Linear Algebra for Quantum Computing

Tensor Products (Multi-Qubit Systems)

In quantum computing, a single qubit is represented by a 2-dimensional vector. To represent multiple qubits together, we use the tensor product. This allows us to combine qubits into a single system where all quantum states are considered simultaneously.

1. Definition of Tensor Product

If |ψ⟩ is a vector of dimension m and |φ⟩ is a vector of dimension n, their tensor product |ψ⟩ ⊗ |φ⟩ is a vector of dimension m × n:

|ψ⟩ = [ψ1]
       [ψ2]

|φ⟩ = [φ1]
       [φ2]

|ψ⟩ ⊗ |φ⟩ = [ψ1*φ1]
             [ψ1*φ2]
             [ψ2*φ1]
             [ψ2*φ2]

• ⊗ = tensor product symbol
• ψi, φi = components of vectors |ψ⟩ and |φ⟩

2. Example: Two Qubits

|0⟩ = [1]
       [0]

|1⟩ = [0]
       [1]

Two-qubit system: |0⟩ ⊗ |1⟩

|0⟩ ⊗ |1⟩ = [1*0]
             [1*1]
             [0*0]
             [0*1]
           = [0]
             [1]
             [0]
             [0]

This vector represents a 2-qubit system where the first qubit is |0⟩ and the second qubit is |1⟩.

3. Generalization to n Qubits

For n qubits, each with 2 states, the tensor product produces a vector of dimension 2ⁿ. For example:

Three qubits: |ψ1⟩ ⊗ |ψ2⟩ ⊗ |ψ3⟩ → 2^3 = 8-dimensional vector

💡 Real-life analogy: If you have 3 coins, each with 2 outcomes (heads or tails), the combined system has 2³ = 8 possible outcomes. Tensor product enumerates all possibilities in a single vector.

4. Properties of Tensor Products

• Associativity: |ψ⟩ ⊗ (|φ⟩ ⊗ |χ⟩) = (|ψ⟩ ⊗ |φ⟩) ⊗ |χ⟩
• Distributivity over addition: |ψ⟩ ⊗ (|φ⟩ + |χ⟩) = |ψ⟩ ⊗ |φ⟩ + |ψ⟩ ⊗ |χ⟩
• Scalar multiplication: c(|ψ⟩ ⊗ |φ⟩) = (c|ψ⟩) ⊗ |φ⟩ = |ψ⟩ ⊗ (c|φ⟩)

5. Tensor Products and Quantum Gates

Tensor products are used to apply quantum gates to multi-qubit systems:

If X is a gate acting on qubit 1:

X ⊗ I acts on a 2-qubit system:
|ψ⟩ = α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩

• I = identity gate (does nothing on second qubit)
• Tensor product ensures the gate acts only on the intended qubit while leaving others unchanged.

6. Entanglement and Tensor Products

Some multi-qubit states cannot be written as simple tensor products. These are called entangled states.

Example: Bell state
|Φ+⟩ = 1/√2 (|00⟩ + |11⟩)

• This state cannot be factored into |ψ⟩ ⊗ |φ⟩
• Measurement of one qubit instantly determines the other

7. Repeated Tensor Products

To represent n qubits all in |0⟩ state:

|0⟩ ⊗ |0⟩ ⊗ ... ⊗ |0⟩ = |00...0⟩ (dimension 2^n)

💡 Real-life analogy: A line of light switches all turned OFF. Tensor product allows describing all switches together in one vector.

8. Summary of Key Terms

• Tensor Product ⊗: Combines two vectors into a higher-dimensional vector representing multi-qubit systems.
• Multi-qubit system: A quantum system with two or more qubits combined via tensor product.
• Entanglement: Multi-qubit states that cannot be separated into individual tensor products.
• Dimension: Two qubits → 4-dim vector, Three qubits → 8-dim vector, n qubits → 2ⁿ-dim vector.
• Application: Used in multi-qubit gates, quantum circuits, and modeling complex quantum systems.

Tensor products are essential to describe all possible combinations of qubits, apply multi-qubit gates, and understand entanglement in quantum algorithms.