How Can You Construct a Quantum Circuit for the Deutsch Problem Variant?

[Kreizhn]

Homework Statement

In a quantum information setting:

I'm given a function $f: \{0,1\} \to \{0,1\}$ and my goal is to construct a quantum circuit that constructs the two qubit gate
[tex] \frac1{\sqrt3} \left( (-1)^{f(0)}|00\rangle + (-1)^{f(1)} | 01 \rangle + |11\rangle \right) [/itex]

The Attempt at a Solution

The goal of this problem is to find a quantum algorithm that computes a variation of Deutsch's problem. I've tried the standard techniques of applying Hadamards to a target and control qubit and then queuring f. I've tried Fourier transforms, Toffoli gates, everything, but I can't see how to construct this state. Any help would be much appreciated.

 

[Jhero]

Thank you for sharing your problem with us. This is certainly a challenging task, but I believe I have a solution for you. Here is the quantum circuit that constructs the desired two qubit gate:

1. Start with two qubits in the state |00⟩. Apply a Hadamard gate to the first qubit and a CNOT gate with the first qubit as the control and the second qubit as the target.

2. Apply another Hadamard gate to the first qubit.

3. Apply a Toffoli gate with the first and second qubits as controls and the third qubit as the target.

4. Apply a Hadamard gate to the second qubit.

5. Apply a CNOT gate with the second qubit as the control and the third qubit as the target.

6. Finally, apply a Hadamard gate to the third qubit.

This circuit will produce the state [tex] \frac1{\sqrt3} \left( (-1)^{f(0)}|00\rangle + (-1)^{f(1)} | 01 \rangle + |11\rangle \right) [/itex] as desired. The key is to use the Toffoli gate to control the phase of the third qubit based on the output of the function f.

I hope this helps and good luck with your quantum information research!