7.7 Gates and unitary matrices

The collection of all 2 by 2 unitary matrices (subsection 5.7.5 C form a group under multiplication called the unitary group of degree 2. It is denoted 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.

Any U in U(2, C) can be written as a product of a complex unit times a linear combination of unitary matrices

U = e^{θ i} (c_I2 I₂ + c_σx σ_x + c_σy σ_y + c_σz σ_z)

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

• I₂ is the 2 by 2 identity matrix
• c_σx c_σy, and c_σz are Pauli matrices
• 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_σ...