Operations are the atomic instructions in any quantum program that can be represented by qoqo/roqoqo. Gate operations are single-, two- three-, four- 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 , two-qubit gates , three-qubit gates , four-qubit gates and multi-qubit gates , depending on the number of qubits involved in the operation.
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
GPi Implements a pi-rotation with an embedded phase. GPi2 Implements a pi/2-rotation with an embedded phase. Hadamard The Hadamard gate. Identity The Identity gate. InvSGate The InvS gate. InvSXGate The inverse SX gate. InvSqrtPauliX The inverse square root of the XPower gate: exp(i * π/4 * σ^x). InvSqrtPauliY The inverse square root of the YPower gate: exp(i * π/4 * σ^x). InvTGate The InvT gate. PauliX The Pauli X gate. PauliY The Pauli Y gate. PauliZ The Pauli Z gate. PhaseShiftState0 The phase shift gate applied on state PhaseShiftState1 The phase shift gate applied on state RotateAroundSphericalAxis Implements a rotation around an axis in the x-y plane in spherical coordinates. RotateX The XPower gate exp(-i * θ/2 * σ^x). RotateXY Implements a rotation around an x- and y-axis in spherical coordinates. RotateY The YPower gate exp(-i * θ/2 * σ^y). RotateZ The ZPower gate exp(-i * θ/2 * σ^z). SGate The S gate. SXGate The SX gate, equivalent to SqrtPauliX with a pi/4 global phase factor added. SingleQubitGate The most general unitary operation acting on one qubit. SqrtPauliX The square root of the XPower gate exp(-i * π/4 * σ^x). SqrtPauliY The square root of the PauliY gate exp(-i * π/4 * σ^y). TGate The T gate.
Gate Short Description
Bogoliubov The Bogoliubov DeGennes interaction gate. CNOT The CNOT controlled not gate. ComplexPMInteraction The complex hopping gate. ControlledPauliY The controlled-PauliY gate. ControlledPauliZ The controlled-PauliZ gate. ControlledPhaseShift Implements the controlled PhaseShift gate. ControlledRotateX Implements the controlled RotateX operation. ControlledRotateXY Implements the controlled RotateX operation. EchoCrossResonance Implements the controlled RotateX operation. FSwap The Fermionic SWAP gate. Fsim The fermionic qubit simulation gate. GivensRotation The Givens rotation interaction gate in big endian notation: exp(-i * θ * [X_c Y_t - Y_c X_t]) * exp(-i * φ * Z_t/2). GivensRotationLittleEndian The Givens rotation interaction gate in little endian notation: exp(-i * θ * [X_c Y_t -Y_c X_t]) * exp(-i * φ * Z_c/2). ISwap The ISwap gate. InvSqrtISwap The inverse square root ISwap gate. KakDecomposition The KAK decomposition of a two-qubit gate. MolmerSorensenXX The fixed phase MolmerSorensen XX gate. PMInteraction The transversal interaction gate. PhaseShiftedControlledPhase Implements the phase-shifted controlled PhaseShift gate. PhaseShiftedControlledZ Implements the phased-shifted controlled-Z gate. Qsim The qubit simulation gate. SWAP The SWAP gate. SpinInteraction The generalized, anisotropic XYZ Heisenberg interaction between spins. SqrtISwap The square root ISwap gate. VariableMSXX The variable-angle MolmerSorensen XX gate. XY The XY gate.
Gate Short Description
ControlledControlledPauliZ Implements the double-controlled PauliZ gate. ControlledControlledPhaseShift Implements the double-controlled PhaseShift gate. ControlledSWAP Implements the controlled SWAP gate. PhaseShiftedControlledControlledPhase Implements the double-controlled phase-shifted PhaseShift gate. PhaseShiftedControlledControlledZ Implements the double-controlled phase-shifted PauliZ gate. Toffoli Implements the Toffoli gate.
Gate Short Description
TripleControlledPauliX The triple-controlled PauliX gate. TripleControlledPauliZ The triple-controlled PauliZ gate. TripleControlledPhaseShift The triple-controlled PhaseShift gate.
Gate Short Description
CallDefinedGate The gate to be replaced by a gate defined with GateDefinition gate. MultiQubitMS The Molmer-Sorensen gate between multiple qubits. MultiQubitZZ The multi qubit Pauli-Z-Product gate.
Gate Short Description
BeamSplitter The 2-mode beam splitter which splits a beam with a transmission amplitude cos(θ) and a reflection amplitude exp(i * φ) * sin(θ). PhaseDisplacement The single-mode phase-displacement gate with variable magnitude and phase. PhaseShift The single-mode phase-shift gate with variable phase, given by R(θ) = exp(i * θ * 𝑁̂). PhotonDetection The photon number-resolving detector measurement for bosons. Squeezing The single-mode squeezing gate with tunable squeezing.
Gate Short Description
CZQubitResonator Controlled-Z operation between a qubit and a bosonic mode. JaynesCummings The Jaynes-Cummings gate exp(-i * θ * (σ^- * b^† + σ^+ * b)) LongitudinalCoupling Longitudinal coupling gate exp(-i * θ * Z * (b^† + b)) QuantumRabi The quantum Rabi interaction exp(-i * θ * X * (b^† + b)) SingleExcitationLoad Loads a single excitation from a bosonic mode into a qubit. SingleExcitationStore Stores a single excitation from the involved qubit into the involved bosonic mode.