QC — Quantum Mechanics Notations

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Photo by G. Crescoli

This is the appendix and the notation for the Quantum Computing series.

Vectors

Vector expressed as a ket |v⟩

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Vector expressed as a bra〈v|

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In component form:

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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.)

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The Dirac notation |0⟩ is just a shorthand of:

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Inner product

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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

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Cross product

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Tensor product

The state space of a composite physical system is the tensor product of the state spaces.

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The definition of the tensor product is:

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For example,

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Here is another example of a system composed of two states.

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Manipulation

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Notation

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Modified from source 1 & source 2

Quantum superpositions (Qubits)

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Example,

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For 3-qubits:

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The vector form for a 3-qubits:

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Properties

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If the vectors are orthogonal (like the bases),

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Credits and reference

An Introduction to Quantum Algorithms

The Role of Interference and Entanglement in Quantum Computing

Written by

Deep Learning

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