7.7 Gates and unitary matrices

The collection of all 2-by-2 unitary matrices (section 5.8) with entries in C form a group under multiplication called the unitary group of degree 2. We denote it by U(2, C). It is a subgroup of GL(2, C), the general linear group of degree 2 over C.

Every 1-qubit gate corresponds to such a unitary matrix. We can create all 2-by-2 unitary matrices from the identity and Pauli matrices. Pauli$matrix matrix$Pauli operator$Pauli

We can write any U(2, C) as a product of a complex unit times a linear combination of unitary matrices

with

where we have the following definitions, properties, and identities:

• 0 ≤ θ < 2π
• c_I2 is in R
• c_σx, c_σy, and c_σz are in C
• |c_I2|² + |c_σx|² + |c_σy|² + |c_σz|² = 1

and

The complex unit only affects the global phase of the qubit state and so...