Excited State and Pauli Operators

3.2.3 Excited State and Pauli Operators

As you may have guessed, a qubit does more than sit around just in the |0⟩ ground state. To put it into the |1⟩ excited state, we need a quantum gate -- so in this section we will introduce gates, and how to use them in the Composer.

Quantum gates, or operations, are typically represented as matrices. A gate that acts on one qubit is represented by a 2×2 unitary matrix. Since quantum operations need to be reversible and preserve probability amplitudes, the matrices must be unitary. The result of the quantum gate is found by multiplying the matrix representing the gate with the vector representing the quantum state.

      |ψ′⟩=U|ψ⟩   where U†U=1   (A† represents the complex conjugation and transpose of any matrix A ).

A common group of gates, known as the Pauli Operators, are represented by the matrices

The Pauli X gate is known as an Xπ -rotation and it takes |0⟩→X|0⟩=|1⟩ ; in other words, it flips the zero to a one, or vice versa (this is why it is also commonly referred to as a bit-flip). You can enter it into the Composer with the score already made for you below! Did you find that (unlike in the previous tutorial's example) now the qubit was in the excited state |1⟩ with high probability? Any deviation from the excited state is likely due to decoherence and imperfect measurements.

In the other examples below, explore what the Pauli Operators do. What do you get when you try a Y or Z gate? Did you find that Y gave you an excited state and Z did not do anything?

Experiment Log:

1.      Pauli X: |0> is zero probability and |1> is 100% probability

2.      Pauli Y: same as above, an excited state.  If use simulation, the downloaded CSV shows:

3.      Pauli Z: again using simulation, nothing changes, the downloaded CSV shows: