QC — Quantum Mechanics Notations

Jonathan Hui
3 min readNov 17, 2018
Photo by G. Crescoli

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

Modified from source 1 & source 2

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

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