QC — Quantum Mechanics Notations
This is the appendix and the notation for the Quantum Computing series.
Vectors
Vector expressed as a ket |v⟩
Vector expressed as a bra〈v|
In component form:
Hilbert space is a vector space with inner product and norm defined and amplitudes belong to complex numbers. (Vector space is just an abstraction of what we learn about vectors in linear algebra.)
The Dirac notation |0⟩ is just a shorthand of:
Inner product
The result of the inner product is a scalar. The result is independent of which computational bases are used to encode |u⟩ and |v⟩.
Norm
Cross product
Tensor product
The state space of a composite physical system is the tensor product of the state spaces.
The definition of the tensor product is:
For example,
Here is another example of a system composed of two states.
Manipulation
Notation
Quantum superpositions (Qubits)
Example,
For 3-qubits:
The vector form for a 3-qubits:
Properties
If the vectors are orthogonal (like the bases),
Credits and reference
An Introduction to Quantum Algorithms
The Role of Interference and Entanglement in Quantum Computing
