# ::: AQT Quantum Gate Definitions

The available quantum gate set for AQT backends consists of the following gates:

• Rx-gate – rotation around the Bloch sphere’s x-axis
• Ry-gate – rotation around the Bloch sphere’s y-axis
• Rz-gate – rotation around the Bloch sphere’s z-axis
• R-gate – rotation around an arbitrary axis on the Bloch sphere’s equatorial plane
• MS-gate – entangling gate of Mølmer-Sørenson-type

## Gate Definitions

All quantum gates are defined in terms of the Pauli matrices,

$\begin{eqnarray} \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} , && \sigma_y = \begin{pmatrix} 0 & -\mathrm{i} \\ \mathrm{i} & 0 \end{pmatrix} , && \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} . \end{eqnarray}$

### Rx-gate

The Rx-gate on qubit $$j$$ with pulse area $$t$$ in units of $$\pi$$ is defined as

$U_{\mathrm{R_x}}^j(t) = \exp\left (-\mathrm{i} t \frac{\pi}{2} \sigma_x^j\right) = \begin{pmatrix} \cos(t\frac{\pi}{2}) && -\mathrm{i}\sin(t \frac{\pi}{2}) \\ -\mathrm{i}\sin(t \frac{\pi}{2}) && \cos(t \frac{\pi}{2}) \end{pmatrix}$

Examples:

$U_{\mathrm{R_x}}^j(0.5) \ket{0_j} =\frac{1}{\sqrt{2}}\left (\ket{0_j} – \mathrm{i}\ket{1_j} \right )$

$U_{\mathrm{R_x}}^j(0.5) \ket{1_j} =\frac{1}{\sqrt{2}}\left ( – \mathrm{i}\ket{0_j} + \ket{1_j} \right )$

The JSON syntax to encode the Rx-gate is:

[["X", t, [j]]]


### Ry-gate

The Ry-gate on qubit $$j$$ with pulse area $$t$$ in units of $$\pi$$ is defined as

$U_{\mathrm{R_y}}^j(t) = \exp\left( -\mathrm{i} t \frac{\pi}{2} \sigma_y^j \right) = \begin{pmatrix} \cos(t \frac{\pi}{2}) && -\sin(t \frac{\pi}{2}) \\ \sin(t \frac{\pi}{2}) && \cos(t \frac{\pi}{2}) \end{pmatrix}$

Examples:

$U_{\mathrm{R_y}}^j(0.5) \ket{0_j} =\frac{1}{\sqrt{2}}\left (\ket{0_j} + \ket{1_j} \right )$

$U_{\mathrm{R_y}}^j(0.5) \ket{1_j} =\frac{1}{\sqrt{2}}\left ( -\ket{0_j} + \ket{1_j} \right )$

The JSON syntax to encode the Ry-gate is

[["Y", t, [j]]]


### Rz-gate

The Rz-gate on qubit $$j$$ with pulse area $$t$$ in units of $$\pi$$ is defined as

$U_{\mathrm{R_z}}^j(t) = \exp\left( -\mathrm{i} t \frac{\pi}{2} \sigma_z^j \right) = \begin{pmatrix} \cos(t \frac{\pi}{2}) -\mathrm{i}\sin(t \frac{\pi}{2}) && 0 \\ 0 && \cos(t \frac{\pi}{2}) +\mathrm{i}\sin(t \frac{\pi}{2}) \end{pmatrix}$

Examples:

$U_{\mathrm{R_z}}^j(1)\frac{1}{\sqrt{2}}\left( \ket{0_j} + \ket{1_j}\right) = \frac{1}{\sqrt{2}}\left (\ket{0_j} – \ket{1_j} \right )$

$U_{\mathrm{R_z}}^j(0.5)\frac{1}{\sqrt{2}}\left (\ket{0_j} + \ket{1_j} \right) =\frac{1}{\sqrt{2}}\left (\ket{0_j} +\mathrm{i}\ket{1_j} \right )$

The JSON syntax to encode the Rz-gate is

[["Z", t, [j]]]


### R-gate

The R-gate on qubit $$j$$ with pulse area $$t$$ and mixing angle $$p$$, both in units of $$\pi$$, is defined as

$U_{\mathrm{R}}^j(t,p) = \exp\left( -\mathrm{i} t \frac{\pi}{2} \left[\sin(p \pi)\sigma_y^j + \cos(p \pi)\sigma_x^j \right] \right) = \begin{pmatrix} \cos(t \frac{\pi}{2}) && -\mathrm{i} e^{-\mathrm{i} p \pi}\sin(t \frac{\pi}{2})\\ -\mathrm{i} e^{\mathrm{i} p \pi}\sin(t \frac{\pi}{2}) && \cos(t \frac{\pi}{2}) \end{pmatrix}$

Examples:

$U_{\mathrm{R}}^j(0.5, 0) \ket{0_j} = U _{\mathrm{R_x}} ^j(0.5) \ket{0_j} =\frac{1}{\sqrt{2}}\left (\ket{0_j} – \mathrm{i}\ket{1_j} \right )$

$U_{\mathrm{R}}^j(0.5, 0.5) \ket{0_j} = U _{\mathrm{R_y}} ^j(0.5) \ket{0_j} =\frac{1}{\sqrt{2}}\left (\ket{0_j} + \ket{1_j} \right )$

The JSON syntax to encode the R-gate is

[["R", t, p, [j]]]


### MS-gate

The MS-gate on qubits $$j$$ and $$k$$ with pulse area $$t$$ in units of $$\pi$$ is defined as

\begin{aligned} U_{\mathrm{MS}}^{j,k} \left (t \right) & = e^{\mathrm{i}t\frac{\pi}{2}}\exp{\left(-\mathrm{i} t \pi {S_x}^2 \right) } = \\ & = \begin{pmatrix} \cos(t \frac{\pi}{2}) && 0 && 0 && -\mathrm{i}\sin(t \frac{\pi}{2}) \\ 0 && \cos(t \frac{\pi}{2}) && -\mathrm{i}\sin(t \frac{\pi}{2}) && 0 \\ 0 && -\mathrm{i}\sin(t\frac{\pi}{2}) && \cos(t \frac{\pi}{2}) && 0 \\ -\mathrm{i}\sin(t \frac{\pi}{2}) && 0 && 0 && \cos(t \frac{\pi}{2}) \end{pmatrix} \end{aligned}

with

$S_x =\frac{1}{2}\left (\sigma_x^j + \sigma_x^k \right)$

A fully-entangling gate between qubit 0 and qubit 1 therefore is

$U_{\mathrm{MS}}^{0,1} \left (0.5 \right) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 && 0 && 0 && -\mathrm{i} \\ 0 && 1 && -\mathrm{i} && 0 \\ 0 && -\mathrm{i} && 1 && 0 \\ -\mathrm{i} && 0 && 0 && 1 \end{pmatrix}$

Examples:

$U_{\mathrm{MS}}^{j,k}(0.5) \ket{0_j0_k} =\frac{1}{\sqrt{2}}\left (\ket{0_j0_k} -\mathrm{i} \ket{1_j1_k} \right )$

$U_{\mathrm{MS}}^{j,k}(0.5) \ket{1_j1_k} =\frac{1}{\sqrt{2}}\left ( -\mathrm{i}\ket{0_j0_k} + \ket{1_j1_k} \right )$

The JSON syntax to encode the MS-gate is

[["MS", t, [j,k]]]


The example for the fully-entangling gate is therefore

[["MS", 0.5, [0,1]]]


As all AQT backends are still under heavy development, please understand that all of the above is subject to change without further notice. We will break your code!

## AQT Tutorials

We provide tutorials on how to use the AQT backends either directly using the JSON definition or with SDKs at the following pages: