The Fredkin gate is a quantum gate that operates on three qubits: one control qubit and two target qubits. When the control qubit is in the \(|1\rangle\) state, the states of the target qubits are swapped (a SWAP operation is performed). Mathematically, it is represented by an 8×8 matrix.

This gate swaps the second and third qubits’ states when the control qubit is in the \(|1\rangle\) state. Let’s express this simply as follows:

\[ |c, x, y\rangle \quad \text{where} \quad c \text{ is the control qubit, and } x \text{ and } y \text{ are the target qubits.} \]

When the Fredkin gate is applied to this state, if \( c = 1 \), the target qubits \( x \) and \( y \) are swapped.

Consider the following input state:

\[ |\psi\rangle = \alpha|000\rangle + \beta|001\rangle + \gamma|010\rangle + \delta|011\rangle + \epsilon|100\rangle + \zeta|101\rangle + \eta|110\rangle + \theta|111\rangle \]

After applying the Fredkin gate:

• If the control qubit is in the \(|0\rangle\) state, the target qubits remain unchanged.
• If the control qubit is in the \(|1\rangle\) state, the target qubits are swapped. Specifically, qubits 2 and 3 are swapped.

The resulting state is:

\[ |\psi’\rangle = \alpha|000\rangle + \beta|001\rangle + \gamma|010\rangle + \delta|011\rangle + \epsilon|100\rangle + \zeta|110\rangle + \eta|101\rangle + \theta|111\rangle \]

The Fredkin gate performs a controlled SWAP operation, meaning that one qubit has the ability to swap the states of the other two qubits. On the Bloch sphere, when the control qubit is \(|1\rangle\), the target qubits are rotated around the X and Y axes.

The Fredkin gate can be synthesized by combining other fundamental gates (such as CNOT, Toffoli, etc.). In quantum hardware systems such as superconducting circuits or photonic systems, the Fredkin gate is implemented by using laser pulses or microwave signals to swap the target qubits based on the state of the control qubit.

The Fredkin gate is used in quantum circuits to perform controlled swap operations. It plays a significant role in reversible computing and quantum circuit design. Additionally, it is used in quantum simulations and algorithms for entanglement and data transfer.

9.1 Important Scenarios for the Fredkin Gate

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

1. Reversible Computing: The Fredkin gate ensures that computations are reversible, preventing information loss. This is important for energy-efficient computation and for performing calculations beyond thermodynamic limits.

2. Quantum Simulations: In quantum system simulations, the Fredkin gate is used when it is necessary to change the positions of specific qubits.

3. Error Correction: The Fredkin gate is used in quantum error correction protocols when specific qubits need to be modified or when entanglement must be created.

The theoretical implications of this gate can be expressed as follows:

• The Fredkin gate plays a critical role in reversible computing and allows quantum computers to perform more efficiently than classical computers. This gate is used to reduce energy consumption and enable quantum systems to perform complex computations. It is also important for the optimization of quantum circuits and error correction.