Quantum computers
Logic gates modify the states of bits. They are devices that classically give binary output values from binary input values. 16 They are deterministic, each type giving the same output for a given input. Linear algebra can represent bit operations. 17 Take the vector ( 1 0 ) to represent the bit value of 0; let ( 0 1 ) refer to 1. It may help to remember the y-component of the vectors correspond to their bit values. Classical bits may only be ( 1 0 ) or ( 0 1 ) , but qubits can occupy any point along the unit circle below. All real-number qubit operations can be conceptualized as moving between these points. The Bloch sphere is the 3- dimensional extension of the unit circle to include the imaginary values that qubits can take.
The single-bit operations are Identity, Negation (NOT), Constant-0 and Constant-1. These can be represented as a 2x2matrix. Identity doesn’t change a bit’s value; Negation swaps (flips) the components; Constant-0 sets the value to 0 and Constant-1 sets the value to 1. The Negation (NOT) matrix is applied to 0, below.
A vector pointing to ( 1 0 ) .
0 1
0 × 0 + 1 × 1 1 × 0 + 0 × 1
1 0
0 1 1 0
(
) (
) = (
) = (
)
The product state fully describes the states of two or more bits. This is achieved using the tensor product. Entangled particles can’t be separated from their product state into two vectors.
𝑦 0 𝑦 1
𝑥 0 𝑦 0 𝑥 0 𝑦 1 𝑥 1 𝑦 0 𝑥 1𝑦 1
𝑥 0 (
)
𝑥 0 𝑥 1
𝑦 0 𝑦 1
(
) ⊗ (
) = (
) = (
)
𝑦 0 𝑦 1
𝑥 1 (
)
0 0 1 0
0 1
1 0
(
) ⊗ (
) = (
)
The CNOT gate depends on two binary inputs so the corresponding matrix is 4x4 and applies to the product state of the two bits. It negates the ‘ target ’ bit if the ‘ control ’ bit is 1.
CNOT Input
𝟎 0
𝟎 1
𝟏 0
𝟏 1
𝟎 0
𝟎 1
𝟏 1
𝟏 0
Output
- Control
- Target
1 0 0 1 0 0 0 0
0 0 0 0 0 1 1 0
1 0 0 1 0 0 0 0
0 0 0 0 0 1 1 0
0 0 1 0
0 0 0 1
0 1
1 0
0 1
0 1
𝐶(1 𝑎𝑛𝑑 0) = (
) ((
) ⊗ (
)) = (
) (
) = (
) = (
) ⊗ (
) = 1 𝑎𝑛𝑑 1
CNOT’s effect on qubits in superposition is more complicated than a bit -flip but still follows the above matrix. The execution of such complicated operations with a single gate allows bits in superposition to be manipulated in many more ways.
16 Kumar (Undated). 17 See Helwer (2018) for all the following mathematics and discussion of gates.
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