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CV Hamiltonians

Here, we provided several auxillary Hamiltonians for CV systems.

Gates

This file contains the Hamiltonian representations of the CV operations found in StrawberryFields.

All Hamiltonians in the file return a tuple containing the Hamiltonian operator (either as a BosonOperator or a QuadOperator) and the time propagation.

displacement(alpha[, mode, hbar])	Returns the Hamiltonian of the displacement operation.
xdisplacement(x[, mode])	Returns the Hamiltonian of the \(x\) displacement operation.
zdisplacement(p[, mode])	Returns the Hamiltonian of the \(p\) displacement operation.
rotation(phi[, mode, hbar])	Returns the Hamiltonian of the rotation operation.
squeezing(r[, phi, mode, hbar])	Returns the Hamiltonian of the squeezing operation.
quadratic_phase([s, mode])	Returns the Hamiltonian of the quadratic phase operation.
beamsplitter([theta, phi, mode1, mode2, hbar])	Returns the Hamiltonian of the beamsplitter operation.
two_mode_squeezing(r[, phi, mode1, mode2, hbar])	Returns the Hamiltonian of the two-mode squeezing operation.
controlled_addition([s, mode1, mode2])	Returns the Hamiltonian of the controlled addition operation.
controlled_phase([s, mode1, mode2])	Returns the Hamiltonian of the controlled phase operation.
cubic_phase([gamma, mode])	Returns the Hamiltonian of the cubic phase operation.
kerr([kappa, mode, hbar])	Returns the Hamiltonian of the Kerr operation.

Code details

sfopenboson.hamiltonians.displacement(alpha, mode=0, hbar=2)

Returns the Hamiltonian of the displacement operation.

The time evolution unitary associated with displacement is

\[D(\alpha) = \exp( \alpha \ad -\alpha^* \a) = \exp(r (e^{i\phi}\ad -e^{-i\phi}\a))\]

where \(\alpha=r e^{i \phi}\) with \(r \geq 0\) and \(\phi \in [0,2 \pi)\).

Therefore, \(U=e^{-iHt/\hbar}\) where \(H = {i}{\hbar}(e^{i\phi}\ad -e^{-i\phi}\a)\) and \(t=r\).

Parameters:

• a (complex) – the displacement in the phase space
• mode (int) – the qumode on which the operation acts
• hbar (float) – the scaling convention chosen in the definition of the quadrature operators: \([\x,\p]=i\hbar\)

Returns:

tuple containing the Hamiltonian representing the operation and the propagation time

Return type:

tuple (BosonOperator, t)

sfopenboson.hamiltonians.xdisplacement(x, mode=0)

Returns the Hamiltonian of the \(x\) displacement operation.

The time evolution unitary associated with \(x\) displacement is

\[X(x) = \exp(-ix\p/\hbar)\]

Therefore, \(U=e^{-iHt/\hbar}\) where \(H = \p\) and \(t=x\).

Parameters:

• x (float) – the position displacement in the phase space
• mode (int) – the qumode on which the operation acts

Returns:

tuple containing the Hamiltonian representing the operation and the propagation time

Return type:

tuple (QuadOperator, t)

sfopenboson.hamiltonians.zdisplacement(p, mode=0)

Returns the Hamiltonian of the \(p\) displacement operation.

The time evolution unitary associated with \(p\) displacement is

\[X(x) = \exp(ip\x/\hbar)\]

Therefore, \(U=e^{-iHt/\hbar}\) where \(H =-\x\) and \(t=p\).

Parameters:

• p (float) – the position displacement in the phase space
• mode (int) – the qumode on which the operation acts

Returns:

tuple containing the Hamiltonian representing the operation and the propagation time

Return type:

tuple (QuadOperator, t)

sfopenboson.hamiltonians.rotation(phi, mode=0, hbar=2)

Returns the Hamiltonian of the rotation operation.

The time evolution unitary associated with rotation is

\[R(\phi) = \exp\left(i \phi \ad \a\right) =\exp\left(i \frac{\phi}{2} \left(\frac{\x^2+ \p^2}{\hbar}-I\right)\right)\]

Therefore, \(U=e^{-iHt/\hbar}\) where \(H = -\hbar\ad\a\) and \(t=\phi\).

Parameters:

• phi (float) – the rotation angle
• mode (int) – the qumode on which the operation acts
• hbar (float) – the scaling convention chosen in the definition of the quadrature operators: \([\x,\p]=i\hbar\)

Returns:

tuple containing the Hamiltonian representing the operation and the propagation time

Return type:

tuple (BosonOperator, t)

sfopenboson.hamiltonians.squeezing(r, phi=0, mode=0, hbar=2)

Returns the Hamiltonian of the squeezing operation.

The time evolution unitary associated with squeezing is

\[S(r,\phi) = \exp\left(\frac{r}{2}\left(e^{-i\phi}\a^2 -e^{i\phi}{\ad}^{2} \right) \right)\]

Therefore, \(U=e^{-iHt/\hbar}\) where \(H = \frac{i\hbar}{2}\left(e^{-i\phi}\a^2 -e^{i\phi}{\ad}^{2}\right)\) and \(t=r\).

Parameters:

• r (float) – the squeezing magnitude
• phi (float) – the quadrature angle in which the squeezing occurs. \(\phi=0\) corresponds to squeezing in the \(\x\) quadrature, and \(\phi=\pi/2\) corresponds to squeezing in the \(\p\) quadrature.
• mode (int) – the qumode on which the operation acts
• hbar (float) – the scaling convention chosen in the definition of the quadrature operators: \([\x,\p]=i\hbar\)

Returns:

tuple containing the Hamiltonian representing the operation and the propagation time

Return type:

tuple (BosonOperator, t)

sfopenboson.hamiltonians.quadratic_phase(s=1, mode=0)

Returns the Hamiltonian of the quadratic phase operation.

The time evolution unitary associated with the quadratic phase is

\[P(s) = \exp\left(i \frac{s}{2 \hbar} \x^2\right)\]

Therefore, \(U=e^{-iHt/\hbar}\) where \(H = -\x^2/2\) and \(t=s\).

Parameters:

• s (float) – the quadratic phase parameter
• mode (int) – the qumode on which the operation acts

Returns:

tuple containing the Hamiltonian representing the operation and the propagation time

Return type:

tuple (QuadOperator, t)

sfopenboson.hamiltonians.beamsplitter(theta=0.7853981633974483, phi=0, mode1=0, mode2=1, hbar=2)

Returns the Hamiltonian of the beamsplitter operation.

The time evolution unitary associated with the beamsplitter is

\[B(\theta,\phi) = \exp\left(\theta (e^{i \phi} \ad_0 \a_1 - e^{-i \phi}\a_0 \ad_1) \right)\]

Therefore, \(U=e^{-iHt/\hbar}\) where \(H(\phi) = {i}{\hbar}\left(e^{i \phi} \ad_0 \a_1 - e^{-i \phi}\a_0 \ad_1\right)\) and \(t=\theta\).

Parameters:

• theta (float) – transmitivity angle \(\theta\) where \(t=\cos(\theta)\)
• phi (float) – phase angle \(\phi\) where \(r=e^{i\phi}\sin(\theta)\)
• mode1 (int) – the first qumode \(\a_0\) on which the operation acts
• mode2 (int) – the second qumode \(\a_1\) on which the operation acts
• hbar (float) – the scaling convention chosen in the definition of the quadrature operators: \([\x,\p]=i\hbar\)

Returns:

tuple containing the Hamiltonian representing the operation and the propagation time

Return type:

tuple (BosonOperator, t)

sfopenboson.hamiltonians.two_mode_squeezing(r, phi=0, mode1=0, mode2=1, hbar=2)

Returns the Hamiltonian of the two-mode squeezing operation.

The time evolution unitary associated with two-mode squeezing is

\[S_2(r,\phi) = \exp\left(r\left(e^{-i\phi}\a_0 \a_1 -e^{i\phi}{\ad_0} \ad_1 \right) \right)\]

Therefore, \(U=e^{-iHt/\hbar}\) where \(H = {i}{\hbar}\left(e^{-i\phi}\a_0 \a_1 -e^{i\phi}{\ad_0} \ad_1\right)\) and \(t=r\).

Parameters:

• r (float) – the squeezing magnitude
• phi (float) – the quadrature in which the squeezing occurs. \(\phi=0\) corresponds to squeezing in the \(\x\) quadrature, and \(\phi=\pi/2\) corresponds to squeezing in the \(\p\) quadrature.
• mode1 (int) – the first qumode \(\a_0\) on which the operation acts
• mode2 (int) – the second qumode \(\a_1\) on which the operation acts
• hbar (float) – the scaling convention chosen in the definition of the quadrature operators: \([\x,\p]=i\hbar\)

Returns:

tuple containing the Hamiltonian representing the operation and the propagation time

Return type:

tuple (BosonOperator, t)

sfopenboson.hamiltonians.controlled_addition(s=1, mode1=0, mode2=1)

Returns the Hamiltonian of the controlled addition operation.

The time evolution unitary associated with controlled addition is

\[CX(s) = \exp\left( -i \frac{s}{\hbar}\x_0\otimes \p_1 \right)\]

Therefore, \(U=e^{-iHt/\hbar}\) where \(H =\x_0\otimes \p_1\) and \(t=s\).

Parameters:

• s (float) – the controlled addition parameter
• mode1 (int) – the first qumode \(\a_0\) on which the operation acts
• mode2 (int) – the second qumode \(\a_1\) on which the operation acts

Returns:

tuple containing the Hamiltonian representing the operation and the propagation time

Return type:

tuple (QuadOperator, t)

sfopenboson.hamiltonians.controlled_phase(s=1, mode1=0, mode2=1)

Returns the Hamiltonian of the controlled phase operation.

The time evolution unitary associated with controlled phase is

\[CZ(s) = \exp\left( i \frac{s}{\hbar}\x_0\otimes \x_1 \right)\]

Therefore, \(U=e^{-iHt/\hbar}\) where \(H = -\x_0\otimes \x_1\) and \(t=s\).

Parameters:

• s (float) – the controlled addition parameter
• mode1 (int) – the first qumode \(\a_0\) on which the operation acts
• mode2 (int) – the second qumode \(\a_1\) on which the operation acts

Returns:

tuple containing the Hamiltonian representing the operation and the propagation time

Return type:

tuple (QuadOperator, t)

sfopenboson.hamiltonians.cubic_phase(gamma=1, mode=0)

Returns the Hamiltonian of the cubic phase operation.

The time evolution unitary associated with the cubic phase is

\[V(\gamma) = \exp\left(i \frac{\gamma}{3 \hbar} \x^3\right)\]

Therefore, \(U=e^{-iHt/\hbar}\) where \(H = -\x^3/3\) and \(t=\gamma\).

Parameters:

• gamma (float) – the cubic phase parameter
• mode (int) – the qumode on which the operation acts

Returns:

tuple containing the Hamiltonian representing the operation and the propagation time

Return type:

tuple (QuadOperator, t)

sfopenboson.hamiltonians.kerr(kappa=1, mode=0, hbar=2)

Returns the Hamiltonian of the Kerr operation.

The time evolution unitary associated with the Kerr gate is

\[K(\kappa) = \exp\left(i \kappa \hat{n}^2 \right)\]

Therefore, \(U=e^{-iHt/\hbar}\) where \(H = -\hat{n}^2\hbar=-(\ad \a)^2\hbar\) and \(t=\kappa\).

Parameters:

• kappa (float) – the Kerr parameter
• mode (int) – the qumode on which the operation acts
• hbar (float) – the scaling convention chosen in the definition of the quadrature operators: \([\x,\p]=i\hbar\)

Returns:

tuple containing the Hamiltonian representing the operation and the propagation time

Return type:

tuple (BosonOperator, t)