CNOT Gate

 Controlled NOT gate: 

⏐x, y⟩ → ⏐x, x ⨂ y⟩

Matrix:  

The CNOT maps 

⏐0 0⟩ → ⏐0 0⟩

⏐0 1⟩ → ⏐0 1⟩ 

⏐1 0⟩ → ⏐1 1⟩

⏐1 1⟩ → ⏐1 0⟩

Proof:

⏐00⟩ = [1 0 0 0], ⏐01⟩ = [0 1 0 0], ⏐10⟩ = [0 0 1 0], ⏐11⟩ = [0 0 0 1], all vectors transposed for easy typing. 

Here ⏐0⟩ == [1 0] and ⏐1⟩ == [0 1] and ⏐ab⟩ = ⏐a⟩ ⨂ ⏐b⟩

Apply matrix multiplications: 

CNOT × ⏐00⟩ → ⏐00⟩

CNOT × ⏐01⟩ → ⏐01⟩

CNOT × ⏐10⟩ → ⏐11⟩

CNOT × ⏐11⟩ → ⏐10⟩

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In math, left Kronecker product ⨂ is

For example: 

Qubit Operation (2)

CNOT gate: to flip iff  (if and only if) the control quit is |1>, otherwise it does nothing. 

Entanglement:

⏐+⟩ == [1/sqrt (2) * (|0> + |1>) == 0.707 |0> + 0.707 |1>

⏐-⟩ == [1/sqrt (2)] * (|0> - |1>)

X (a, b) = (b, a), NOT gate

|a, b> == |b, a>, see below qubit swap:

[a |0> + b |1>] |0> == a|00> + b|10>

|𝝍> == a |0> + b |1>

|𝝍>|𝝍> == a^2|00> + ab|01> + ab|10> + b^2|11>

Bell state (entanglement):

Linear Algebra - Qubit Operations

Vectors are commonly written in column format. Sometimes, we also use shorthand format such as (3, 4).

In quantum computing, the state |0> corresponds to vector (1, 0), and |1> corresponds to vector (0, 1). 

Commonly used quantum gates, quantum circuit symbols, and math representations: 

The Bloch sphere representation of X (NOT), H (Hadamard), and Z, S, T (phase) gates. 

We can see that 

• X gate rotates along X axis 180 degree (or less depending on the initial angle to Z axis). (NOT) 
• H gate rotates along Y  axis 90 degree (or less). 
• Z, S, T gates rotate along Z axis at certain degree. (phase)

PGP Encryption

Developed by Phil Zimmerman in 1991. It combines symmetric, asymmetric (public key) encryption, hash, and digital signature all together providing confidentiality, integrity, and authentication. Good for email. The algorithm:

At the sender (Alice) side: 

1. Alice: Message M is hashed
2. Sign (encrypt) the hash value with her private key (EP)
3. Compress (zip) the message + signed hash value
4. Use a session key Ks to encrypt zipped output (ES)
5. Use Bob's public key to encrypt session key (EP)
6. Send encrypted message and encrypted session key 

At the recipient (Bob) side:

1. Use Bob's private key to decrypt session key (DP)
2. Use session key Ks to decrypt the zipped message
3. unzip, got message and signed hash value
4. Use Alice's public key to decrypt the hash value 
5. Computing hash value for the message received
6. Compare with hash value Alice computed.

Below are the PGP diagram:

Mathematically:

Note:

Image copyright: Author of this post. Free to use but reference is required. 

Quantum Circuit

 A quantum circuit is a computational routine consisting of coherent quantum operations on quantum data, such as qubits, and concurrent real-time classical computation. It is an ordered sequence of quantum gates, measurements, and resets, which may be conditioned on and use data from the real-time classical computation. A set of quantum gates is said to be universal if any unitary transformation of the quantum data can be efficiently approximated arbitrarily well as a sequence of gates in the set. Any quantum program can be represented by a sequence of quantum circuits and non-concurrent classical computation.

A quantum gate is a reversible (unitary) operation applied to one or more qubits.

Electronic computer: program --> instructions (operand and data) - binary bits
Quantum computer: program --> quantum circuits (quantum gate and quantum data) - qubits