8.3 Multi-qubit gates

A quantum gate operation on one qubit has a 2-by-2 unitary square matrix relative to some basis, as we saw in section 5.9. For two qubits, the matrix is 4-by-4. For ten, it is 2¹⁰-by-2¹⁰, which is 1,024-by-1,024. We now look at how to work with common lower-dimensional gates, allowing you to extrapolate to larger ones.

We can tensor any two 1-qubit gates to create a 2-qubit gate. For example,

Let’s examine what we get when we tensor together multiple 1-qubit Hadamard H gates.

8.3.1 The quantum H^⊗n gate

We start by looking at what applying a Hadamard H to each qubit in a 2-qubit system means. The H gate has the matrix gate$H`gate-style gate$H<sup>⊗n</sup>`gate-style gate$Hadamard H`gate-style gate$H<sup>⊗1</sup>`gate-style

operating on C². Starting with the two qubit states

applying H to each qubit means to compute

which is the same as...