The Controlled-Z (CZ) gate is a quantum gate that operates on two qubits: one control qubit and one target qubit. When the control qubit is in the \(|1\rangle\) state, the Z gate (phase shift) is applied to the target qubit.

This gate applies a -1 phase shift to the \(|11\rangle\) state while leaving the other states unchanged. Let’s describe this process mathematically:

Consider an input state \(|\psi\rangle\) for a series of qubits, and let’s apply the CZ gate to these qubits.

\[ |\psi\rangle = \alpha|00\rangle + \beta|01\rangle + \gamma|10\rangle + \delta|11\rangle \]

When the control qubit is in the \(|1\rangle\) state and the target qubit is also in the \(|1\rangle\) state, a phase shift of \(-1\) is applied to \(|11\rangle\), resulting in:

\[ CZ |\psi\rangle = \alpha|00\rangle + \beta|01\rangle + \gamma|10\rangle – \delta|11\rangle \]

The CZ gate applies a phase shift to the target qubit when the control qubit is in the \(|1\rangle\) state. On the Bloch sphere, this results in a 180° rotation of the target qubit around the Z axis. The CZ gate is used to create entanglement and precisely control phase shifts.

Physically, the application of the CZ gate is typically performed in superconducting quantum circuits using microwave signals. In ion traps, laser pulses are used to apply the phase shift effect of the control qubit on the target qubit. This process requires precise timing and frequency control to achieve accurate phase shifts.

The CZ gate is used in quantum algorithms to implement phase shifts and to facilitate interactions between qubits. It is particularly useful for managing interference effects and creating entanglement.

11.1 Important Scenarios for the CZ Gate

There are three important scenarios where the CZ gate is essential:

1. Quantum Fourier Transform: The CZ gate is used in the Fourier transform to perform phase shift operations. This allows specific phases to be applied to specific qubits during the transformation process.

2. Entanglement Creation: The CZ gate is used to entangle qubits and to use this entanglement to control and manage interference effects. Entangled states, such as Bell states, can be created using combinations of the CZ and Hadamard gates.

3. Phase Shift Algorithms: The CZ gate is used to create specific phase relationships between qubits, and these phase relationships are used for measuring the results in quantum algorithms.

The CZ gate is a crucial component in quantum algorithms for phase control and entanglement. The creation of entangled states allows the use of quantum parallelism, and interference effects influence the measurement outcomes. The CZ gate is essential for improving the efficiency of quantum circuits and ensuring the accuracy of quantum algorithm results.