Quantum Computing Linear Algebra Q&A (Complete)

Section 1: Vectors & Inner Products

Q1: What is a vector in quantum computing?

A: A vector represents a quantum state with amplitudes for each basis state, e.g., |ψ⟩ = α|0⟩ + β|1⟩ = [α, β]ᵀ. Real-life: Arrow in space showing probability direction and magnitude.

Q2: What is the inner product of two vectors?

A: ⟨φ|ψ⟩ = Σ φ_i* ψ_i, measures overlap/probability amplitude. Real-life: Projection of one arrow onto another; probability of state transition.

Q3: How do we normalize a vector?

A: ||ψ|| = sqrt(⟨ψ|ψ⟩) = 1. Ensures total probability = 1. Real-life: Adjusting arrow length to 1 to represent full certainty.

Q4: What does orthogonality mean?

A: ⟨φ|ψ⟩ = 0 → vectors independent, no overlap. Real-life: Perpendicular arrows or independent musical notes.

Q5: What is a basis in vector space?

A: Set of vectors to represent any vector in space. Example: |0⟩ and |1⟩ for a single qubit. Real-life: X/Y axes on a graph; any vector can be decomposed along axes.

Q6: What is a dual vector?

A: Bra ⟨ψ| corresponds to ket |ψ⟩. Used in inner products. Real-life: Mirror image of arrow to measure alignment.

Q7: How do you measure angle between vectors?

A: cos θ = Re(⟨φ|ψ⟩) / (||φ|| ||ψ||). Real-life: Angle between arrows; helps understand similarity of quantum states.

Q8: What is a vector norm?

A: ||v|| = sqrt(Σ |v_i|²). Represents length of vector. Real-life: Physical length of an arrow.

Q9: Can vectors be complex?

A: Yes, quantum vectors usually have complex numbers. Real-life: Arrow in 2D plane: x = real, y = imaginary.

Q10: How are probabilities computed from vectors?

A: Probability = |amplitude|² = |α|². Real-life: Chance of spinning arrow pointing along desired direction.

Section 2: Complex Numbers

Q11: Why are complex numbers used?

A: Encode magnitude & phase; interference depends on phase. Real-life: Water waves: height = magnitude, timing = phase.

Q12: How to take conjugate?

A: z = a + bi → z* = a - bi. Real-life: Reverse spinning direction of arrow.

Q13: What is magnitude of complex number?

A: |z| = sqrt(a²+b²). Real-life: Length of arrow in 2D plane.

Q14: What is phase?

A: θ = arctan(b/a). Determines interference. Real-life: Angle of arrow; affects combination with other arrows.

Q15: How do complex numbers appear in qubits?

A: |ψ⟩ = α|0⟩ + β|1⟩ with α, β ∈ ℂ. Real-life: Two spinning arrows representing superposition.

Q16: How do complex numbers affect measurement?

A: Probabilities = |α|², |β|². Phase affects interference in algorithms. Real-life: Two overlapping waves create constructive or destructive patterns.

Q17: How do you multiply complex numbers?

A: (a+bi)(c+di) = (ac-bd) + (ad+bc)i. Real-life: Combine rotations and scaling of arrows.

Q18: How are complex conjugates used in inner products?

A: ⟨φ|ψ⟩ = Σ φ_i* ψ_i ensures probabilities are real. Real-life: Flip one arrow to measure projection correctly.

Q19: What is Euler’s formula?

A: e^(iθ) = cosθ + i sinθ. Phase representation in quantum gates. Real-life: Rotating arrow in circle; angle = θ.

Q20: How does phase affect interference?

A: |ψ1⟩ + e^(iθ)|ψ2⟩ → amplitude can cancel or enhance depending on θ. Real-life: Two waves meeting: crest + crest = bigger, crest + trough = cancel.

Section 3: Tensor Products

Q21: What is tensor product?

A: |ψ⟩ ⊗ |φ⟩ = [ψ1φ1, ψ1φ2, ψ2φ1, ψ2φ2]ᵀ. Real-life: Two dice combined → all outcome combinations.

Q22: How does tensor product scale?

A: n qubits → 2ⁿ dimension. Real-life: 3 coins → 8 combinations: HHH, HHT…

Q23: Tensor product with gates?

A: Apply gate to one qubit: U⊗I. Real-life: Press button on one machine while others stay unchanged.

Q24: What is entanglement?

A: Cannot factor as tensor products. Example: |Φ+⟩ = (|00⟩ + |11⟩)/√2 Real-life: Magic dice: first die = second die automatically.

Q25: Tensor product inner product?

A: ⟨φ1⊗φ2|ψ1⊗ψ2⟩ = ⟨φ1|ψ1⟩ × ⟨φ2|ψ2⟩. Real-life: Probability of two independent dice multiplied.

Q26: Can tensor product create entanglement?

A: Only multi-qubit gates (like CNOT) applied to tensor products can entangle.

Q27: Real-life analogy for tensor products?

A: Combine multiple coins, dice, or spinning arrows to see all combinations.

Q28: How to represent multi-qubit states?

A: |ψ1⟩⊗|ψ2⟩⊗…⊗|ψn⟩. Real-life: Combination of multiple arrows/dice outcomes.

Section 4: Hermitian, Unitary, Projection Operators

Q29: What is a Hermitian matrix?

A: H†=H, eigenvalues real, observables. Real-life: Measuring weight or temperature.

Q30: What is a Unitary matrix?

A: U†U=I, preserves norm, reversible. Real-life: Rotating rigid arrow; length unchanged.

Q31: Projection operator?

A: P=|φ⟩⟨φ|, P²=P, projects vector onto |φ⟩. Real-life: Flashlight shadow along direction.

Q32: Measurement probability?

A: P=|⟨φ|ψ⟩|². Real-life: Chance of shadow aligning with light.

Q33: Examples of Hermitian & Unitary matrices?

• Pauli X,Y,Z (Hermitian & Unitary)
• Hadamard H (Unitary)
• Phase S,T (Unitary)

Q34: How do these operators relate?

• Hermitian → measurement
• Unitary → evolution/gates
• Projection → probability post-measurement

Q35: Can all Hermitian matrices be diagonalized?

Yes, eigenvectors orthogonal, eigenvalues real.

Q36: Can all unitary matrices create entanglement?

No, only multi-qubit gates.

Q37: Why projection operators are idempotent?

P²=P → projecting twice gives same result.

Q38: Eigenvalues of Hermitian matrices?

Always real → measurable.

Q39: Eigenvalues of Unitary matrices?

Magnitude = 1, λ = e^(iθ). Real-life: Rotating arrow in unit circle.

Q40: How to compute probability with projection?

P = ⟨ψ|P|ψ⟩ = |⟨φ|ψ⟩|².

Q41: How to reverse a unitary operation?

Apply U† = U⁻¹. Real-life: Undoing rotation of a rigid object.

Q42: How does tensor product affect operators?

Apply to multi-qubit: (A⊗B)(|ψ⟩⊗|φ⟩)=A|ψ⟩⊗B|φ⟩. Real-life: Press two buttons independently; outputs combine.

Q43: What is commutation?

[A,B] = AB - BA. Determines if observables can be measured simultaneously. Real-life: Two machines interfere if order matters.

Q44: What is anti-commutation?

{A,B} = AB + BA = 0. Relevant for Pauli matrices. Real-life: Two arrows cancel each other in combination.

Q45: What is completeness relation?

Σ |φ_i⟩⟨φ_i| = I. All possible states sum to identity. Real-life: All shadows cover whole area; nothing left out.

Q46: What is trace of a matrix?

Tr(A) = sum of diagonal elements. For Hermitian, sum of eigenvalues. Real-life: Total weight of object decomposed along axes.

Q47: What is determinant?

det(U) for unitary = e^(iθ). Determines phase factor. Real-life: Scaling of volume under transformation.

Q48: How do phases affect multi-qubit gates?

Phase differences interfere; critical in Grover/Shor. Real-life: Timing waves for constructive interference.

Q49: How to check if operator is Hermitian?

A†=A → yes. Check eigenvalues real. Real-life: Measuring property gives real number.

Q50: How to check if operator is Unitary?

U†U=I → yes. Norm preserved. Real-life: Rotation without changing arrow length.

Q51: How does measurement collapse the state?

After measuring |φ⟩, |ψ⟩ → |φ⟩ with probability |⟨φ|ψ⟩|². Real-life: Shadow fixes arrow along measured direction.

Q52: How to describe multiple qubits combined?

Tensor product + unitary gates + entanglement. Real-life: Multiple spinning arrows/dice outcomes combined to simulate system.