How can the controlled swap gate be used to compute the AND gate in a reversible manner?

/ / Published in Quantum Information, EITC/QI/QIF Quantum Information Fundamentals, Introduction to Quantum Computation, Reversible computation, Examination review

The controlled swap gate, also known as the Fredkin gate, is a fundamental gate in reversible computation that can be used to compute the AND gate in a reversible manner. Reversible computation is a computational paradigm where every operation is reversible, meaning that the input can be uniquely reconstructed from the output. This is in contrast to classical computation, where irreversible operations are common.

To understand how the controlled swap gate can be used to compute the AND gate reversibly, let's first examine the behavior of the controlled swap gate. The controlled swap gate takes three qubits as input: two control qubits and one target qubit. If the first control qubit is in the state |1⟩, it swaps the states of the second control qubit and the target qubit. Otherwise, it leaves the states unchanged.

The truth table for the controlled swap gate is as follows:

|Control 1|Control 2|Target |Output |
|———|———|———|———|
|0 |0 |0 |0 |
|0 |0 |1 |1 |
|0 |1 |0 |0 |
|0 |1 |1 |1 |
|1 |0 |0 |0 |
|1 |0 |1 |1 |
|1 |1 |0 |1 |
|1 |1 |1 |0 |

Now, let's consider how we can use the controlled swap gate to compute the AND gate reversibly. The AND gate takes two input bits and outputs 1 if both input bits are 1, and 0 otherwise. In reversible computation, we need to ensure that the input bits can be uniquely reconstructed from the output bits.

To compute the AND gate using the controlled swap gate, we can set the first control qubit of the controlled swap gate to the logical AND of the two input bits, and the second control qubit and the target qubit to the input bits themselves. The output of the controlled swap gate will then be the result of the AND gate, and the input bits can be uniquely reconstructed from the output bits.

Here is an example circuit that demonstrates how the controlled swap gate can be used to compute the AND gate:

     ┌───┐
q_0: ┤ H ├───────■───────
     ├───┤       │
q_1: ┤ H ├───────┼───────
     ├───┤       │       
q_2: ┤ X ├──■────┼───────
     ├───┤  │    │       
q_3: ┤ X ├──┼────┼───────
     └───┘┌─┴─┐┌─┴─┐┌───┐
q_4: ─────┤ X ├┤ X ├┤ X ├
          └───┘└───┘└───┘

In this circuit, q_0 and q_1 are the input bits, and q_4 is the output bit. The H gates are Hadamard gates, which put the qubits into a superposition of states. The X gates are Pauli-X gates, which flip the state of a qubit. The controlled swap gate is represented by the boxes labeled "X" in the circuit.

By applying this circuit to the input bits, we can compute the AND gate reversibly, with the output bit q_4 containing the result of the AND operation. The input bits q_0 and q_1 can be uniquely reconstructed from the output bit q_4, making this computation reversible.

The controlled swap gate can be used to compute the AND gate in a reversible manner by setting the first control qubit to the logical AND of the input bits and using the second control qubit and the target qubit to represent the input bits themselves. The output of the controlled swap gate will then be the result of the AND gate, and the input bits can be uniquely reconstructed from the output bits.

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