Quantum Computing - Hermitian & Unitary Matrices

Objectives: Quantum Computing - Hermitian & Unitary Matrices

Quantum Computing - Hermitian & Unitary Matrices

Hermitian & Unitary Matrices in Quantum Computing

1. Introduction

In quantum computing, matrices represent operations (quantum gates) on qubits. Two key types of matrices are:

  • Hermitian matrices: Represent observable quantities (things you can measure, like spin or energy).
  • Unitary matrices: Represent quantum gates that evolve qubit states without losing probability.
💡 Real-life analogy: Hermitian = ruler for measuring properties, Unitary = rotation device that changes direction without changing size. ---

2. Hermitian Matrices

A matrix H is Hermitian if it equals its own conjugate transpose:

H† = H
  • H† (H dagger) = conjugate transpose of H
  • Conjugate transpose = transpose + complex conjugate

Example:

H = [2  i]
    [-i 3]

H† = [2  -i]
      [i  3]

H† = H → Hermitian

Properties of Hermitian Matrices

  • All eigenvalues are real → corresponds to measurable quantities.
  • Eigenvectors corresponding to different eigenvalues are orthogonal.
  • Used to represent observables (like spin, energy).
💡 Real-life analogy: Imagine measuring the weight of objects. The weight (observable) is always real. Hermitian matrices ensure the measured result is always a real number.

Applications in Quantum Computing

  • Represent measurements: e.g., Pauli matrices X, Y, Z are Hermitian.
  • Used in Hamiltonians to describe energy and system evolution.
  • Eigenvalues = possible measurement outcomes; eigenvectors = states where measurement is certain.
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3. Unitary Matrices

A matrix U is unitary if:

U† U = U U† = I
  • I = identity matrix
  • U† = conjugate transpose
💡 Real-life analogy: A unitary matrix is like spinning a perfectly rigid object in space. It rotates, but size (probability) is preserved.

Example:

Hadamard gate H = 1/√2 [1  1]
                        [1 -1]

H† = H (because real and symmetric)
H† H = I → Unitary

Properties of Unitary Matrices

  • Preserve vector norms: ||U|ψ⟩|| = |||ψ⟩|| → probability preserved
  • Reversible: U† = U⁻¹ → all quantum operations are reversible
  • Eigenvalues have magnitude 1 → e^(iθ)

Applications in Quantum Computing

  • All quantum gates are unitary → ensures total probability = 1
  • Used in constructing quantum algorithms (Hadamard, CNOT, Pauli gates)
  • Ensures reversibility of quantum operations (necessary for coherent evolution)
💡 Real-life analogy: Imagine rotating a rigid object: the shape and size do not change (norm preserved), just like unitary operations preserve probability. ---

4. Common Hermitian and Unitary Matrices

Pauli Matrices (Hermitian & Unitary)

X = [0 1]
    [1 0]

Y = [0 -i]
    [i  0]

Z = [1  0]
    [0 -1]
  • Used as basic quantum gates and observables
  • Eigenvalues = ±1 → possible measurement outcomes

Hadamard Gate (Unitary)

H = 1/√2 [1  1]
          [1 -1]
  • Creates superposition: |0⟩ → (|0⟩ + |1⟩)/√2
  • Unitary: H† H = I

Phase Gate (Unitary)

S = [1 0]
    [0 i]

T = [1 0]
    [0 e^(iπ/4)]
  • Imparts phase shift without changing magnitude
  • Used in interference and quantum algorithms
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5. Eigenvalues and Eigenvectors

Hermitian matrices:

  • Eigenvalues = possible measurement outcomes (real)
  • Eigenvectors = states where measurement outcome is certain

Unitary matrices:

  • Eigenvalues = e^(iθ), magnitude 1
  • Eigenvectors = basis states that rotate but norm preserved
💡 Real-life analogy: Eigenvectors = special directions of a spinning top; along these directions, the behavior is predictable. ---

6. Summary of Key Terms

  • Hermitian matrix: H† = H, represents measurable observables, eigenvalues real
  • Unitary matrix: U† U = I, represents quantum gates, preserves probability, reversible
  • Eigenvalue: Result of measurement or rotation along eigenvector
  • Eigenvector: Quantum state associated with eigenvalue
  • Pauli matrices: X, Y, Z → basic gates/observables
  • Hadamard gate: Creates superposition, unitary
  • Phase gate: Imparts phase, unitary
  • Reversibility: All quantum gates are unitary → no information lost

Hermitian matrices define what you can measure; Unitary matrices define how your qubits evolve. Understanding both is essential for designing and analyzing quantum circuits.

Reference Book: N/A

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