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Quantum Circuit Practice Problems

Solve circuit construction challenges under strict automated verification. Filter problems by specific quantum algorithm, foundational topics, or difficulty levels.

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Filter By Quantum AlgorithmShowing 50 of 50 problems
Topic:
Q#StatusProblemAlgorithmCategoryDifficultyQubitsAction
1
Equal Superposition State |+⟩
Prepare a 1-qubit system into an equal superposition state ∣+⟩=12(∣0⟩+∣1⟩)|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) starting from ∣0⟩|0\rangle.
Single-Qubit OperationsEasy1QSolve
2
Minus Superposition | - ⟩ State
Prepare the orthogonal superposition state ∣−⟩=12(∣0⟩−∣1⟩)|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle) starting from ∣0⟩|0\rangle.
Single-Qubit OperationsEasy1QSolve
3
Combined Bit and Phase Flip -|1⟩
Starting from ∣0⟩|0\rangle, map the qubit state to −∣1⟩-|1\rangle.
Single-Qubit OperationsEasy1QSolve
4
Circular Polarization |i+⟩ State
Prepare the positive Y-eigenstate ∣i+⟩=12(∣0⟩+i∣1⟩)|i+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + i|1\rangle) from ∣0⟩|0\rangle.
Single-Qubit OperationsEasy1QSolve
5
Negative Y-Eigenstate |i-⟩ Preparation
Prepare the negative Y-eigenstate ∣i−⟩=12(∣0⟩−i∣1⟩)|i-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - i|1\rangle) starting from ∣0⟩|0\rangle.
Single-Qubit OperationsEasy1QSolve
6
T-Gate π/4 Relative Phase Shift
Create the state 12(∣0⟩+eiπ/4∣1⟩)\frac{1}{\sqrt{2}}(|0\rangle + e^{i\pi/4}|1\rangle) on the equatorial plane of the Bloch sphere.
Single-Qubit OperationsEasy1QSolve
7
Inverse Phase Operation (S† from S & Z)
Synthesize the adjoint phase gate S†=(100−i)S^\dagger = \begin{pmatrix}1 & 0 \\ 0 & -i\end{pmatrix} using only S and Z gates.
Single-Qubit OperationsEasy1QSolve
8
Hadamard Self-Inverse Identity (H · H = I)
Demonstrate that the Hadamard transformation is unitary and its own inverse (H†=HH^\dagger = H).
Single-Qubit OperationsEasy1QSolve
9
Pauli-X Involution (X · X = I)
Demonstrate that applying two consecutive bit-flip operations restores the original state.
Single-Qubit OperationsEasy1QSolve
10
Clifford Sequence H · S · H
Construct the composite unitary transformation U=HSHU = H S H on ∣0⟩|0\rangle.
Single-Qubit OperationsMedium1QSolve
11
Pauli-Y Combined Bit & Phase Operation
Apply the Pauli-Y operator Y=(0−ii0)Y = \begin{pmatrix}0 & -i \\ i & 0\end{pmatrix} directly on state ∣0⟩|0\rangle.
Single-Qubit OperationsEasy1QSolve
12
T-Gate Squaring Identity (T · T = S)
Demonstrate that two consecutive π/4\pi/4 phase rotations equal a π/2\pi/2 Phase gate (T2=ST^2 = S).
Single-Qubit OperationsEasy1QSolve
13
Z-Basis Eigenstate Phase Modulation
Demonstrate that the Pauli-Z gate leaves ∣0⟩|0\rangle unchanged and maps ∣1⟩|1\rangle to −∣1⟩-|1\rangle.
Single-Qubit OperationsEasy1QSolve
14
Maximally Entangled Bell Pair |Φ+⟩
Construct the canonical Bell state ∣Φ+⟩=12(∣00⟩+∣11⟩)|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) on 2 qubits initialized to ∣00⟩|00\rangle.
Quantum EntanglementMedium2QSolve
15
Bell State |Φ-⟩ Generation
Construct the orthogonal Bell state ∣Φ−⟩=12(∣00⟩−∣11⟩)|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle) starting from ∣00⟩|00\rangle.
Quantum EntanglementEasy2QSolve
16
Bell State |Ψ+⟩ Generation
Generate the symmetric entangled state ∣Ψ+⟩=12(∣01⟩+∣10⟩)|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle) on 2 qubits starting from ∣00⟩|00\rangle.
Quantum EntanglementMedium2QSolve
17
Singlet State |Ψ-⟩ Generation
Prepare the antisymmetric singlet state ∣Ψ−⟩=12(∣01⟩−∣10⟩)|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle) starting from ∣00⟩|00\rangle.
Quantum EntanglementMedium2QSolve
18
3-Qubit Greenberger–Horne–Zeilinger (GHZ) State
Generate the genuine tripartite entangled GHZ state ∣GHZ⟩=12(∣000⟩+∣111⟩)|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}(|000\rangle + |111\rangle) on 3 qubits.
Quantum EntanglementMedium3QSolve
19
3-Qubit GHZ Parity Verification
Generate the 3-qubit GHZ state and verify computational basis measurements strictly yield only |000⟩ or |111⟩.
Quantum EntanglementMedium3QSolve
20
3-Qubit GHZ |GHZ-⟩ State with Negative Phase
Prepare the state ∣GHZ−⟩=12(∣000⟩−∣111⟩)|\text{GHZ}^-\rangle = \frac{1}{\sqrt{2}}(|000\rangle - |111\rangle) across 3 qubits.
Quantum EntanglementMedium3QSolve
21
2-Qubit Cluster State via Controlled-Z
Create the 2-qubit graph state ∣G2⟩=12(∣00⟩+∣01⟩+∣10⟩−∣11⟩)|G_2\rangle = \frac{1}{2}(|00\rangle + |01\rangle + |10\rangle - |11\rangle) essential for measurement-based quantum computing.
Quantum EntanglementMedium2QSolve
22
2-Qubit Symmetric Superposition (|01⟩ + |10⟩)
Prepare the symmetric zero-net-spin subspace state 12(∣01⟩+∣10⟩)\frac{1}{\sqrt{2}}(|01\rangle + |10\rangle) without any component of ∣00⟩|00\rangle or ∣11⟩|11\rangle.
Quantum EntanglementMedium2QSolve
23
Entangling Superposition with Phase Kickback
Generate the entangled state 12(∣00⟩−∣01⟩)\frac{1}{\sqrt{2}}(|00\rangle - |01\rangle) and observe phase kickback under CX.
Quantum EntanglementMedium2QSolve
24
Deutsch-Jozsa Constant Oracle (f(x) = 0)
Implement the Deutsch-Jozsa test for a Constant-0 function f(x)=0f(x)=0 (identity oracle).
Quantum ParallelismMedium2QSolve
25
Deutsch-Jozsa Balanced Oracle DiscriminationPremium
Implement the Deutsch-Jozsa algorithm for the balanced oracle f(x)=xf(x) = x (represented by a CNOT from input Q0 to ancilla Q1).
Quantum ParallelismHard2QSolve
26
Deutsch Oracle: Constant-1 Function f(x) = 1
Implement Deutsch algorithm for oracle f(x)=1f(x)=1, which applies an unconditional bit-flip (Pauli-X) on the ancilla qubit.
Quantum ParallelismMedium2QSolve
27
Deutsch Oracle: Balanced Inverted Function f(x) = ¬xPremium
Construct the Deutsch algorithm circuit for f(x)=1−xf(x) = 1 - x, an inverted balanced oracle.
Quantum ParallelismHard2QSolve
28
Bernstein-Vazirani 1-Bit Hidden String s = "1"
Extract the 1-bit secret string s=1s = 1 in a single query using the Bernstein-Vazirani algorithm.
Quantum ParallelismMedium2QSolve
29
Bernstein-Vazirani 2-Bit Hidden String s = "11"Premium
Find the secret 2-bit string s=11s = 11 with a single quantum oracle evaluation on 3 qubits (Q0, Q1 inputs, Q2 ancilla).
Quantum ParallelismHard3QSolve
30
Bernstein-Vazirani 2-Bit Hidden String s = "10"Premium
Find the secret 2-bit string s=10s = 10 with a single quantum query on 3 qubits (Q0, Q1 inputs, Q2 ancilla).
Quantum ParallelismHard3QSolve
31
Bernstein-Vazirani 2-Bit Hidden String s = "01"Premium
Find the secret 2-bit string s=01s = 01 with a single quantum query on 3 qubits (Q0, Q1 inputs, Q2 ancilla).
Quantum ParallelismHard3QSolve
32
Grover's 2-Qubit Search for |11⟩Premium
Implement a 2-qubit Grover Search circuit to find the marked database entry ∣11⟩|11\rangle.
Amplitude AmplificationHard2QSolve
33
Grover's Search for Marked State |00⟩Premium
Construct a Grover Search instance configured to search and amplify the target ∣00⟩|00\rangle.
Amplitude AmplificationHard2QSolve
34
Grover's Search for Marked State |01⟩Premium
Construct a 2-qubit Grover Search circuit configured to find and amplify target state ∣01⟩|01\rangle (Q0=1, Q1=0).
Amplitude AmplificationHard2QSolve
35
Grover's Search for Marked State |10⟩Premium
Construct a 2-qubit Grover Search circuit configured to find and amplify target state ∣10⟩|10\rangle (Q0=0, Q1=1).
Amplitude AmplificationHard2QSolve
36
Phase Oracle Inversion for |11⟩
Implement the phase oracle Uf=I−2∣11⟩⟨11∣U_f = I - 2|11\rangle\langle 11| that flips the sign of the marked item ∣11⟩|11\rangle.
Amplitude AmplificationEasy2QSolve
37
Phase Oracle Inversion for |00⟩
Implement the phase oracle U0=I−2∣00⟩⟨00∣U_0 = I - 2|00\rangle\langle 00| that marks ∣00⟩|00\rangle with a −1-1 relative phase.
Amplitude AmplificationMedium2QSolve
38
Phase Oracle Inversion for |01⟩
Construct the phase-marking operator Uf=I−2∣01⟩⟨01∣U_f = I - 2|01\rangle\langle 01| for target state ∣01⟩|01\rangle.
Amplitude AmplificationMedium2QSolve
39
Phase Oracle Inversion for |10⟩
Construct the phase-marking operator Uf=I−2∣10⟩⟨10∣U_f = I - 2|10\rangle\langle 10| for target state ∣10⟩|10\rangle.
Amplitude AmplificationMedium2QSolve
40
Swap Protocol via 3 CNOT Gates
Exchange the quantum states of two qubits WITHOUT using the SWAP gate.
Circuit IdentitiesMedium2QSolve
41
H-X-H Conjugation Identity (H · X · H = Z)
Verify the foundational Clifford identity that conjugating Pauli-X with Hadamard gates converts a bit flip into a phase flip: HXH=ZH X H = Z.
Circuit IdentitiesEasy1QSolve
42
H-Z-H Conjugation Identity (H · Z · H = X)
Demonstrate that conjugating Pauli-Z with Hadamard converts a phase flip into a bit flip: HZH=XH Z H = X.
Circuit IdentitiesEasy1QSolve
43
Controlled-Z Synthesis from CNOT & Hadamards
Synthesize a Controlled-Z gate between Qubit 0 and Qubit 1 using only Hadamard gates and a standard CNOT.
Circuit IdentitiesMedium2QSolve
44
CNOT Direction Reversal with Hadamards
Reverse the direction of a CNOT gate (making Q1 control and Q0 target) using 4 Hadamard gates: CX1,0=(H⊗H)CX0,1(H⊗H)CX_{1,0} = (H \otimes H) CX_{0,1} (H \otimes H).
Circuit IdentitiesMedium2QSolve
45
SWAP Gate Self-Inverse Identity
Verify that applying the SWAP gate twice is equivalent to the identity operation (SWAP2=ISWAP^2 = I).
Circuit IdentitiesEasy2QSolve
46
Quantum Teleportation of Superposition |+⟩Premium
Teleport an unknown quantum state ∣psi⟩=∣+⟩|psi\rangle = |+\rangle on Qubit 0 to Bob’s Qubit 2 using an entangled Bell pair between Q1 and Q2.
Quantum CommunicationHard3QSolve
47
Superdense Coding: 2-Bit Transmission "11"Premium
Encode classical bits (1, 1) using Alice’s local unitary on an entangled Bell pair and verify Bob can decode it with 100% certainty.
Quantum CommunicationHard2QSolve
48
Superdense Coding: Encode Bits "01"Premium
Transmit the 2-bit classical message "01" by applying Alice’s local Pauli-Z operation to an entangled Bell pair.
Quantum CommunicationHard2QSolve
49
Superdense Coding: Encode Bits "10"Premium
Transmit the 2-bit classical message "10" by applying Alice’s local Pauli-X operation to an entangled Bell pair.
Quantum CommunicationHard2QSolve
50
Superdense Coding: Encode Bits "00"
Transmit the 2-bit classical message "00" (identity operation) across an EPR channel.
Quantum CommunicationMedium2QSolve