Quantum Computing - Linear Algebra: Complex Numbers in Vectors

Objectives: Quantum Computing - Linear Algebra: Complex Numbers in Vectors

Quantum Computing - Linear Algebra: Complex Numbers in Vectors

Linear Algebra for Quantum Computing

Complex Numbers in Quantum Vectors

In quantum computing, complex numbers are used to represent the amplitudes of qubit states. A complex number has a real part and an imaginary part, written as:

z = a + bi
  • a = real part
  • b = imaginary part
  • i = √(-1), the imaginary unit

💡 Real-life analogy: Think of complex numbers as a combination of two movements: forward/backward (real) and side-to-side (imaginary). Together they describe a direction in 2D space.

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1. Complex Conjugate

The complex conjugate of z = a + bi is:

z* = a - bi

It flips the sign of the imaginary part. Complex conjugates are used in computing inner products of quantum states.

⟨φ|ψ⟩ = Σ φ_i* ψ_i

Here φ_i* is the complex conjugate of φ_i.

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2. Magnitude of a Complex Number

The magnitude (or absolute value) of z = a + bi is:

|z| = sqrt(a^2 + b^2)

💡 Example: z = 3 + 4i → |z| = sqrt(3² + 4²) = 5

💡 Real-life analogy: Imagine moving 3 meters east and 4 meters north. The direct distance to your destination is 5 meters, which is the magnitude.

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3. Phase of a Complex Number

Complex numbers have a phase (angle) θ, given by:

θ = arctan(b / a)

This phase is crucial in quantum computing because it affects interference patterns in quantum algorithms.

💡 Real-life analogy: Imagine two pendulums swinging. The angle difference between their swings is like the phase difference of two quantum amplitudes. Constructive and destructive interference depends on this phase. ---

4. Complex Numbers in Quantum Vectors

Quantum states are expressed as vectors with complex components:

|ψ⟩ = [α]
       [β]
  • α = α_r + α_i i (real + imaginary)
  • β = β_r + β_i i

Each amplitude represents the "weight" of that basis state. The probability of measuring |0⟩ is |α|², and for |1⟩ is |β|².

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5. Example of a Qubit with Complex Numbers

|ψ⟩ = (1/√2)|0⟩ + (i/√2)|1⟩

Vector form:
|ψ⟩ = [1/√2]
       [i/√2]

Probability of measuring |0⟩ = |1/√2|² = 1/2
Probability of measuring |1⟩ = |i/√2|² = 1/2

💡 Real-life analogy: Imagine a coin spinning in a tilted plane (not just heads or tails). Its tilt represents the phase. Measuring it “collapses” the coin to either heads or tails with equal probability.

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6. Operations with Complex Numbers

  • Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
  • Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
  • Conjugate multiplication: z * z* = |z|²
💡 Real-life analogy: Combining two light waves: the amplitude and phase of each wave combine to give interference patterns (constructive/destructive), similar to multiplying complex numbers in quantum amplitudes. ---

7. Why Complex Numbers Are Essential in Quantum Computing

  • They represent **probability amplitudes** of qubits.
  • Phases allow **interference**, which quantum algorithms exploit (e.g., Grover’s search, Shor’s factoring).
  • Used in defining **unitary operators** (quantum gates), which always involve complex numbers.

🔹 Key takeaway: Complex numbers encode both probability magnitude (|α|²) and phase information (θ), both essential for quantum computation.

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8. Summary of Key Terms

  • Complex number: a + bi, where a = real, b = imaginary
  • Complex conjugate: flips the sign of imaginary part, z* = a - bi
  • Magnitude: |z| = sqrt(a² + b²), gives probability amplitude size
  • Phase: θ = arctan(b / a), determines interference
  • Qubit amplitudes: complex numbers α, β describe the quantum state
  • Probability: P(|0⟩) = |α|², P(|1⟩) = |β|²

Next, we will cover Tensor Products, which allow combining multiple qubits into a single system. Tensor products use vectors and complex numbers extensively.

Reference Book: N/A

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