Bridge Gate Identity

qc/hardware qc/gates

A circuit identity that implements a non-local CX(0,2) using only nearest-neighbor gates on a chain 0—1—2, without moving any state (unlike SWAP):

$CX(0,2)=CX(0,1)\cdot CX(1,2)\cdot CX(0,1)\cdot CX(1,2)$

Qubit 1 acts as a “bridge” — temporarily disturbed, then restored, while the entangling effect propagates from qubit 0 to qubit 2.

Cost: 4 nearest-neighbor CX gates to replace 1 non-local CX — a concrete “tax” for ignoring hardware topology. (After cancelling an adjacent identical CX with a neighboring gate in context, this can sometimes simplify to 3.)

Order matters, and it’s context-dependent: there are two mirror-image valid forms —

• CX(0,1)·CX(1,2)·CX(0,1)·CX(1,2) (starts near)
• CX(1,2)·CX(0,1)·CX(1,2)·CX(0,1) (starts far)

Both are mathematically equal to CX(0,2), but which one composes correctly depends on what gate already precedes the substitution point in your circuit — a gate that looks reusable from an earlier line usually isn’t; it was serving a different purpose (e.g. GHZ-building) and the bridge identity needs its own full, self-contained 4-gate block.

Related

• Transpilation
• CX Gate and Entanglement
• GHZ States — used to build nearest-neighbor GHZ circuits without SWAPs

Self-Check

• Could you explain what the “bridge” qubit is doing in this identity, and why it ends up unchanged?
• Why is this identity sometimes preferable to a SWAP-based approach?
• Why do the two mirror-image forms of the identity matter — why isn’t either one always correct?