Quantum – Theory – Single, Multipart and Mega Qubits

This post covers the features of single and multipart qubits.

Single Qubit Features

• A qubit is described by the equation (a|jˆ>  + b|j^v>) and is derived from the generic formula for a quantum state: |ψ> = cos (𝜭/2) |jˆ>  + e^iϕ sin( 𝜭/2) |j^v>.
• The σx gate inverts a qubit. For example: it allows |j^{^}> to change to |j^v>, ^–|j^{^}> to change to ^–|j^v> and vice versa.
• The top row and bottom row of a qubit’s matrix can be multiples of either 0, ^{^}, ^v, i^{^} or i^v Opposite Values.
• A qubit of the equivalent of  5|j^{^}> + 3|j^v> qubits is normalised  in the equation: 
  • 5/√(5² + 3²)|j^{^}> + 3/√(5² + 3²)|j^v>
  • = 5/√(34)|j^{^}> + 3/√34)|j^v>

Flip Operation

• The flip  operation is the short for the σf gate, denoted by ^–, and changes the signs of both elements of the qubit. For example: 
  • ^–|j^{^}> = ^–=

• ^{– –}|j^{^}> = ^{– –}  =^–  =

• ^–|^{^}> = ^–=

• ^–|^v> = ^–=

• ^–|i^{^}> = ^–=

• ^–|i^v> = ^–=

• Flipped qubits cancel each other out. For example: |j^v> + ^–|j^v> = 0

Multipart Qubit Features

• Quantum computers use single qubits to form multipart qubits in multipart qubit circuits
• Quantum computers execute programs using multipart qubit circuits.
• Represent the multipart qubit |j^vj^{^}j^{^}> in a matrix as follows:
  • 
  • Note that the next post on matrices shows how the tensor product is used to set up multipart qubits

• |j^vj^{^}j^{^}> ≠ |j^{^}j^vj^{^}> ≠ |j^{^}j^{^}j^v>
• However, |^–j^vj^{^}j^{^}> = |j^v-j^{^}j^{^}> = |j^vj^{^-}j^{^}> = ^–|j^vj^{^}j^{^}>
  • A column of bits forms a multipart qubit. An odd number of flips, ^–, results in a flip sign preceding a multipart qubit. For example:
    • |^–j^vj^{^}j^{^}> = |j^v-j^{^}j^{^}> = |j^vj^{^-}j^{^}> = ^–|j^vj^{^}j^{^}>
    • =
    • Note that the post on next post on matrices shows the calculation of this multipart qubit.

• In the multipart qubits |00100> + ^–|01100> the second part is flipped. Determine the flip sign of a part of a multipart qubit by the elements of the qubit. An odd number of ^v Opposite Signs require a flip to proceed the part.
• Primary and dependent single qubits make up a multipart qubit

Superposition and Entanglement and Mega Qubits

• A qubit splitter  changes |j^{^}> or |j^v>  into superposition and forms a mega qubit. For example: A Hadamard gate splits |j^{^}> into the mega qubit |j^{^}> + |j^v> and splits |j^v> into the mega qubit |j^{^}> + ^–|j^v>.
• Only the primary qubits of a multipart qubit are put in superposition because the others will get there after the implementation of dependencies.
• The number of possible independent states is 2ⁿ where n is the number of primary qubits. These possible independent states form a mega qubit.
• Entangled qubits are orthogonal within each qubit and between states. For example: In the entangled multipart qubit: |ψ> = 1^{^}/√2(|j^{^}j^{^}> + ^–|j^vj^v>):
  • The first qubit Q0 state is |j^{^}> +  ^–|j^v> as follows:  
    • Q0 =   
    • The two elements within Q0 are orthogonal
  • The second qubit Q1 represents |j^{^}> + ^–|j^v> as follows:
    • Q1 =    
    • The two elements within Q1 are orthogonal
  • The states |j^{^}j^{^}> and ^–|j^vj^v> are orthogonal because
    • <j^{^}j^{^}|.^–|j^vj^v> = [1^{^} 0 0 0]  

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