sf.ops.Sgate¶
-
class
Sgate
(r, phi=0.0)[source]¶ Bases:
strawberryfields.ops.Gate
Phase space squeezing gate.
\[S(z) = \exp\left(\frac{1}{2}(z^* a^2 -z {a^\dagger}^2)\right)\]where \(z = r e^{i\phi}\).
- Parameters
r (float) – squeezing amount
phi (float) – squeezing phase angle \(\phi\)
Details and Conventions
Definition
\[\begin{split}& S(z) = \exp\left(\frac{1}{2}\left(z^* \a^2-z {\ad}^{2} \right) \right) = \exp\left(\frac{r}{2}\left(e^{-i\phi}\a^2 -e^{i\phi}{\ad}^{2} \right) \right)\\ & S^\dagger(z) \a S(z) = \a \cosh(r) -\ad e^{i \phi} \sinh r\\ & S^\dagger(z) \ad S(z) = \ad \cosh(r) -\a e^{-i \phi} \sinh(r)\end{split}\]where \(z=r e^{i \phi}\) with \(r \geq 0\) and \(\phi \in [0,2 \pi)\).
The squeeze gate affects the position and momentum operators as
\[S^\dagger(z) \x_{\phi} S(z) = e^{-r}\x_{\phi}, ~~~ S^\dagger(z) \p_{\phi} S(z) = e^{r}\p_{\phi}\]The Fock basis decomposition of displacement and squeezing operations was analysed by Krall [30], and the following quantity was calculated,
\[\begin{split}f_{n,m}(r,\phi,\beta)&=\bra{n}\exp\left(\frac{r}{2}\left(e^{i \phi} \a^2 -e^{-i \phi} \ad \right) \right) D(\beta) \ket{m} = \bra{n}S(z^*) D(\beta) \ket{m}\\ &=\sqrt{\frac{n!}{\mu m!}} e^{\frac{\beta ^2 \nu ^*}{2\mu }-\frac{\left| \beta \right| ^2}{2}} \sum_{i=0}^{\min(m,n)}\frac{\binom{m}{i} \left(\frac{1}{\mu \nu }\right)^{i/2}2^{\frac{i-m}{2} +\frac{i}{2}-\frac{n}{2}} \left(\frac{\nu }{\mu }\right)^{n/2} \left(-\frac{\nu ^*}{\mu }\right)^{\frac{m-i}{2}} H_{n-i}\left(\frac{\beta }{\sqrt{2} \sqrt{\mu \nu }}\right) H_{m-i}\left(-\frac{\alpha ^*}{\sqrt{2}\sqrt{-\mu \nu ^*}}\right)}{(n-i)!}\end{split}\]where \(\nu=e^{- i\phi} \sinh(r), \mu=\cosh(r), \alpha=\beta \mu - \beta^* \nu\).
Two important special cases of the last formula are obtained when \(r \to 0\) and when \(\beta \to 0\):
For \(r \to 0\) we can take \(\nu \to 1, \mu \to r, \alpha \to \beta\) and use the fact that for large \(x \gg 1\) the leading order term of the Hermite polynomials is \(H_n(x) = 2^n x^n +O(x^{n-2})\) to obtain
\[f_{n,m}(0,\phi,\beta) = \bra{n}D(\beta) \ket{m}=\sqrt{\frac{n!}{ m!}} e^{-\frac{\left| \beta \right| ^2}{2}} \sum_{i=0}^{\min(m,n)} \frac{(-1)^{m-i}}{(n-i)!} \binom{m}{i} \beta^{n-i} (\beta^*)^{m-i}\]On the other hand if we let \(\beta\to 0\) we use the fact that
\[\begin{split}H_n(0) =\begin{cases}0, & \mbox{if }n\mbox{ is odd} \\ (-1)^{\tfrac{n}{2}} 2^{\tfrac{n}{2}} (n-1)!! , & \mbox{if }n\mbox{ is even} \end{cases}\end{split}\]to deduce that \(f_{n,m}(r,\phi,0)\) is zero if \(n\) is even and \(m\) is odd or vice versa.
When writing the Bloch-Messiah reduction [25][31] of a Gaussian state in the Fock basis one often needs the following matrix element
\[\bra{k} D(\alpha) R(\theta) S(r) \ket{l} = e^{i \theta l } \bra{k} D(\alpha) S(r e^{2i \theta}) \ket{l} = e^{i \theta l} f^*_{l,k}(-r,-2\theta,-\alpha)\]Attributes
Returns a copy of the gate with the self.dagger flag flipped.
Extra dependencies due to parameters that depend on measurements.
-
H
¶ Returns a copy of the gate with the self.dagger flag flipped.
H stands for hermitian conjugate.
- Returns
formal inverse of this gate
- Return type
Gate
-
measurement_deps
¶ Extra dependencies due to parameters that depend on measurements.
- Returns
dependencies
- Return type
set[RegRef]
-
ns
= 1¶
Methods
apply
(reg, backend, **kwargs)Ask a backend to execute the operation on the current register state right away.
decompose
(reg, **kwargs)Decompose the operation into elementary operations supported by the backend API.
merge
(other)Merge the operation with another (acting on the exact same set of subsystems).
-
apply
(reg, backend, **kwargs)¶ Ask a backend to execute the operation on the current register state right away.
Like
Operation.apply()
, but takes into account the special nature of p[0] and applies self.dagger.- Returns
Gates do not return anything, return value is None
- Return type
None
-
decompose
(reg, **kwargs)¶ Decompose the operation into elementary operations supported by the backend API.
Like
Operation.decompose()
, but applies self.dagger.
-
merge
(other)¶ Merge the operation with another (acting on the exact same set of subsystems).
Note
For subclass overrides: merge may return a newly created object, or self, or other, but it must never modify self or other because the same Operation objects may be also used elsewhere.
- Parameters
other (Operation) – operation to merge this one with
- Returns
other * self. The return value None represents the identity gate (doing nothing).
- Return type
Operation, None
- Raises
MergeFailure – if the two operations cannot be merged