controlled 3 qubits 1 parameter
Deutsch
The original universal three-qubit gate, applying $iR_x(2\theta)$ to the target when both controls are $|1\rangle$.
The Deutsch gate is the doubly-controlled rotation David Deutsch introduced in 1989 to prove that a single three-qubit gate can be universal for quantum computation. When both controls are set it applies $i R_x(2\theta)$ to the target; otherwise it does nothing. In the computational basis, $D(\theta)$ is
$$ \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & i\cos\theta & \sin\theta \\ 0 & 0 & 0 & 0 & 0 & 0 & \sin\theta & i\cos\theta \end{bmatrix} $$
Special values
| $\theta$ | Gate |
|---|---|
| $\pi/2$ | Toffoli |
Properties
- Universality: when $\theta/\pi$ is irrational, repeated applications of $D(\theta)$ alone approximate any unitary to arbitrary accuracy (using ancilla qubits) — the historical prototype for all universality results that followed.
- Since $D(\theta)$ contains the Toffoli, it inherits classical (reversible) universality too.
Usage
- Mostly of theoretical and historical interest; no hardware implements it natively, and modern universality proofs use small discrete gate sets (e.g. Clifford + $T$) instead.
The unitary, drawn
Every entry of the matrix as a disc. The area is the transition probability $|U_{jk}|^2$; the colour and the hand are the phase. Rows are outputs $\langle j|$, columns inputs $|k\rangle$, in the computational basis. Each parameter has a slider; play runs through the family along the path of the Quirk link ($\theta = 2\pi t$).
Phase
Disc area $= |U_{jk}|^2$. Hover an entry for its value, or take the JSON.
SDK support
How to call this gate in each framework. Hover a ⓘ for a note; the support matrix compares every gate at once.
| SDK | Name |
|---|---|
| Qiskit | — |
| PennyLane | — |
| Cirq | — |
| Q# | — |
| PyQuil | — |
| Braket | — |
| BQSKit | — |
| Qibo | qibo.gates.DEUTSCH |
| pytket | — ⓘ |
| Stim | — |
| OpenQASM | — |