quantum gate directory
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Symbol
Qubits: $3$ Parameters: $1$
Description:
The original universal three-qubit gate, applying $iR_x(2\theta)$ to the target when both controls are $|1\rangle$.
Alternate notations:
- $D(\theta)$
- $\Lambda^2(iR_x(2\theta))$
SDK Support
| SDK | Name |
|---|---|
| Qiskit | — |
| PennyLane | — |
| Cirq | — |
| Q# | — |
| PyQuil | — |
| Braket | — |
| BQSKit | — |
| Qibo |
qibo.gates.DEUTSCH
|
| pytket | — ⓘ |
| Stim | — |
| OpenQASM | — |
Reference
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:
$$ D(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & i\cos\theta & \sin\theta \\ 0 & 0 & 0 & 0 & 0 & 0 & \sin\theta & i\cos\theta \end{bmatrix} $$
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.
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