Gates#
Gates can be characterized by a unitary operator \(U\), which evolves the quantum state in the following manner
Gates with at most quadratic Hamiltonians are called linear gates. Evolution of the ladder operators by linear gates could be expressed in the form
where \(S_{(c)} \in \operatorname{Sp}(2d, \mathbb{R})\), \(d \in \mathbb{C}^{2d}\),
where \(a_1^\dagger, ..., a_d^\dagger\) and \(a_1, \dots, a_d\) are the creation and annihilation operators, respectively.
Most of the gates defined here are linear gates, which can be characterized by \(S_{(c)}\) and \(d\).
- class Interferometer(matrix: ndarray)#
Applies a general interferometer gate.
The general unitary operator can be written as
\[I = \exp \left ( i \sum_{i, j = 1}^d H_{i j} a_i^\dagger a_j \right ),\]where the parameter U and \(H\) is related by
\[U = \exp \left ( i H \right ).\]The evolution of the ladder operators can be described by
\[\begin{split}S_{(c)} = \begin{bmatrix} U & 0_{d \times d} \\ 0_{d \times d} & U^* \end{bmatrix},\end{split}\]where \(U \in \mathbb{C}^{d \times d}\) is a unitary matrix.
- Parameters:
matrix (numpy.ndarray) – The unitary interferometer matrix, corresponding to a passive transformation on the one-particle subspace.
- class Beamsplitter(theta: float = 0.0, phi: float = 0.7853981633974483)#
Applies a beamsplitter gate.
The general unitary operator can be written as
\[B_{ij} (\theta, \phi) = \exp \left ( \theta e^{i \phi} a^\dagger_i a_j - \theta e^{- i \phi} a^\dagger_j a_i \right ).\]The symplectic representation of the beamsplitter gate is
\[\begin{split}S_{(c)} = \begin{bmatrix} t & -r^* & & \\ r & t & & \\ & & t & -r \\ & & r^* & t \end{bmatrix},\end{split}\]where \(t = \cos(\theta)\) and \(r = e^{i \phi} \sin(\theta)\).
- class Phaseshifter(phi: float)#
Applies a rotation or a phaseshifter gate.
The unitary operator corresponding to the phaseshifter gate on the \(i\)-th mode is
\[R_i (\phi) = \exp \left ( i \phi a_i^\dagger a_i \right ).\]The symplectic representation of the phaseshifter gate is
\[\begin{split}S_{(c)} = \begin{bmatrix} e^{i \phi} & 0 \\ 0 & e^{- i \phi} \end{bmatrix}.\end{split}\]- Parameters:
phi (float) – The angle of the rotation.
- class MachZehnder(int_: float, ext: float)#
Mach-Zehnder interferometer.
The Mach-Zehnder interferometer is equivalent to
\[MZ(\phi_{int}, \phi_{ext}) = B(\pi/4, \pi/2) (R(\phi_{int}) \oplus \mathbb{1}) B(\pi/4, \pi/2) (R(\phi_{ext}) \oplus \mathbb{1}).\]The symplectic representation of the Mach-Zehnder interferometer is
\[\begin{split}S_{(c)} = \begin{bmatrix} e^{i \phi_{ext} } (e^{i \phi_{int} } - 1) & i (e^{i \phi_{int} } + 1) & & \\ i e^{i \phi_{ext} } (e^{i \phi_{int} } + 1) & 1 - e^{i \phi_{int} } & & \\ & & e^{-i \phi_{ext} } (e^{-i \phi_{int} } - 1) & -i (e^{-i \phi_{int} } + 1) \\ & & -i e^{-i \phi_{ext} } (e^{-i \phi_{int} } + 1) & 1 - e^{-i \phi_{int} } \end{bmatrix},\end{split}\]where \(\phi_{int}, \phi_{ext} \in \mathbb{R}\).
- class Fourier#
Applies a Fourier gate. It simply transforms the quadratures as follows:
\[\begin{split}-\hat{p} = \operatorname{\textit{F}} \hat{x} \operatorname{\textit{F}}^\dagger \\ \hat{x} = \operatorname{\textit{F}} \hat{p} \operatorname{\textit{F}}^\dagger\end{split}\]The unitary operator corresponding to the Fourier gate on the \(i\)-th mode is
\[F_{i} = \exp \left ( i \frac{\pi}{2} a_i^\dagger a_i \right ).\]The symplectic representation of the Fourier gate is
\[\begin{split}S_{(c)} = \begin{bmatrix} i & 0 \\ 0 & -i \\ \end{bmatrix}.\end{split}\]Note
Corresponds to the
Phaseshiftergate with \(\phi = \pi/2\).
- class GaussianTransform(passive: ndarray, active: ndarray)#
Applies a Gaussian transformation gate.
The symplectic representation of the Gaussian gate is
\[\begin{split}S_{(c)} = \begin{bmatrix} P & A \\ A^* & P^* \end{bmatrix},\end{split}\]where \(P, A \in C^{d \times d}\) fulfilling the relations
\[\begin{split}P P^\dagger - A A^\dagger &= I_{d \times d} \\ P A^T &= A P^T\end{split}\]- Parameters:
passive (numpy.ndarray) – The passive submatrix of the symplectic matrix corresponding to the transformation.
active (numpy.ndarray) – The active submatrix of the symplectic matrix corresponding to the transformation.
- Raises:
InvalidParameters – Raised if the parameters do not form a symplectic matrix.
- class Squeezing(r: float, phi: float = 0.0)#
Applies the squeezing gate.
The unitary operator corresponding to the squeezing gate is
\[S_{i} (z) = \exp \left ( \frac{1}{2}(z^* a_i^2 - z a_i^{\dagger 2} ) \right ).\]The symplectic representation of the squeezing gate is
\[\begin{split}S_{(c)} = \begin{bmatrix} \cosh r & - e^{i \phi} \sinh r \\ - e^{- i \phi} \sinh r & \cosh r \end{bmatrix}.\end{split}\]The unitary squeezing operator is
\[S(z) = \exp \left ( \frac{1}{2}(z^* a_i^2 - z a_i^{\dagger 2}) \right ),\]where \(z = re^{i\phi}\).
- class QuadraticPhase(s: float)#
Applies the quadratic phase instruction to the state.
The unitary operator corresponding to the Fourier gate is
\[QP_{i} (s) = \exp \left ( i \frac{s}{2 \hbar} x_i^2 \right ).\]The symplectic representation of the quadratic phase gate is
\[\begin{split}S_{(c)} = \begin{bmatrix} 1 + i \frac{s}{2} & i \frac{s}{2} \\ -i \frac{s}{2} & 1 - i \frac{s}{2} \end{bmatrix}.\end{split}\]
- class Squeezing2(r: float, phi: float = 0.0)#
Applies the two-mode squeezing gate to the state.
The unitary operator corresponding to the two-mode squeezing gate is
\[S_{ij} (z) = \exp \left ( \frac{1}{2}(z^* a_i a_j - z a_i^\dagger a_j^\dagger ) \right ).\]The symplectic representation of the two-mode squeezing gate is
\[\begin{split}S_{(c)} = \begin{bmatrix} \cosh r & 0 & 0 & e^{i \phi} \sinh r \\ 0 & \cosh r & e^{i \phi} \sinh r & 0 \\ 0 & e^{- i \phi} \sinh r & \cosh r & 0 \\ e^{- i \phi} \sinh r & 0 & 0 & \cosh r \end{bmatrix}.\end{split}\]
- class ControlledX(s: float)#
Applies the controlled X gate to the state.
The unitary operator corresponding to the controlled X gate is
\[CX_{ij} (s) = \exp \left ( -i \frac{s}{\hbar} x_i p_j \right ).\]The symplectic representation of the controlled X gate is
\[\begin{split}S_{(z)} = \begin{bmatrix} 1 & - \frac{s}{2} & 0 & \frac{s}{2} \\ \frac{s}{2} & 1 & \frac{s}{2} & 0 \\ 0 & \frac{s}{2} & 1 & - \frac{s}{2} \\ \frac{s}{2} & 0 & \frac{s}{2} & 1 \end{bmatrix}.\end{split}\]
- class ControlledZ(s: float)#
Applies the controlled Z gate to the state.
The unitary operator corresponding to the controlled Z gate is
\[CZ_{ij} (s) = \exp \left ( -i \frac{s}{\hbar} x_i x_j \right ).\]The symplectic representation of the controlled Z gate is
\[\begin{split}S_{(z)} = \begin{bmatrix} 1 & i \frac{s}{2} & 0 & i \frac{s}{2} \\ i \frac{s}{2} & 1 & i \frac{s}{2} & 0 \\ 0 & -i \frac{s}{2} & 1 & - i \frac{s}{2} \\ -i \frac{s}{2} & 0 & -i \frac{s}{2} & 1 \end{bmatrix}.\end{split}\]
- class Displacement(r: float, phi: float | None = 0.0)#
Phase space displacement instruction.
Evolves the ladder operators by
\[\begin{split}\xi \mapsto \xi + \begin{bmatrix} \alpha \\ \alpha^* \end{bmatrix},\end{split}\]where \(\alpha \in \mathbb{C}^{d}\).
One must either specify \(\alpha\) only, or the combination of \(r\) and \(\phi\). When \(r\) and \(\phi\) are the given parameters, \(\alpha\) is calculated via:
\[\alpha = r e^{i\phi}\]
- class PositionDisplacement(x: float)#
Position displacement gate. It affects only the \(\hat{x}\) quadrature.
Note
The specified displacement is automatically scaled by \(\sqrt{2 \hbar}\).
- Parameters:
x (float) – The position displacement.
- class MomentumDisplacement(p: float)#
Momentum displacement gate. It only affects the \(\hat{p}\) quadrature.
Note
The specified displacement is automatically scaled by \(\sqrt{2 \hbar}\).
- Parameters:
p (float) – The momentum displacement.
- class CubicPhase(gamma: float)#
Cubic Phase gate.
The definition of the Cubic Phase gate is
\[\operatorname{CP}(\gamma) = e^{i \hat{x}^3 \frac{\gamma}{3 \hbar}}\]The Cubic Phase gate transforms the annihilation operator as
\[\operatorname{CP}^\dagger(\gamma) \hat{a} \operatorname{CP}(\gamma) = \hat{a} + i\frac{\gamma(\hat{a} +\hat{a}^\dagger)^2}{2\sqrt{2/\hbar}}\]It transforms the \(\hat{p}\) quadrature as follows:
\[\operatorname{CP}^\dagger(\gamma) \hat{p} \operatorname{CP}(\gamma) = \hat{p} + \gamma \hat{x}^2.\]Note
This is a non-linear gate, therefore it couldn’t be used with
GaussianState. Using this gate requires a high cutoff to make the more accurate simulation.- Parameters:
gamma (float) – The Cubic Phase parameter.
- class Kerr(xi: float)#
Kerr gate.
The definition of the Kerr gate is
\[K_i (\xi) = \exp \left ( i \xi n_i n_i \right ).\]The Kerr gate transforms the annihilation operator as
\[K(\xi) a K(\xi)^\dagger = a \exp(- i \xi (1 + 2 n)).\]Note
This is a non-linear gate, therefore it couldn’t be used with
GaussianState.- Parameters:
xi (float) – The magnitude of the Kerr nonlinear term.
- class CrossKerr(xi: float)#
Cross-Kerr gate.
The definition of the Cross-Kerr gate is
\[CK_{ij} (\xi) = \exp \left ( i \xi n_i n_j \right )\]The Cross-Kerr gate transforms the annihilation operators as
\[\begin{split}CK_{ij} (\xi) a_i CK_{ij} (\xi) &= a_i \exp(- i \xi n_j) \\ CK_{ij} (\xi) a_j CK_{ij} (\xi) &= a_j \exp(- i \xi n_i)\end{split}\]Note
This is a non-linear gate, therefore it couldn’t be used with
GaussianState.- Parameters:
xi (float) – The magnitude of the Cross-Kerr nonlinear term.
- class Graph(adjacency_matrix: ndarray, mean_photon_number: float = 1.0)#
Applies a graph given its adjacency matrix, see this article for more details. It decomposes the adjacency matrix of a graph into single mode squeezers and interferometers.
- Parameters:
adjacency_matrix (numpy.ndarray) – A symmetric matrix with a size of \(N \times N\) and it can be real or complex.
mean_photon_number (float, optional) – The mean photon number \(\bar{n}\) for a mode. Defaults to \(1.0\).
- Raises:
InvalidParameter – If the adjacency matrix is not invertible or not symmetric.