DCX (double-CNOT) gate

Definition

In the $q[0]$-leftmost basis ordering used throughout this catalog, composing $\mathrm{CX}(q_0\!\to\!q_1)$ then $\mathrm{CX}(q_1\!\to\!q_0)$ gives

$$\mathrm{DCX} = \begin{pmatrix} 1&0&0&0\\0&0&1&0\\0&0&0&1\\0&1&0&0 \end{pmatrix}$$

in the ordered basis $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$ (frameworks using a little-endian statevector ordering will print an index-permuted version of this same matrix).

Action on basis states

$$\mathrm{DCX}|00\rangle = |00\rangle, \qquad \mathrm{DCX}|01\rangle = |11\rangle, \qquad \mathrm{DCX}|10\rangle = |01\rangle, \qquad \mathrm{DCX}|11\rangle = |10\rangle$$

$|00\rangle$ is a fixed point; the other three basis states form a single 3-cycle $|01\rangle\to|11\rangle\to|10\rangle\to|01\rangle$, verified by direct basis-state tracking through both CNOTs.

Decomposition

$$\mathrm{DCX} = \mathrm{CX}(q_1\!\to\!q_0)\cdot \mathrm{CX}(q_0\!\to\!q_1)$$

This is not really a "decomposition" in the sense of compiling a hardware-hard gate down to easier ones — two reversed CNOTs is the definition of DCX. The gate exists as a named primitive because this exact two-CNOT pattern recurs often enough in routing and partial-SWAP contexts to be worth naming directly.

Key identities

• $\mathrm{DCX}$ is Clifford, since it is a product of two Clifford ($\mathrm{CX}$) gates.
• $\mathrm{DCX}^3 = I\otimes I$ (period 3, from the 3-cycle structure), but $\mathrm{DCX}^2 \ne I\otimes I$ — unlike $\mathrm{CX}$, $\mathrm{CZ}$, or $\mathrm{SWAP}$, $\mathrm{DCX}$ is not self-inverse: $\mathrm{DCX}^\dagger = \mathrm{DCX}^2 \ne \mathrm{DCX}$.
• $\mathrm{DCX}$ differs from $\mathrm{SWAP}$ (three alternating CNOTs) by exactly one CNOT: $\mathrm{SWAP} = \mathrm{CX}(q_0\!\to\!q_1)\cdot\mathrm{DCX}$, so appending one more CNOT to DCX completes a full SWAP.
• Because DCX is built from only two CNOTs rather than three, it is cheaper than a full SWAP whenever a circuit only needs the specific 3-cycle permutation rather than a genuine two-qubit exchange.