# Decompositions¶

Note

In Strawberry Fields we use the convention \(\hbar=2\) by default, but other
conventions can also be chosen by setting the global variable `sf.hbar`

at the beginning of a session.
In this document we keep \(\hbar\) explicit.

Contents

This page contains some common continuous-variable (CV) decompositions. However, before beginning, it is useful to state some important properties of Gaussian unitaries and the symplectic group.

## Gaussian unitaries¶

Gaussian states are those with positive Wigner distributions, and are fully characterised by their first and second moments; the means vector \(\hat{\mathbf{r}}=(\x_1,\dots,\x_N,\p_1,\dots,\p_N)\), and the covariance matrix \(V_{ij}=\frac{1}{2}\langle\Delta r_i\Delta r_j + \Delta r_i\Delta r_j\rangle\) respectively (for more details, see Gaussian States).

Gaussian unitaries, it then follows, are quantum operations that retain the Gaussian character of the state; i.e., the set of unitary transformations \(U\) that transform Gaussian states into Gaussian states:

In the Hilbert space, these are represented by unitaries of the form \(U=e^{-iH/2}\) with Hamiltonians \(H\) that are at most second-order polynomials in the quadrature operators \(\x\) and \(\p\). In order to preserve the commutation relations \([r_i,r_j]=i\hbar\Omega\), where

is the symplectic matrix, the set of Gaussian
unitary transformations \(U\) in the Hilbert space is represented by *real symplectic
transformations* on the first and second moments of the Gaussian state:

Here, \(\mathbf{d}\in\mathbb{R}^{2N}\), and \(S\in\mathbb{R}^{2N\times 2N}\) is a symplectic matrix satisfying the condition \(S\Omega S^T=S\).

## Williamson decomposition¶

Definition

For every positive definite real matrix \(V\in\mathbb{R}^{2N\times 2N}\), there exists a symplectic matrix \(S\) and diagonal matrix \(D\) such that

where \(D=\text{diag}(\nu_1,\dots,\nu_N,\nu_1,\dots,\nu_N)\), and \(\{\nu_i\}\) are the eigenvalues of \(|i\Omega V|\), where \(||\) represents the element-wise absolute value.

Tip

*Implemented in Strawberry Fields as a state preparation decomposition by*
`strawberryfields.ops.Gaussian`

The Williamson decomposition allows an arbitrary Gaussian covariance matrix to be decomposed into a symplectic transformation acting on the state described by the diagonal matrix \(D\).

The matrix \(D\) can always be decomposed further into a set of thermal states with mean photon number given by

### Pure states¶

In the case where \(V\) represents a pure state (\(|V|-(\hbar/2)^{2N}=0\)), the Williamson decomposition outputs \(D=\frac{1}{2}\hbar I_{2N}\); that is, a symplectic transformation \(S\) acting on the vacuum. It follows that the original covariance matrix can therefore be recovered simply via \(V=\frac{\hbar}{2}SS^T\).

Note

\(V\) must be a valid quantum state satisfying the uncertainty principle: \(V+\frac{1}{2}i\hbar\Omega\geq 0\). If this is not the case, the Williamson decomposition will return non-physical thermal states with \(\bar{n}_i<0\).

## Bloch-Messiah (or Euler) decomposition¶

Definition

For every symplectic matrix \(S\in\mathbb{R}^{2N\times 2N}\), there exists orthogonal symplectic matrices \(O_1\) and \(O_2\), and diagonal matrix \(Z\), such that

where \(Z=\text{diag}(e^{-r_1},\dots,e^{-r_N},e^{r_1},\dots,e^{r_N})\) represents a set of one mode squeezing operations with parameters \((r_1,\dots,r_N)\).

Tip

*Implemented in Strawberry Fields as a gate decomposition by*
`strawberryfields.ops.GaussianTransform`

Gaussian symplectic transforms can be grouped into two main types; passive transformations (those which preserve photon number) and active transformations (those which do not). Compared to active transformation, passive transformations have an additional constraint - they must preserve the trace of the covariance matrix, \(\text{Tr}(SVS^T)=\text{Tr}(V)\); this only occurs when the symplectic matrix \(S\) is also orthogonal (\(SS^T=\I\)).

The Bloch-Messiah decomposition therefore allows any active symplectic transformation to be decomposed into two passive Gaussian transformations \(O_1\) and \(O_2\), sandwiching a set of one-mode squeezers, an active transformation.

### Acting on the vacuum¶

In the case where the symplectic matrix \(S\) is applied to a vacuum state \(V=\frac{\hbar}{2}\I\), the action of \(O_2\) cancels out due to its orthogonality:

As such, a symplectic transformation acting on the vacuum is sufficiently characterised by single mode squeezers followed by a passive Gaussian transformation (\(S = O_1 Z\)).

## Rectangular decomposition¶

The rectangular decomposition allows any passive Gaussian transformation to be decomposed into a series of beamsplitters and rotation gates.

Definition

For every real orthogonal symplectic matrix

the corresponding unitary matrix \(U=X+iY\in\mathbb{C}^{N\times N}\) representing a multiport interferometer can be decomposed into a set of \(N(N-1)/2\) beamsplitters and single mode rotations with circuit depth of \(N\).

For more details, see [1].

Tip

*Implemented in Strawberry Fields as a gate decomposition by*
`strawberryfields.ops.Interferometer`

Note

The rectangular decomposition as formulated by Clements [1] uses a different beamsplitter convention to Strawberry Fields:

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