Rewriting quantum gates as CNOT+rotations

[wizzart]

Homework Statement

The goal is to 'decompose' common 2 qubit quantum gates such as the pi/8 gate into a sequence of CNOTS and single qubit rotations. I have the book by Nielsen and Chuang and the info is sortof in there (universality proof of CNOT), but I don't get how to apply it, i.e. how to calculate in a systematic way what rotations to add to the CNOT.

I would be very grateful is someone could help me out here. I'm of course interested in the decomposition of a particular gate, but even more so in a consistent method of how to handle this kind of problem. Thanks in advance!

For the relevant matrices see for instance http://en.wikipedia.org/wiki/Quantum_gate"

In addition: I now that H N H (Hadamard NOT Hadamard) gives the S-gate (1 0 / 0 -1), which in turn is the square of the pi/8 gate, but I can't generalize this. My best guess would be to combine the Hadamards with additional pi-rotations, but I have no clue if this is correct and if so what axis to choose.

 

[Jhero]

it is important to have a systematic method for solving problems and finding solutions. In the case of decomposing 2 qubit quantum gates, there are a few steps that can be followed to help you in this process.

1. Start by understanding the gate you want to decompose. Look at its matrix representation and try to understand what it does to the quantum state. This will give you a better idea of what operations you need to perform in order to achieve the same result using CNOTs and single qubit rotations.

2. Use the universality proof of CNOT to your advantage. This proof states that any quantum gate can be decomposed into a sequence of CNOTs and single qubit rotations. This means that you can use CNOT as a building block to construct any other gate.

3. Look for patterns in the matrix representation of the gate. For example, if the gate has a lot of zeros in its matrix, it might be possible to use CNOTs to cancel out those zeros and simplify the decomposition process.

4. Use the concept of controlled operations. This means that you can perform a certain operation on a qubit only if another qubit is in a specific state. This can be used to your advantage when decomposing gates, as it allows you to combine multiple operations into a single CNOT.

5. Use the concept of phase kickback. This is a technique where a phase shift is applied to one qubit, but the effect is seen on another qubit due to the entanglement between the two qubits. This can be used to simplify the decomposition of gates.

6. Keep in mind that there may not be a unique solution to decomposing a gate. Different sequences of CNOTs and single qubit rotations may achieve the same result. It is important to find a decomposition that is efficient and easy to implement.

Overall, the key is to have a good understanding of the gate you want to decompose and to use the properties and concepts of quantum gates to your advantage. With practice and experience, you will develop your own systematic approach to solving these types of problems.