What Are the Unitaries in the Clifford Group?

[haaj86]

I am reading some book on quantum computing, and it mentions the Clifford group. I understand the Pauli Group and the idea that Clifford group consists of unitaries that map the elements in the Pauli group back to the Pauli group, but what are these unitaries exactly? Can you list some of them. Also it says that for a single qubit there are 24 unitaries? that just seems too much. And finally, the unitaries are generated by the Hadamard gate, phase gate and the CNOT gate, can somebody show me how this is possible.

I think if you can simply list the exact form of the unitaries I will be able to answer my own questions, but if you can't bother to do so then please tell me where to find them.

 

[azntoon]

The Clifford group comprises all the unitaries that map the elements in the Pauli group back to the Pauli group. This means that the Clifford group should contain 24 elements for a single qubit, since the Pauli group contains 4 elements. The 24 element unitaries of the Clifford group are generated by the Hadamard gate, phase gate and CNOT gate, and the exact form of these unitaries is as follows: Hadamard gate: H = 1/√2 ( |0⟩⟨0| + |1⟩⟨1| ) + 1/√2 ( |0⟩⟨1| + |1⟩⟨0| ) Phase gate: P = |0⟩⟨0| + e^(iπ/2) |1⟩⟨1| CNOT gate: CNOT = |00⟩⟨00| + |01⟩⟨01| + |10⟩⟨11| + |11⟩⟨10| These three gates can then be combined with each other to generate the other 21 unitaries of the Clifford group, such as the controlled-Z gate (CZ), the controlled-S gate (CS) and the controlled-T gate (CT). There are also additional gates, such as the Toffoli gate, which are not part of the Clifford group but can be used to generate additional unitaries.

 

[Entropia]

The Clifford group is a mathematical group that plays a crucial role in quantum computing. It consists of a set of unitary operations that map elements of the Pauli group back to the Pauli group. In other words, these unitaries preserve the properties of the Pauli group.

Some examples of unitaries in the Clifford group include the Hadamard gate, phase gate, CNOT gate, and the SWAP gate. These unitaries are also known as single-qubit Clifford gates and are used to manipulate a single qubit in a quantum system.

It is true that for a single qubit, there are 24 unitaries in the Clifford group. This may seem like a large number, but it is necessary for the full set of operations to be able to map all elements of the Pauli group back to itself.

As for how these unitaries are generated by the Hadamard gate, phase gate, and CNOT gate, it involves a mathematical concept known as the Gottesman-Knill theorem. This theorem states that any operation in the Clifford group can be decomposed into a combination of these three gates. However, the exact form of the unitaries may vary depending on the specific implementation and notation used.

If you are interested in learning more about the Clifford group and its unitaries, I would recommend referring to academic resources such as research papers or textbooks on quantum computing. These sources will provide a more comprehensive and detailed explanation of the Clifford group and its properties.