Operations are the atomic instructions in any quantum program that can be represented by qoqo/roqoqo. Gate operations are single-, two- or multi-qubit unitary operations that apply a unitary transformation and can be executed on any universal quantum computer. Mathematically, a gate can be represented by a unitary matrix.
A list of the gate operations available in qoqo and roqoqo with their mathematical description is provided in this section. We differentiate between single-qubit gates acting on a single qubit, two-qubit gates applied on a pair of qubits and multi-qubit gates affecting a series of qubits.
A rotation angle is usually annotated with \( \theta \) and its corresponding argument is theta.
For the phase angle, the symbol \( \varphi \) is used.
The rotation angle \( \phi \) in the x-y plane is addressed by the argument name phi.
\( \sigma_x \), \( \sigma_y \), \( \sigma_z \) are the Pauli matrices X, Y, Z
\[
\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} := X, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} := Y, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} := Z
\].
Gate Short Description
Hadamard The Hadamard gate, to create a superposition of states, and so to change the basis. InvSqrtPauliX The inverse square root of the PauliX gate \( e^{i \frac{\theta}{4} \sigma_x} \). PauliX The Pauli X gate, a rotation with a fixed angle of \( \frac{\pi}{2} \), corresponds to a "flip" on x-axis. PauliY The Pauli Y gate, a rotation with a fixed angle of \( \frac{\pi}{2} \), corresponds to a "flip" on y-axis. PauliZ The Pauli Z gate, a rotation with a fixed angle of \( \frac{\pi}{2} \), corresponds to a "flip" on z-axis. PhaseShiftState0 Rotation around z-axis by angle \(\theta\) applied on state \( \left |0 \right> \) results in a phase shift compared to RotateZ gate. PhaseShiftState1 Rotation around z-axis by angle \(\theta\) applied on state \( \left|1 \right> \) results in phase shift compared to RotateZ gate. RotateAroundSphericalAxis Implements a rotation around an axis in spherical coordinates. RotateX The rotation gate around x-axis \( e^{-i \frac{\theta}{2} \sigma_x} \). RotateXY Implements a rotation around an axis in the x-y plane, where the axis is defined by an angle/spherical coordinates. RotateY The rotation gate around y-axis \( e^{-i \frac{\theta}{2} \sigma_y} \). RotateZ The rotation gate around z-axis \( e^{-i \frac{\theta}{2} \sigma_z} \). SGate The S gate. SqrtPauliX The square root of the PauliX gate \( e^{-i \frac{\theta}{4} \sigma_x} \). TGate The T gate.
Gate Short Description
Bogoliubov The Bogoliubov DeGennes interaction gate. CNOT The controlled not gate, e.g. to entangle two qubits. ComplexPMInteraction The complex hopping gate. ControlledPauliY The controlled PauliY gate. ControlledPauliZ The controlled PauliZ gate. ControlledPhaseShift The controlled phase shift gate. Fsim The fermionic qubit simulation gate. FSwap The fermionic SWAP gate. GivensRotation The Givens rotation interaction gate in big endian notation: \(e^{-\mathrm{i} \theta (X_c Y_t - Y_c X_t)}\cdot e^{-i \phi Z_t/2} \). GivensRotationLittleEndian The Givens rotation interaction gate in little-endian notation: \(e^{-\mathrm{i} \theta (X_c Y_t - Y_c X_t)}\cdot e^{-i \phi Z_c/2} \). InvSqrtISwap The inverse square root of the ISwap gate. ISwap The complex swap gate. MolmerSorensenXX The fixed-phase Mølmer–Sørensen XX gate. PhaseShiftedControlledZ The phased-shifted controlled-Z gate. PMInteraction The transversal interaction gate. Qsim The qubit simulation gate. SpinInteraction The generalized, anisotropic XYZ Heisenberg interaction between spins. SqrtISwap The square root of the ISwap gate. SWAP The swap gate, to switch the positions of two qubits. VariablesMSXX The variable-angle Mølmer–Sørensen XX gate. XY The XY gate.
Gate Short Description
MultiQubitMS The Mølmer–Sørensen gate between multiple qubits. MultiQubitZZ The multi-qubit PauliZ-product gate.