How to make Toffoli gate using matrix form in multi qubits system?
Posted: Tue Aug 15, 2023 4:14 am
The Toffoli gate is a three-qubit gate that acts as a controlled-controlled-NOT gate (CCX gate). It flips the target qubit if both control qubits are in the |1⟩ state. The matrix representation of the Toffoli gate can be derived by considering its action on the computational basis states.
In a three-qubit system, the matrix representation of the Toffoli gate is a 8x8 matrix. The order of qubits in the matrix representation usually follows the convention of qubit indices increasing from right to left.
The Toffoli gate is represented by the following matrix:
In this matrix, the rows and columns correspond to the computational basis states: |000⟩, |001⟩, ..., |111⟩. The Toffoli gate applies the NOT (X) operation to the target qubit (the last qubit) if both control qubits (the first two qubits) are in the |1⟩ state.
Implementing the Toffoli gate on real quantum hardware or in a quantum simulator, the actual gate decomposition may involve a sequence of single-qubit and two-qubit gates due to hardware limitations and gate decompositions. The matrix representation is a convenient way to understand the action of the gate on different basis states, but the actual implementation might differ based on the available gates and qubit interactions.
In a three-qubit system, the matrix representation of the Toffoli gate is a 8x8 matrix. The order of qubits in the matrix representation usually follows the convention of qubit indices increasing from right to left.
The Toffoli gate is represented by the following matrix:
Code: Select all
| 1 0 0 0 0 0 0 0 |
| 0 1 0 0 0 0 0 0 |
| 0 0 1 0 0 0 0 0 |
| 0 0 0 1 0 0 0 0 |
| 0 0 0 0 1 0 0 0 |
| 0 0 0 0 0 1 0 0 |
| 0 0 0 0 0 0 0 1 |
| 0 0 0 0 0 0 1 0 |
Implementing the Toffoli gate on real quantum hardware or in a quantum simulator, the actual gate decomposition may involve a sequence of single-qubit and two-qubit gates due to hardware limitations and gate decompositions. The matrix representation is a convenient way to understand the action of the gate on different basis states, but the actual implementation might differ based on the available gates and qubit interactions.