as we found before.
1.4 Unitary Operations and Single-Qubit Gates
We refer to a transformation from one quantum state to another as a gate. The effect of a single qubit gate is to change α and β into a new mixture α′ and β′:
(1.12)
This can be written as a matrix equation
(1.13)
(1.14)
Since the length of the state vector must always be unity, we are only allowed to use matrices U that conserve the length of the vector. In other words, ⟨ψ′|ψ′⟩ = ⟨ψ|ψ⟩ = 1. This puts a very important constraint on the matrix U:
(1.15)
using the following observation:
(1.16)
Since ⟨ψ|ψ⟩ = 1, we conclude that
(1.17)
where I is the identity matrix
(1.18)
Matrices that satisfy this requirement are called unitary matrices. We can view these matrices as performing an operation on a qubit by changing the mixture of basis states. Consequently, the matrices U are also referred to as unitary operators.
The identity matrix I can be considered to be the simplest “gate” and leaves the state vector unchanged. Classically, the NOT gate is the only non-trivial single-bit gate. In contrast, there are many non-trivial single qubit quantum gates (technically, the number of 2×2 unitary matrices is unlimited). The most common non-trivial single qubit gates are the Pauli-X (X), Pauli-Y (Y), Pauli-Z (Z), and Hadamard (H) gates defined as follows:
(1.19)
(1.20)
(1.21)
(1.22)
To get an understanding of what these gates do, consider applying an X gate to the “ground” state |0⟩:
(1.23)
Similarly,
(1.24)We see then that the X gate is a “bit flip” gate, and transforms |0⟩ into |1⟩ and vice versa. This, then is the analog of the classical NOT gate. You should verify the following results from applying Y,Z, and H gates:
(1.25)(1.26)
(1.27)
In addition, it is interesting to note that each one of these matrices is its own Hermitian conjugate. Consequently, these four gates have the property that applying them twice gives the identity matrix: