Linear Algebra for Quantum Computing-Vectors & Inner Products

Objectives: Linear Algebra for Quantum Computing-Vectors & Inner Products

Quantum Computing - Linear Algebra: Vectors & Inner Products

Linear Algebra for Quantum Computing

Vectors & Inner Products

Vectors are one of the most fundamental concepts in quantum computing. In simple terms, a vector is an object that has both a magnitude (size) and a direction. In quantum computing, vectors represent quantum states, which describe the state of a qubit or a system of qubits.

1. What is a Vector?

A vector in quantum computing is often represented as a column of numbers, usually complex numbers (numbers that have a real and an imaginary part). Example:

|ψ⟩ = α|0⟩ + β|1⟩
  • Here, α and β are complex numbers called amplitudes.
  • |0⟩ and |1⟩ are the basic quantum states called basis vectors.

In vector form:

|0⟩ = [1]
       [0]

|1⟩ = [0]
       [1]

|ψ⟩ = [α]
       [β]

Real-life analogy: Think of |ψ⟩ as the position of a small robot in a 2D grid. α tells how much it points towards "0" direction, β tells how much it points towards "1" direction. The robot can be anywhere along the line connecting |0⟩ and |1⟩, just like a qubit can be in a superposition of states.

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2. Dirac Notation (Bra-Ket)

Quantum computing uses a special notation:

  • |ψ⟩ = ket = column vector (represents a quantum state)
  • ⟨ψ| = bra = row vector (complex conjugate transpose of ket)

Example:

If |ψ⟩ = [α]
          [β]

Then ⟨ψ| = [α* β*]  (α* means complex conjugate of α)

💡 Real-life example: Imagine a musical chord played on a piano. |ψ⟩ represents the combination of notes, and ⟨ψ| represents how another listener would "measure" or detect those notes. The inner product ⟨φ|ψ⟩ tells how similar two chords are.

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3. Vector Addition and Scalar Multiplication

  • Vector Addition: Add corresponding components
  •   [a1]   [b1]   [a1 + b1]
      [a2] + [b2] = [a2 + b2]
      
  • Scalar Multiplication: Multiply each component by a number (real or complex)
  •   c * [a1] = [c * a1]
          [a2]   [c * a2]
      

💡 Real-life example: Suppose you mix red and blue paints in a certain ratio (scalar multiplication) and then combine it with yellow paint (vector addition). The resulting color depends on both operations, just like combining quantum states changes the system.

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4. Inner Product (Dot Product)

The inner product measures the overlap between two vectors, which in quantum computing corresponds to probability amplitude.

⟨φ|ψ⟩ = φ₁*ψ₁ + φ₂*ψ₂ + ... + φn*ψn
  • φ₁*, φ₂* … = complex conjugate of φ vector components
  • ψ₁, ψ₂ … = components of ψ vector

Example:

|ψ⟩ = [1 + i]
       [2]

|φ⟩ = [3]
       [i]

⟨φ|ψ⟩ = 3*(1 - i) + i*2
        = 3 - 3i + 2i
        = 3 - i

💡 Real-life analogy: Think of two arrows on a map. The inner product tells you how much one arrow points in the same direction as the other. In quantum computing, it tells the probability that |ψ⟩ will be measured as |φ⟩.

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5. Norm (Length) of a Vector

The norm (or magnitude) of a vector |ψ⟩ is:

||ψ|| = sqrt(⟨ψ|ψ⟩)

For a qubit |ψ⟩ = α|0⟩ + β|1⟩:

||ψ|| = sqrt(|α|² + |β|²)

Quantum states are always normalized, meaning:

||ψ|| = 1

💡 Real-life example: Imagine adjusting the volume of two speakers in a room so that the total sound is "balanced" at a fixed loudness. Normalizing ensures the quantum state has total probability 1.

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6. Orthogonality

Two vectors |ψ⟩ and |φ⟩ are orthogonal if:

⟨φ|ψ⟩ = 0

💡 Real-life analogy: Think of two musical notes that are completely different; they produce no overlap when played together. In quantum computing, orthogonal states are independent and can represent distinct classical outcomes.

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7. Summary of Key Terms

  • Vector: Represents a quantum state, has direction and magnitude.
  • Ket |ψ⟩: Column vector representing a state.
  • Bra ⟨ψ|: Row vector, complex conjugate of ket.
  • Inner product ⟨φ|ψ⟩: Measures overlap; gives probability amplitude.
  • Norm ||ψ||: Length of vector; quantum states are normalized to 1.
  • Orthogonal: Two vectors with zero inner product, representing independent states.

🔹 This is the foundation for understanding qubits, quantum gates, superposition, and measurement probabilities in quantum computing. Every operation you perform on qubits depends on vectors and their inner products.

Reference Book: N/A

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