5.2.2 The Hadamard gate

The Hadamard gate is a single-qubit gate defined by a matrix

In a circuit diagram representing a quantum algorithm, the symbol for the Hadamard gate is shown in Figure 8.

Figure 8 The symbol used in a quantum circuit for a Hadamard gate,

Show description|Hide description

The figure shows a horizontal line on the left leading to a square box with a capital letter H in it, then a horizontal line leading from the box to the right.

Figure 8 The symbol used in a quantum circuit for a Hadamard gate,

Consider the action of a Hadamard gate on logical state .

First, the gate and logical state are written as matrices,

Next, the matrices are multiplied to obtain the final state matrix

Finally, the final state matrix is rewritten in terms of the logical states and :

You can see that the final output state, is a superposition state. This calculation shows that a Hadamard gate allows the transformation of the logical qubit state into a superposition state.

The effect on a general state can be determined by combining the results of the action of a Hadamard gate on logical states and .

All these results are summarised in the truth table for the Hadamard gate in Table 4.