a) 1/2
b) 1/√2
c) 1/4
d) 1
Answer: b) 1/√2
85. If a quantum system is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value⟩ ⟩ ⟩
of the observable A = 2|0 0| + 3|1 1|?⟩⟨ ⟩⟨
a) 5/3
b) 8/3
c) 7/3
d) 11/3
Answer: b) 8/3
86. What is the trace distance between the pure states |0 and (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
87. If the Hadamard gate H is applied to each qubit in the state |00 , what is the resulting state?⟩
a) (1/2)(|00 + |01 + |10 + |11 )⟩⟩⟩⟩
b) (1/√2)(|00 + |11 )⟩ ⟩
c) (1/2)(|00 - |01 - |10 + |11 )⟩ ⟩ ⟩ ⟩
d) |++⟩
Answer: a) (1/2)(|00 + |01 + |10 + |11 )⟩ ⟩ ⟩ ⟩
88. What is the quantum circuit depth of a quantum Fourier transform on n qubits using the
standard decomposition?
a) n
b) n^2
c) log(n)
d) 2^n
Answer: b) n^2
What is the probability of measuring |0 in the state |ψ = cos(θ/2)|0 + sin(θ/2)|1 ?⟩ ⟩ ⟩ ⟩
a) cos^2(θ/2)
b) sin^2(θ/2)
c) cos(θ/2)
d) sin(θ/2)
Answer: a) cos^2(θ/2)
55. If a qubit is in the state |ψ = α|0 + β|1 , what is the expectation value of the Pauli-Z ⟩ ⟩ ⟩
operator?
a) α^2 - β^2
b) α^2 + β^2
c) 2αβ
d) α - β
Answer: a) α^2 - β^2
56. What is the dimension of the Hilbert space for a system of 3 qubits?
a) 3
b) 6
c) 8
d) 16
Answer: c) 8
57. If the CNOT gate is applied to the state (1/√2)(|00 + |11 ), what is the resulting state?⟩ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: a) (1/√2)(|00 + |11 )⟩ ⟩
58. What is the probability of finding a marked item in a database of N items after one Grover
iteration?
a) 1/N
b) 4/N
c) 1/√N
d) 4/√N
Answer: b) 4/N
59. If |ψ = α|0 + β|1 is a pure state, what is the condition for α and β?⟩ ⟩ ⟩
a) α + β = 1
b) α^2 + β^2 = 1
c) α = β
d) α^2 - β^2 = 1
Answer: b) α^2 + β^2 = 1
60. What is the action of the Pauli-X gate on the state |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) |+⟩
b) |-⟩
c) |0⟩
d) |1⟩
Answer: a) |+⟩
61. If a quantum system is in the state |ψ = (1/√2)|0 + (i/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |0 ?⟩
a) 0
b) 1/4
c) 1/2
d) 1
Answer: c) 1/2
62. What is the trace distance between the states ρ = |0 0| and σ = |+ +|?⟩⟨ ⟩⟨
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
63. If the Hadamard gate H is applied to the state |ψ = α|0 + β|1 , what is the resulting state?⟩ ⟩ ⟩
a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
b) ((α-β)/√2)|0 + ((α+β)/√2)|1⟩ ⟩
c) (α/√2)|0 + (β/√2)|1⟩ ⟩
d) (β/√2)|0 + (α/√2)|1⟩ ⟩
Answer: a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
64. What is the fidelity between the pure states |ψ = (1/√2)(|0 + |1 ) and |φ = (1/√2)(|0 - |1 )?⟩ ⟩ ⟩ ⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: b) 1/2
65. If a qubit is in the state |ψ = cos(θ)|0 + sin(θ)|1 , what is the expectation value of the Pauli-⟩ ⟩ ⟩
X operator?
a) cos(θ)
b) sin(θ)
c) sin(2θ)
d) cos(2θ)
Answer: c) sin(2θ)
66. What is the Schmidt rank of the state (1/√3)(|00 + |11 + |22 )?⟩ ⟩ ⟩
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
67. If the phase gate S is applied twice to a qubit in state |+ , what is the resulting state?⟩
a) |+⟩
b) |-⟩
c) i|+⟩
d) -|+⟩
Answer: b) |-⟩
68. What is the von Neumann entropy of the density matrix ρ = (1/2)|0 0| + (1/2)|1 1|?⟩⟨ ⟩⟨
a) 0
b) log(2)
c) 1/2
d) 1
Answer: b) log(2)
69. If a quantum algorithm requires O(√N) queries to solve a problem, how many queries would
be needed for a problem size of 1,000,000?
a) 10
b) 100
c) 1,000
d) 10,000
Answer: c) 1,000
70. What is the probability of measuring |1 in the state |ψ = (3/5)|0 + (4/5)|1 ?⟩ ⟩ ⟩ ⟩
a) 3/5
b) 4/5
c) 9/25
d) 16/25
Answer: d) 16/25
71. If the CNOT gate is applied to the state (1/√2)(|0 + |1 ) |0 , what is the resulting state?⟩ ⟩ ⊗ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: b) (1/√2)(|00 + |10 )⟩ ⟩
72. What is the quantum circuit depth of a sequential application of n single-qubit gates?
a) 1
b) n
c) log(n)
d) 2^n
Answer: b) n
73. If |ψ = α|0 + β|1 and |φ = γ|0 + δ|1 are two qubit states, what is their inner product ψ|⟩ ⟩ ⟩ ⟩ ⟩ ⟩ ⟨
φ ?⟩
a) αγ + βδ
b) α*γ + β*δ
c) |α|^2 + |β|^2
d) |γ|^2 + |δ|^2
Answer: b) α*γ + β*δ
74. What is the gate complexity of the quantum Fourier transform on n qubits?
a) O(n)
b) O(n log n)
c) O(n^2)
d) O(2^n)
Answer: b) O(n log n)
75. If a qubit is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value of the ⟩ ⟩ ⟩
Pauli-Y operator?
a) 0
b) i√2/3
c) -i√2/3
d) 2/3
Answer: c) -i√2/3
76. What is the tensor product of the Pauli-X and Pauli-Z matrices?
a) [[0, 1], [1, 0]]
b) [[0, -i], [i, 0]]
c) [[0, 1], [-1, 0]]
d) [[1, 0], [0, -1]]
Answer: c) [[0, 1], [-1, 0]]
77. If the Toffoli gate is applied to the state |110 , what is the resulting state?⟩
a) |110⟩
b) |111⟩
c) |100⟩
d) |101⟩
Answer: b) |111⟩
78. What is the trace of the matrix representation of the Hadamard gate?
a) 0
b) 1
c) √2
d) 2
Answer: b) 1
79. If a quantum system is in the state |ψ = (1/√2)|0 + (1/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: d) 1
80. What is the dimension of the Hilbert space for a system of n qudits with d levels each?
a) n
b) d
c) n^d
d) d^n
Answer: d) d^n
81. If the density matrix of a qubit is ρ = (1/2)I + (1/2)σz, where I is the identity matrix and σz is
the Pauli-Z matrix, what is the probability of measuring |0 ?⟩
a) 1/4
b) 1/2
c) 3/4
d) 1
Answer: c) 3/4
82. What is the quantum circuit depth of a CNOT gate followed by a Hadamard gate on the
target qubit?
a) 1
b) 2
c) 3
d) 4
Answer: b) 2
83. If |ψ = α|0 + β|1 is a pure state, what is the purity Tr(ρ^2) of its density matrix ρ?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1
d) 2
Answer: c) 1
84. What is the probability of finding the correct answer in Grover's algorithm after π√N/4
iterations for a database of size N?
a) 1/2
b) 1/√2
c) 1/4
d) 1
Answer: b) 1/√2
85. If a quantum system is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value⟩ ⟩ ⟩
of the observable A = 2|0 0| + 3|1 1|?⟩⟨ ⟩⟨
a) 5/3
b) 8/3
c) 7/3
d) 11/3
Answer: b) 8/3
86. What is the trace distance between the pure states |0 and (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
87. If the Hadamard gate H is applied to each qubit in the state |00 , what is the resulting state?⟩
a) (1/2)(|00 + |01 + |10 + |11 )⟩⟩⟩⟩
b) (1/√2)(|00 + |11 )⟩ ⟩
c) (1/2)(|00 - |01 - |10 + |11 )⟩ ⟩ ⟩ ⟩
d) |++⟩
Answer: a) (1/2)(|00 + |01 + |10 + |11 )⟩ ⟩ ⟩ ⟩
88. What is the quantum circuit depth of a quantum Fourier transform on n qubits using the
standard decomposition?
a) n
b) n^2
c) log(n)
d) 2^n
Answer: b) n^2
What is the probability of measuring |0 in the state |ψ = cos(θ/2)|0 + sin(θ/2)|1 ?⟩ ⟩ ⟩ ⟩
a) cos^2(θ/2)
b) sin^2(θ/2)
c) cos(θ/2)
d) sin(θ/2)
Answer: a) cos^2(θ/2)
55. If a qubit is in the state |ψ = α|0 + β|1 , what is the expectation value of the Pauli-Z ⟩ ⟩ ⟩
operator?
a) α^2 - β^2
b) α^2 + β^2
c) 2αβ
d) α - β
Answer: a) α^2 - β^2
56. What is the dimension of the Hilbert space for a system of 3 qubits?
a) 3
b) 6
c) 8
d) 16
Answer: c) 8
57. If the CNOT gate is applied to the state (1/√2)(|00 + |11 ), what is the resulting state?⟩ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: a) (1/√2)(|00 + |11 )⟩ ⟩
58. What is the probability of finding a marked item in a database of N items after one Grover
iteration?
a) 1/N
b) 4/N
c) 1/√N
d) 4/√N
Answer: b) 4/N
59. If |ψ = α|0 + β|1 is a pure state, what is the condition for α and β?⟩ ⟩ ⟩
a) α + β = 1
b) α^2 + β^2 = 1
c) α = β
d) α^2 - β^2 = 1
Answer: b) α^2 + β^2 = 1
60. What is the action of the Pauli-X gate on the state |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) |+⟩
b) |-⟩
c) |0⟩
d) |1⟩
Answer: a) |+⟩
61. If a quantum system is in the state |ψ = (1/√2)|0 + (i/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |0 ?⟩
a) 0
b) 1/4
c) 1/2
d) 1
Answer: c) 1/2
62. What is the trace distance between the states ρ = |0 0| and σ = |+ +|?⟩⟨ ⟩⟨
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
63. If the Hadamard gate H is applied to the state |ψ = α|0 + β|1 , what is the resulting state?⟩ ⟩ ⟩
a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
b) ((α-β)/√2)|0 + ((α+β)/√2)|1⟩ ⟩
c) (α/√2)|0 + (β/√2)|1⟩ ⟩
d) (β/√2)|0 + (α/√2)|1⟩ ⟩
Answer: a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
64. What is the fidelity between the pure states |ψ = (1/√2)(|0 + |1 ) and |φ = (1/√2)(|0 - |1 )?⟩ ⟩ ⟩ ⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: b) 1/2
65. If a qubit is in the state |ψ = cos(θ)|0 + sin(θ)|1 , what is the expectation value of the Pauli-⟩ ⟩ ⟩
X operator?
a) cos(θ)
b) sin(θ)
c) sin(2θ)
d) cos(2θ)
Answer: c) sin(2θ)
66. What is the Schmidt rank of the state (1/√3)(|00 + |11 + |22 )?⟩ ⟩ ⟩
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
67. If the phase gate S is applied twice to a qubit in state |+ , what is the resulting state?⟩
a) |+⟩
b) |-⟩
c) i|+⟩
d) -|+⟩
Answer: b) |-⟩
68. What is the von Neumann entropy of the density matrix ρ = (1/2)|0 0| + (1/2)|1 1|?⟩⟨ ⟩⟨
a) 0
b) log(2)
c) 1/2
d) 1
Answer: b) log(2)
69. If a quantum algorithm requires O(√N) queries to solve a problem, how many queries would
be needed for a problem size of 1,000,000?
a) 10
b) 100
c) 1,000
d) 10,000
Answer: c) 1,000
70. What is the probability of measuring |1 in the state |ψ = (3/5)|0 + (4/5)|1 ?⟩ ⟩ ⟩ ⟩
a) 3/5
b) 4/5
c) 9/25
d) 16/25
Answer: d) 16/25
71. If the CNOT gate is applied to the state (1/√2)(|0 + |1 ) |0 , what is the resulting state?⟩ ⟩ ⊗ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: b) (1/√2)(|00 + |10 )⟩ ⟩
72. What is the quantum circuit depth of a sequential application of n single-qubit gates?
a) 1
b) n
c) log(n)
d) 2^n
Answer: b) n
73. If |ψ = α|0 + β|1 and |φ = γ|0 + δ|1 are two qubit states, what is their inner product ψ|⟩ ⟩ ⟩ ⟩ ⟩ ⟩ ⟨
φ ?⟩
a) αγ + βδ
b) α*γ + β*δ
c) |α|^2 + |β|^2
d) |γ|^2 + |δ|^2
Answer: b) α*γ + β*δ
74. What is the gate complexity of the quantum Fourier transform on n qubits?
a) O(n)
b) O(n log n)
c) O(n^2)
d) O(2^n)
Answer: b) O(n log n)
75. If a qubit is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value of the ⟩ ⟩ ⟩
Pauli-Y operator?
a) 0
b) i√2/3
c) -i√2/3
d) 2/3
Answer: c) -i√2/3
76. What is the tensor product of the Pauli-X and Pauli-Z matrices?
a) [[0, 1], [1, 0]]
b) [[0, -i], [i, 0]]
c) [[0, 1], [-1, 0]]
d) [[1, 0], [0, -1]]
Answer: c) [[0, 1], [-1, 0]]
77. If the Toffoli gate is applied to the state |110 , what is the resulting state?⟩
a) |110⟩
b) |111⟩
c) |100⟩
d) |101⟩
Answer: b) |111⟩
78. What is the trace of the matrix representation of the Hadamard gate?
a) 0
b) 1
c) √2
d) 2
Answer: b) 1
79. If a quantum system is in the state |ψ = (1/√2)|0 + (1/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: d) 1
80. What is the dimension of the Hilbert space for a system of n qudits with d levels each?
a) n
b) d
c) n^d
d) d^n
Answer: d) d^n
81. If the density matrix of a qubit is ρ = (1/2)I + (1/2)σz, where I is the identity matrix and σz is
the Pauli-Z matrix, what is the probability of measuring |0 ?⟩
a) 1/4
b) 1/2
c) 3/4
d) 1
Answer: c) 3/4
82. What is the quantum circuit depth of a CNOT gate followed by a Hadamard gate on the
target qubit?
a) 1
b) 2
c) 3
d) 4
Answer: b) 2
83. If |ψ = α|0 + β|1 is a pure state, what is the purity Tr(ρ^2) of its density matrix ρ?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1
d) 2
Answer: c) 1
84. What is the probability of finding the correct answer in Grover's algorithm after π√N/4
iterations for a database of size N?
a) 1/2
b) 1/√2
c) 1/4
d) 1
Answer: b) 1/√2
85. If a quantum system is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value⟩ ⟩ ⟩
of the observable A = 2|0 0| + 3|1 1|?⟩⟨ ⟩⟨
a) 5/3
b) 8/3
c) 7/3
d) 11/3
Answer: b) 8/3
86. What is the trace distance between the pure states |0 and (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
87. If the Hadamard gate H is applied to each qubit in the state |00 , what is the resulting state?⟩
a) (1/2)(|00 + |01 + |10 + |11 )⟩⟩⟩⟩
b) (1/√2)(|00 + |11 )⟩ ⟩
c) (1/2)(|00 - |01 - |10 + |11 )⟩ ⟩ ⟩ ⟩
d) |++⟩
Answer: a) (1/2)(|00 + |01 + |10 + |11 )⟩ ⟩ ⟩ ⟩
88. What is the quantum circuit depth of a quantum Fourier transform on n qubits using the
standard decomposition?
a) n
b) n^2
c) log(n)
d) 2^n
Answer: b) n^2
What is the probability of measuring |0 in the state |ψ = cos(θ/2)|0 + sin(θ/2)|1 ?⟩ ⟩ ⟩ ⟩
a) cos^2(θ/2)
b) sin^2(θ/2)
c) cos(θ/2)
d) sin(θ/2)
Answer: a) cos^2(θ/2)
55. If a qubit is in the state |ψ = α|0 + β|1 , what is the expectation value of the Pauli-Z ⟩ ⟩ ⟩
operator?
a) α^2 - β^2
b) α^2 + β^2
c) 2αβ
d) α - β
Answer: a) α^2 - β^2
56. What is the dimension of the Hilbert space for a system of 3 qubits?
a) 3
b) 6
c) 8
d) 16
Answer: c) 8
57. If the CNOT gate is applied to the state (1/√2)(|00 + |11 ), what is the resulting state?⟩ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: a) (1/√2)(|00 + |11 )⟩ ⟩
58. What is the probability of finding a marked item in a database of N items after one Grover
iteration?
a) 1/N
b) 4/N
c) 1/√N
d) 4/√N
Answer: b) 4/N
59. If |ψ = α|0 + β|1 is a pure state, what is the condition for α and β?⟩ ⟩ ⟩
a) α + β = 1
b) α^2 + β^2 = 1
c) α = β
d) α^2 - β^2 = 1
Answer: b) α^2 + β^2 = 1
60. What is the action of the Pauli-X gate on the state |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) |+⟩
b) |-⟩
c) |0⟩
d) |1⟩
Answer: a) |+⟩
61. If a quantum system is in the state |ψ = (1/√2)|0 + (i/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |0 ?⟩
a) 0
b) 1/4
c) 1/2
d) 1
Answer: c) 1/2
62. What is the trace distance between the states ρ = |0 0| and σ = |+ +|?⟩⟨ ⟩⟨
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
63. If the Hadamard gate H is applied to the state |ψ = α|0 + β|1 , what is the resulting state?⟩ ⟩ ⟩
a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
b) ((α-β)/√2)|0 + ((α+β)/√2)|1⟩ ⟩
c) (α/√2)|0 + (β/√2)|1⟩ ⟩
d) (β/√2)|0 + (α/√2)|1⟩ ⟩
Answer: a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
64. What is the fidelity between the pure states |ψ = (1/√2)(|0 + |1 ) and |φ = (1/√2)(|0 - |1 )?⟩ ⟩ ⟩ ⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: b) 1/2
65. If a qubit is in the state |ψ = cos(θ)|0 + sin(θ)|1 , what is the expectation value of the Pauli-⟩ ⟩ ⟩
X operator?
a) cos(θ)
b) sin(θ)
c) sin(2θ)
d) cos(2θ)
Answer: c) sin(2θ)
66. What is the Schmidt rank of the state (1/√3)(|00 + |11 + |22 )?⟩ ⟩ ⟩
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
67. If the phase gate S is applied twice to a qubit in state |+ , what is the resulting state?⟩
a) |+⟩
b) |-⟩
c) i|+⟩
d) -|+⟩
Answer: b) |-⟩
68. What is the von Neumann entropy of the density matrix ρ = (1/2)|0 0| + (1/2)|1 1|?⟩⟨ ⟩⟨
a) 0
b) log(2)
c) 1/2
d) 1
Answer: b) log(2)
69. If a quantum algorithm requires O(√N) queries to solve a problem, how many queries would
be needed for a problem size of 1,000,000?
a) 10
b) 100
c) 1,000
d) 10,000
Answer: c) 1,000
70. What is the probability of measuring |1 in the state |ψ = (3/5)|0 + (4/5)|1 ?⟩ ⟩ ⟩ ⟩
a) 3/5
b) 4/5
c) 9/25
d) 16/25
Answer: d) 16/25
71. If the CNOT gate is applied to the state (1/√2)(|0 + |1 ) |0 , what is the resulting state?⟩ ⟩ ⊗ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: b) (1/√2)(|00 + |10 )⟩ ⟩
72. What is the quantum circuit depth of a sequential application of n single-qubit gates?
a) 1
b) n
c) log(n)
d) 2^n
Answer: b) n
73. If |ψ = α|0 + β|1 and |φ = γ|0 + δ|1 are two qubit states, what is their inner product ψ|⟩ ⟩ ⟩ ⟩ ⟩ ⟩ ⟨
φ ?⟩
a) αγ + βδ
b) α*γ + β*δ
c) |α|^2 + |β|^2
d) |γ|^2 + |δ|^2
Answer: b) α*γ + β*δ
74. What is the gate complexity of the quantum Fourier transform on n qubits?
a) O(n)
b) O(n log n)
c) O(n^2)
d) O(2^n)
Answer: b) O(n log n)
75. If a qubit is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value of the ⟩ ⟩ ⟩
Pauli-Y operator?
a) 0
b) i√2/3
c) -i√2/3
d) 2/3
Answer: c) -i√2/3
76. What is the tensor product of the Pauli-X and Pauli-Z matrices?
a) [[0, 1], [1, 0]]
b) [[0, -i], [i, 0]]
c) [[0, 1], [-1, 0]]
d) [[1, 0], [0, -1]]
Answer: c) [[0, 1], [-1, 0]]
77. If the Toffoli gate is applied to the state |110 , what is the resulting state?⟩
a) |110⟩
b) |111⟩
c) |100⟩
d) |101⟩
Answer: b) |111⟩
78. What is the trace of the matrix representation of the Hadamard gate?
a) 0
b) 1
c) √2
d) 2
Answer: b) 1
79. If a quantum system is in the state |ψ = (1/√2)|0 + (1/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: d) 1
80. What is the dimension of the Hilbert space for a system of n qudits with d levels each?
a) n
b) d
c) n^d
d) d^n
Answer: d) d^n
81. If the density matrix of a qubit is ρ = (1/2)I + (1/2)σz, where I is the identity matrix and σz is
the Pauli-Z matrix, what is the probability of measuring |0 ?⟩
a) 1/4
b) 1/2
c) 3/4
d) 1
Answer: c) 3/4
82. What is the quantum circuit depth of a CNOT gate followed by a Hadamard gate on the
target qubit?
a) 1
b) 2
c) 3
d) 4
Answer: b) 2
83. If |ψ = α|0 + β|1 is a pure state, what is the purity Tr(ρ^2) of its density matrix ρ?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1
d) 2
Answer: c) 1
84. What is the probability of finding the correct answer in Grover's algorithm after π√N/4
iterations for a database of size N?
a) 1/2
b) 1/√2
c) 1/4
d) 1
Answer: b) 1/√2
85. If a quantum system is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value⟩ ⟩ ⟩
of the observable A = 2|0 0| + 3|1 1|?⟩⟨ ⟩⟨
a) 5/3
b) 8/3
c) 7/3
d) 11/3
Answer: b) 8/3
86. What is the trace distance between the pure states |0 and (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
87. If the Hadamard gate H is applied to each qubit in the state |00 , what is the resulting state?⟩
a) (1/2)(|00 + |01 + |10 + |11 )⟩⟩⟩⟩
b) (1/√2)(|00 + |11 )⟩ ⟩
c) (1/2)(|00 - |01 - |10 + |11 )⟩ ⟩ ⟩ ⟩
d) |++⟩
Answer: a) (1/2)(|00 + |01 + |10 + |11 )⟩ ⟩ ⟩ ⟩
88. What is the quantum circuit depth of a quantum Fourier transform on n qubits using the
standard decomposition?
a) n
b) n^2
c) log(n)
d) 2^n
Answer: b) n^2
What is the probability of measuring |0 in the state |ψ = cos(θ/2)|0 + sin(θ/2)|1 ?⟩ ⟩ ⟩ ⟩
a) cos^2(θ/2)
b) sin^2(θ/2)
c) cos(θ/2)
d) sin(θ/2)
Answer: a) cos^2(θ/2)
55. If a qubit is in the state |ψ = α|0 + β|1 , what is the expectation value of the Pauli-Z ⟩ ⟩ ⟩
operator?
a) α^2 - β^2
b) α^2 + β^2
c) 2αβ
d) α - β
Answer: a) α^2 - β^2
56. What is the dimension of the Hilbert space for a system of 3 qubits?
a) 3
b) 6
c) 8
d) 16
Answer: c) 8
57. If the CNOT gate is applied to the state (1/√2)(|00 + |11 ), what is the resulting state?⟩ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: a) (1/√2)(|00 + |11 )⟩ ⟩
58. What is the probability of finding a marked item in a database of N items after one Grover
iteration?
a) 1/N
b) 4/N
c) 1/√N
d) 4/√N
Answer: b) 4/N
59. If |ψ = α|0 + β|1 is a pure state, what is the condition for α and β?⟩ ⟩ ⟩
a) α + β = 1
b) α^2 + β^2 = 1
c) α = β
d) α^2 - β^2 = 1
Answer: b) α^2 + β^2 = 1
60. What is the action of the Pauli-X gate on the state |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) |+⟩
b) |-⟩
c) |0⟩
d) |1⟩
Answer: a) |+⟩
61. If a quantum system is in the state |ψ = (1/√2)|0 + (i/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |0 ?⟩
a) 0
b) 1/4
c) 1/2
d) 1
Answer: c) 1/2
62. What is the trace distance between the states ρ = |0 0| and σ = |+ +|?⟩⟨ ⟩⟨
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
63. If the Hadamard gate H is applied to the state |ψ = α|0 + β|1 , what is the resulting state?⟩ ⟩ ⟩
a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
b) ((α-β)/√2)|0 + ((α+β)/√2)|1⟩ ⟩
c) (α/√2)|0 + (β/√2)|1⟩ ⟩
d) (β/√2)|0 + (α/√2)|1⟩ ⟩
Answer: a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
64. What is the fidelity between the pure states |ψ = (1/√2)(|0 + |1 ) and |φ = (1/√2)(|0 - |1 )?⟩ ⟩ ⟩ ⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: b) 1/2
65. If a qubit is in the state |ψ = cos(θ)|0 + sin(θ)|1 , what is the expectation value of the Pauli-⟩ ⟩ ⟩
X operator?
a) cos(θ)
b) sin(θ)
c) sin(2θ)
d) cos(2θ)
Answer: c) sin(2θ)
66. What is the Schmidt rank of the state (1/√3)(|00 + |11 + |22 )?⟩ ⟩ ⟩
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
67. If the phase gate S is applied twice to a qubit in state |+ , what is the resulting state?⟩
a) |+⟩
b) |-⟩
c) i|+⟩
d) -|+⟩
Answer: b) |-⟩
68. What is the von Neumann entropy of the density matrix ρ = (1/2)|0 0| + (1/2)|1 1|?⟩⟨ ⟩⟨
a) 0
b) log(2)
c) 1/2
d) 1
Answer: b) log(2)
69. If a quantum algorithm requires O(√N) queries to solve a problem, how many queries would
be needed for a problem size of 1,000,000?
a) 10
b) 100
c) 1,000
d) 10,000
Answer: c) 1,000
70. What is the probability of measuring |1 in the state |ψ = (3/5)|0 + (4/5)|1 ?⟩ ⟩ ⟩ ⟩
a) 3/5
b) 4/5
c) 9/25
d) 16/25
Answer: d) 16/25
71. If the CNOT gate is applied to the state (1/√2)(|0 + |1 ) |0 , what is the resulting state?⟩ ⟩ ⊗ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: b) (1/√2)(|00 + |10 )⟩ ⟩
72. What is the quantum circuit depth of a sequential application of n single-qubit gates?
a) 1
b) n
c) log(n)
d) 2^n
Answer: b) n
73. If |ψ = α|0 + β|1 and |φ = γ|0 + δ|1 are two qubit states, what is their inner product ψ|⟩ ⟩ ⟩ ⟩ ⟩ ⟩ ⟨
φ ?⟩
a) αγ + βδ
b) α*γ + β*δ
c) |α|^2 + |β|^2
d) |γ|^2 + |δ|^2
Answer: b) α*γ + β*δ
74. What is the gate complexity of the quantum Fourier transform on n qubits?
a) O(n)
b) O(n log n)
c) O(n^2)
d) O(2^n)
Answer: b) O(n log n)
75. If a qubit is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value of the ⟩ ⟩ ⟩
Pauli-Y operator?
a) 0
b) i√2/3
c) -i√2/3
d) 2/3
Answer: c) -i√2/3
76. What is the tensor product of the Pauli-X and Pauli-Z matrices?
a) [[0, 1], [1, 0]]
b) [[0, -i], [i, 0]]
c) [[0, 1], [-1, 0]]
d) [[1, 0], [0, -1]]
Answer: c) [[0, 1], [-1, 0]]
77. If the Toffoli gate is applied to the state |110 , what is the resulting state?⟩
a) |110⟩
b) |111⟩
c) |100⟩
d) |101⟩
Answer: b) |111⟩
78. What is the trace of the matrix representation of the Hadamard gate?
a) 0
b) 1
c) √2
d) 2
Answer: b) 1
79. If a quantum system is in the state |ψ = (1/√2)|0 + (1/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: d) 1
80. What is the dimension of the Hilbert space for a system of n qudits with d levels each?
a) n
b) d
c) n^d
d) d^n
Answer: d) d^n
81. If the density matrix of a qubit is ρ = (1/2)I + (1/2)σz, where I is the identity matrix and σz is
the Pauli-Z matrix, what is the probability of measuring |0 ?⟩
a) 1/4
b) 1/2
c) 3/4
d) 1
Answer: c) 3/4
82. What is the quantum circuit depth of a CNOT gate followed by a Hadamard gate on the
target qubit?
a) 1
b) 2
c) 3
d) 4
Answer: b) 2
83. If |ψ = α|0 + β|1 is a pure state, what is the purity Tr(ρ^2) of its density matrix ρ?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1
d) 2
Answer: c) 1
84. What is the probability of finding the correct answer in Grover's algorithm after π√N/4
iterations for a database of size N?
a) 1/2
b) 1/√2
c) 1/4
d) 1
Answer: b) 1/√2
85. If a quantum system is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value⟩ ⟩ ⟩
of the observable A = 2|0 0| + 3|1 1|?⟩⟨ ⟩⟨
a) 5/3
b) 8/3
c) 7/3
d) 11/3
Answer: b) 8/3
86. What is the trace distance between the pure states |0 and (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
87. If the Hadamard gate H is applied to each qubit in the state |00 , what is the resulting state?⟩
a) (1/2)(|00 + |01 + |10 + |11 )⟩⟩⟩⟩
b) (1/√2)(|00 + |11 )⟩ ⟩
c) (1/2)(|00 - |01 - |10 + |11 )⟩ ⟩ ⟩ ⟩
d) |++⟩
Answer: a) (1/2)(|00 + |01 + |10 + |11 )⟩ ⟩ ⟩ ⟩
88. What is the quantum circuit depth of a quantum Fourier transform on n qubits using the
standard decomposition?
a) n
b) n^2
c) log(n)
d) 2^n
Answer: b) n^2
What is the probability of measuring |0 in the state |ψ = cos(θ/2)|0 + sin(θ/2)|1 ?⟩ ⟩ ⟩ ⟩
a) cos^2(θ/2)
b) sin^2(θ/2)
c) cos(θ/2)
d) sin(θ/2)
Answer: a) cos^2(θ/2)
55. If a qubit is in the state |ψ = α|0 + β|1 , what is the expectation value of the Pauli-Z ⟩ ⟩ ⟩
operator?
a) α^2 - β^2
b) α^2 + β^2
c) 2αβ
d) α - β
Answer: a) α^2 - β^2
56. What is the dimension of the Hilbert space for a system of 3 qubits?
a) 3
b) 6
c) 8
d) 16
Answer: c) 8
57. If the CNOT gate is applied to the state (1/√2)(|00 + |11 ), what is the resulting state?⟩ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: a) (1/√2)(|00 + |11 )⟩ ⟩
58. What is the probability of finding a marked item in a database of N items after one Grover
iteration?
a) 1/N
b) 4/N
c) 1/√N
d) 4/√N
Answer: b) 4/N
59. If |ψ = α|0 + β|1 is a pure state, what is the condition for α and β?⟩ ⟩ ⟩
a) α + β = 1
b) α^2 + β^2 = 1
c) α = β
d) α^2 - β^2 = 1
Answer: b) α^2 + β^2 = 1
60. What is the action of the Pauli-X gate on the state |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) |+⟩
b) |-⟩
c) |0⟩
d) |1⟩
Answer: a) |+⟩
61. If a quantum system is in the state |ψ = (1/√2)|0 + (i/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |0 ?⟩
a) 0
b) 1/4
c) 1/2
d) 1
Answer: c) 1/2
62. What is the trace distance between the states ρ = |0 0| and σ = |+ +|?⟩⟨ ⟩⟨
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
63. If the Hadamard gate H is applied to the state |ψ = α|0 + β|1 , what is the resulting state?⟩ ⟩ ⟩
a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
b) ((α-β)/√2)|0 + ((α+β)/√2)|1⟩ ⟩
c) (α/√2)|0 + (β/√2)|1⟩ ⟩
d) (β/√2)|0 + (α/√2)|1⟩ ⟩
Answer: a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
64. What is the fidelity between the pure states |ψ = (1/√2)(|0 + |1 ) and |φ = (1/√2)(|0 - |1 )?⟩ ⟩ ⟩ ⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: b) 1/2
65. If a qubit is in the state |ψ = cos(θ)|0 + sin(θ)|1 , what is the expectation value of the Pauli-⟩ ⟩ ⟩
X operator?
a) cos(θ)
b) sin(θ)
c) sin(2θ)
d) cos(2θ)
Answer: c) sin(2θ)
66. What is the Schmidt rank of the state (1/√3)(|00 + |11 + |22 )?⟩ ⟩ ⟩
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
67. If the phase gate S is applied twice to a qubit in state |+ , what is the resulting state?⟩
a) |+⟩
b) |-⟩
c) i|+⟩
d) -|+⟩
Answer: b) |-⟩
68. What is the von Neumann entropy of the density matrix ρ = (1/2)|0 0| + (1/2)|1 1|?⟩⟨ ⟩⟨
a) 0
b) log(2)
c) 1/2
d) 1
Answer: b) log(2)
69. If a quantum algorithm requires O(√N) queries to solve a problem, how many queries would
be needed for a problem size of 1,000,000?
a) 10
b) 100
c) 1,000
d) 10,000
Answer: c) 1,000
70. What is the probability of measuring |1 in the state |ψ = (3/5)|0 + (4/5)|1 ?⟩ ⟩ ⟩ ⟩
a) 3/5
b) 4/5
c) 9/25
d) 16/25
Answer: d) 16/25
71. If the CNOT gate is applied to the state (1/√2)(|0 + |1 ) |0 , what is the resulting state?⟩ ⟩ ⊗ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: b) (1/√2)(|00 + |10 )⟩ ⟩
72. What is the quantum circuit depth of a sequential application of n single-qubit gates?
a) 1
b) n
c) log(n)
d) 2^n
Answer: b) n
73. If |ψ = α|0 + β|1 and |φ = γ|0 + δ|1 are two qubit states, what is their inner product ψ|⟩ ⟩ ⟩ ⟩ ⟩ ⟩ ⟨
φ ?⟩
a) αγ + βδ
b) α*γ + β*δ
c) |α|^2 + |β|^2
d) |γ|^2 + |δ|^2
Answer: b) α*γ + β*δ
74. What is the gate complexity of the quantum Fourier transform on n qubits?
a) O(n)
b) O(n log n)
c) O(n^2)
d) O(2^n)
Answer: b) O(n log n)
75. If a qubit is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value of the ⟩ ⟩ ⟩
Pauli-Y operator?
a) 0
b) i√2/3
c) -i√2/3
d) 2/3
Answer: c) -i√2/3
76. What is the tensor product of the Pauli-X and Pauli-Z matrices?
a) [[0, 1], [1, 0]]
b) [[0, -i], [i, 0]]
c) [[0, 1], [-1, 0]]
d) [[1, 0], [0, -1]]
Answer: c) [[0, 1], [-1, 0]]
77. If the Toffoli gate is applied to the state |110 , what is the resulting state?⟩
a) |110⟩
b) |111⟩
c) |100⟩
d) |101⟩
Answer: b) |111⟩
78. What is the trace of the matrix representation of the Hadamard gate?
a) 0
b) 1
c) √2
d) 2
Answer: b) 1
79. If a quantum system is in the state |ψ = (1/√2)|0 + (1/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: d) 1
80. What is the dimension of the Hilbert space for a system of n qudits with d levels each?
a) n
b) d
c) n^d
d) d^n
Answer: d) d^n
81. If the density matrix of a qubit is ρ = (1/2)I + (1/2)σz, where I is the identity matrix and σz is
the Pauli-Z matrix, what is the probability of measuring |0 ?⟩
a) 1/4
b) 1/2
c) 3/4
d) 1
Answer: c) 3/4
82. What is the quantum circuit depth of a CNOT gate followed by a Hadamard gate on the
target qubit?
a) 1
b) 2
c) 3
d) 4
Answer: b) 2
83. If |ψ = α|0 + β|1 is a pure state, what is the purity Tr(ρ^2) of its density matrix ρ?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1
d) 2
Answer: c) 1
84. What is the probability of finding the correct answer in Grover's algorithm after π√N/4
iterations for a database of size N?
a) 1/2
b) 1/√2
c) 1/4
d) 1
Answer: b) 1/√2
85. If a quantum system is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value⟩ ⟩ ⟩
of the observable A = 2|0 0| + 3|1 1|?⟩⟨ ⟩⟨
a) 5/3
b) 8/3
c) 7/3
d) 11/3
Answer: b) 8/3
86. What is the trace distance between the pure states |0 and (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
87. If the Hadamard gate H is applied to each qubit in the state |00 , what is the resulting state?⟩
a) (1/2)(|00 + |01 + |10 + |11 )⟩⟩⟩⟩
b) (1/√2)(|00 + |11 )⟩ ⟩
c) (1/2)(|00 - |01 - |10 + |11 )⟩ ⟩ ⟩ ⟩
d) |++⟩
Answer: a) (1/2)(|00 + |01 + |10 + |11 )⟩ ⟩ ⟩ ⟩
88. What is the quantum circuit depth of a quantum Fourier transform on n qubits using the
standard decomposition?
a) n
b) n^2
c) log(n)
d) 2^n
Answer: b) n^2
What is the probability of measuring |0 in the state |ψ = cos(θ/2)|0 + sin(θ/2)|1 ?⟩ ⟩ ⟩ ⟩
a) cos^2(θ/2)
b) sin^2(θ/2)
c) cos(θ/2)
d) sin(θ/2)
Answer: a) cos^2(θ/2)
55. If a qubit is in the state |ψ = α|0 + β|1 , what is the expectation value of the Pauli-Z ⟩ ⟩ ⟩
operator?
a) α^2 - β^2
b) α^2 + β^2
c) 2αβ
d) α - β
Answer: a) α^2 - β^2
56. What is the dimension of the Hilbert space for a system of 3 qubits?
a) 3
b) 6
c) 8
d) 16
Answer: c) 8
57. If the CNOT gate is applied to the state (1/√2)(|00 + |11 ), what is the resulting state?⟩ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: a) (1/√2)(|00 + |11 )⟩ ⟩
58. What is the probability of finding a marked item in a database of N items after one Grover
iteration?
a) 1/N
b) 4/N
c) 1/√N
d) 4/√N
Answer: b) 4/N
59. If |ψ = α|0 + β|1 is a pure state, what is the condition for α and β?⟩ ⟩ ⟩
a) α + β = 1
b) α^2 + β^2 = 1
c) α = β
d) α^2 - β^2 = 1
Answer: b) α^2 + β^2 = 1
60. What is the action of the Pauli-X gate on the state |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) |+⟩
b) |-⟩
c) |0⟩
d) |1⟩
Answer: a) |+⟩
61. If a quantum system is in the state |ψ = (1/√2)|0 + (i/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |0 ?⟩
a) 0
b) 1/4
c) 1/2
d) 1
Answer: c) 1/2
62. What is the trace distance between the states ρ = |0 0| and σ = |+ +|?⟩⟨ ⟩⟨
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
63. If the Hadamard gate H is applied to the state |ψ = α|0 + β|1 , what is the resulting state?⟩ ⟩ ⟩
a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
b) ((α-β)/√2)|0 + ((α+β)/√2)|1⟩ ⟩
c) (α/√2)|0 + (β/√2)|1⟩ ⟩
d) (β/√2)|0 + (α/√2)|1⟩ ⟩
Answer: a) ((α+β)/√2)|0 + ((α-β)/√2)|1⟩ ⟩
64. What is the fidelity between the pure states |ψ = (1/√2)(|0 + |1 ) and |φ = (1/√2)(|0 - |1 )?⟩ ⟩ ⟩ ⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: b) 1/2
65. If a qubit is in the state |ψ = cos(θ)|0 + sin(θ)|1 , what is the expectation value of the Pauli-⟩ ⟩ ⟩
X operator?
a) cos(θ)
b) sin(θ)
c) sin(2θ)
d) cos(2θ)
Answer: c) sin(2θ)
66. What is the Schmidt rank of the state (1/√3)(|00 + |11 + |22 )?⟩ ⟩ ⟩
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
67. If the phase gate S is applied twice to a qubit in state |+ , what is the resulting state?⟩
a) |+⟩
b) |-⟩
c) i|+⟩
d) -|+⟩
Answer: b) |-⟩
68. What is the von Neumann entropy of the density matrix ρ = (1/2)|0 0| + (1/2)|1 1|?⟩⟨ ⟩⟨
a) 0
b) log(2)
c) 1/2
d) 1
Answer: b) log(2)
69. If a quantum algorithm requires O(√N) queries to solve a problem, how many queries would
be needed for a problem size of 1,000,000?
a) 10
b) 100
c) 1,000
d) 10,000
Answer: c) 1,000
70. What is the probability of measuring |1 in the state |ψ = (3/5)|0 + (4/5)|1 ?⟩ ⟩ ⟩ ⟩
a) 3/5
b) 4/5
c) 9/25
d) 16/25
Answer: d) 16/25
71. If the CNOT gate is applied to the state (1/√2)(|0 + |1 ) |0 , what is the resulting state?⟩ ⟩ ⊗ ⟩
a) (1/√2)(|00 + |11 )⟩ ⟩
b) (1/√2)(|00 + |10 )⟩ ⟩
c) (1/√2)(|01 + |10 )⟩ ⟩
d) (1/√2)(|01 + |11 )⟩ ⟩
Answer: b) (1/√2)(|00 + |10 )⟩ ⟩
72. What is the quantum circuit depth of a sequential application of n single-qubit gates?
a) 1
b) n
c) log(n)
d) 2^n
Answer: b) n
73. If |ψ = α|0 + β|1 and |φ = γ|0 + δ|1 are two qubit states, what is their inner product ψ|⟩ ⟩ ⟩ ⟩ ⟩ ⟩ ⟨
φ ?⟩
a) αγ + βδ
b) α*γ + β*δ
c) |α|^2 + |β|^2
d) |γ|^2 + |δ|^2
Answer: b) α*γ + β*δ
74. What is the gate complexity of the quantum Fourier transform on n qubits?
a) O(n)
b) O(n log n)
c) O(n^2)
d) O(2^n)
Answer: b) O(n log n)
75. If a qubit is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value of the ⟩ ⟩ ⟩
Pauli-Y operator?
a) 0
b) i√2/3
c) -i√2/3
d) 2/3
Answer: c) -i√2/3
76. What is the tensor product of the Pauli-X and Pauli-Z matrices?
a) [[0, 1], [1, 0]]
b) [[0, -i], [i, 0]]
c) [[0, 1], [-1, 0]]
d) [[1, 0], [0, -1]]
Answer: c) [[0, 1], [-1, 0]]
77. If the Toffoli gate is applied to the state |110 , what is the resulting state?⟩
a) |110⟩
b) |111⟩
c) |100⟩
d) |101⟩
Answer: b) |111⟩
78. What is the trace of the matrix representation of the Hadamard gate?
a) 0
b) 1
c) √2
d) 2
Answer: b) 1
79. If a quantum system is in the state |ψ = (1/√2)|0 + (1/√2)|1 , what is the probability of ⟩ ⟩ ⟩
measuring |+ = (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: d) 1
80. What is the dimension of the Hilbert space for a system of n qudits with d levels each?
a) n
b) d
c) n^d
d) d^n
Answer: d) d^n
81. If the density matrix of a qubit is ρ = (1/2)I + (1/2)σz, where I is the identity matrix and σz is
the Pauli-Z matrix, what is the probability of measuring |0 ?⟩
a) 1/4
b) 1/2
c) 3/4
d) 1
Answer: c) 3/4
82. What is the quantum circuit depth of a CNOT gate followed by a Hadamard gate on the
target qubit?
a) 1
b) 2
c) 3
d) 4
Answer: b) 2
83. If |ψ = α|0 + β|1 is a pure state, what is the purity Tr(ρ^2) of its density matrix ρ?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1
d) 2
Answer: c) 1
84. What is the probability of finding the correct answer in Grover's algorithm after π√N/4
iterations for a database of size N?
a) 1/2
b) 1/√2
c) 1/4
d) 1
Answer: b) 1/√2
85. If a quantum system is in the state |ψ = (1/√3)|0 + (√2/√3)|1 , what is the expectation value⟩ ⟩ ⟩
of the observable A = 2|0 0| + 3|1 1|?⟩⟨ ⟩⟨
a) 5/3
b) 8/3
c) 7/3
d) 11/3
Answer: b) 8/3
86. What is the trace distance between the pure states |0 and (1/√2)(|0 + |1 )?⟩ ⟩ ⟩
a) 0
b) 1/2
c) 1/√2
d) 1
Answer: c) 1/√2
87. If the Hadamard gate H is applied to each qubit in the state |00 , what is the resulting state?⟩
a) (1/2)(|00 + |01 + |10 + |11 )⟩⟩⟩⟩
b) (1/√2)(|00 + |11 )⟩ ⟩
c) (1/2)(|00 - |01 - |10 + |11 )⟩ ⟩ ⟩ ⟩
d) |++⟩
Answer: a) (1/2)(|00 + |01 + |10 + |11 )⟩ ⟩ ⟩ ⟩
88. What is the quantum circuit depth of a quantum Fourier transform on n qubits using the
standard decomposition?
a) n
b) n^2
c) log(n)
d) 2^n
Answer: b) n^2
89. If a qubit is in the state |ψ = cos(θ/2)|0 + e^(iφ)sin(θ/2)|1 , what is the expectation value of ⟩ ⟩ ⟩
the Pauli-Y operator?
a) sin(θ)sin(φ)
b) sin(θ)cos(φ)
c) cos(θ)sin(φ)
d) cos(θ)cos(φ)
Answer: a) sin(θ)sin(φ)
90. What is the Schmidt decomposition of the state (1/√3)(|00 + |01 + |10 )?⟩ ⟩ ⟩
a) (1/√3)(|0 + |1 ) |0 + (1/√3)|0 |1⟩ ⟩ ⊗ ⟩ ⟩ ⊗ ⟩
b) (1/√2)|0 (1/√2)(|⟩ ⊗
