Introduction to Quantum Computing

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Hilbert Space

Hilbert space is a mathematical concept used to describe the state of a quantum mechanical systems. It is a vector space equipped with an inner product allowing for the definition of the concepts like length distance and orthogonality which are crucial for understanding the quantum phenomena.

Bloch Sphere

The Bloch sphere is a useful way to represent the state of a single qubit. A qubit can be visualized as a vector starting at the center of the sphere and pointing to some location on its surface. Each axis of the sphere has a specific meaning:

1. The x-axis corresponds to the real part of the qubit’s amplitude.
2. The y-axis corresponds to the imaginary part of the amplitude.
3. The z-axis represents the probability of finding the qubit in state ∣0⟩ or ∣1⟩ when measured.

Dirac's Notations (Bra–Ket Notation)

Dirac’s notation, also known as bra–ket notation, is a powerful and compact mathematical language used in quantum mechanics and quantum computation.

Ket ( |ψ⟩ ): Represents a column vector describing the state of a quantum system. Bra ( ⟨ψ| ): The Hermitian conjugate (complex conjugate transpose) of a ket, represented as a row vector.

Properties

• Inner Product ( ⟨ϕ|ψ⟩ ): A complex number that gives the probability amplitude for the system in state |ψ⟩ to be found in state |ϕ⟩.
• Outer Product ( |ψ⟩⟨ϕ| ): Produces a matrix (operator) that acts on quantum states.
• Normalization: A quantum state must satisfy ⟨ψ|ψ⟩ = 1, meaning the total probability is 1.
• Orthogonality: Two states |ψ⟩ and |ϕ⟩ are orthogonal if ⟨ϕ|ψ⟩ = 0, meaning they correspond to mutually exclusive outcomes.
• Completeness: A set of basis states spans the entire Hilbert space:
  ∑_i |i⟩⟨i| = I
  This means any quantum state can be expressed as a linear combination of basis states.