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# SWAP Gate

**SWAP** exchanges the states of two qubits. If qubit 1 is in state $|a\rangle$ and qubit 2 is in state $|b\rangle$, after SWAP they exchange.

Matrix:

$$\text{SWAP} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$

Action: $|ab\rangle \to |ba\rangle$ (swap basis state labels).

## Properties

- **Self-inverse**: $\text{SWAP}^2 = I$
- **Symmetric**: swaps two qubits
- **Reduces to identity** on identical qubits (trivial for two copies of same state)

## Decomposition

SWAP can be built from three CNOT gates:

$$\text{SWAP} = \text{CX}_{01} \text{CX}_{10} \text{CX}_{01}$$

This decomposition is commonly used since CNOT is more readily available.

## Uses

- **Qubit mapping**: adjust layout to match hardware connectivity; move qubits closer for two-qubit gates
- **Limited connectivity**: enable operations between distant qubits on sparse graphs (at cost of extra gates)
- **Circuit reordering**: logically swap qubits to reduce circuit depth
- **Quantum simulation**: SWAP networks simulate particle dynamics on lattice geometries

## Trade-offs

- **Expensive**: requires 3 CNOT gates (vs 1 for controlled operations)
- **Avoidance**: redesign algorithms to avoid SWAP via gate reordering
- **Necessary**: chips like 1D chains require SWAPs to implement arbitrary algorithms

## Implementation

- **Superconducting qubits**: three CNOT pulses; gate time ~60–300 ns; some platforms have dedicated SWAP
- **Trapped ions**: composed from CX gates
- **Cost note**: dedicated SWAP implementations faster than three CNOTs