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      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:

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