Implement two-mode summing unitary

[Fe-r-oz]

This PR aims to fix #38 by implementing two-mode summing unitary.

Hi, Andrew! Thank you for your help and guidance! I am not sure why I landed to page 123 😅 of https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.88.09790. As you suggestion, I need to follow the information given much earlier in the paper!

The SUM gate can be realized using beamsplitter and squeeze, following the Bloch-Messiah decomposition:

$$\begin{align} \text{SUM}(\lambda) &= \text{BS}(\pi + 2\theta, -\pi/2) \left[ s(r) \otimes s(-r) \right] \text{BS}(2\theta, -\pi/2), \tag{209} \\\ \sinh r &= \frac{\lambda}{2}, \tag{210} \\\ \cos(2\theta) &= \tanh(r), \tag{211} \\\ \sin(2\theta) &= -\text{sech}(r). \tag{212} \end{align}$$

Therefore, we can use the already implemented operators squeeze and beamsplitter and use them to implement the twosumgate. The name follows the convention of similar to twosqueeze. I hope that this attempt is better than the previous attempt at least 😅 beamsplitter appears to not have the parameter to extra phase shift to I have excluded $-\pi/2$

julia> using Gabs;  using Gabs: twosumgate

julia> basis = QuadBlockBasis(2)
QuadBlockBasis(2)

julia> lambda = 1.0
1.0

julia> twosumgate(basis, lambda)
GaussianUnitary for 2 modes.
  symplectic basis: QuadBlockBasis
displacement: 4-element Vector{Float64}:
 0.0
 0.0
 0.0
 0.0
symplectic: 4×4 Matrix{Float64}:
  0.17082   0.894427   0.0       0.0
 -0.894427  1.17082    0.0       0.0
  0.0       0.0        1.17082   0.894427
  0.0       0.0       -0.894427  0.17082

Edit:

I was wondering whether the following can be implemented as a test to check whether the correctness of approach. It seems that excluding extra phase $\phi$ of $\pi$ in $BS(\theta, \phi)$ changes the symplectic matrix 😅. According to 199, the two mode squeeze can be written after Bloch- Messiah decomposition as

$$TMS(r, \frac{\pi}{2}) = BS\left(\frac{\pi}{2}, 0\right) \left[s(r) \otimes s(r)\right] BS\left(\frac{\pi}{2}, \pi\right)$$

which can be written as excluding extra phase $\phi$ of $\pi$ in $BS(\theta, \phi)$:

function _test_twomode_squeeze(basis, r, theta)
    transmit1 = theta/2
    BS1 = beamsplitter(basis, transmit1)
    S1 = squeeze(QuadPairBasis(basis.nmodes - 1), r, 0.0)
    S2 = squeeze(QuadPairBasis(basis.nmodes - 1), r, 0.0)
    S_tensor = S1 ⊗ S2
    transmit2 = theta/2
    BS2 = beamsplitter(basis, transmit2)
    final_symplectic = BS2.symplectic * S_tensor.symplectic * BS1.symplectic
    final_disp = BS2.disp + S_tensor.disp + BS1.disp
    return final_disp, final_symplectic
end

julia> r, theta = rand(Float64), rand(Float64)
(0.2695878972298862, 0.38699466312747455)

julia> basis = QuadPairBasis(2)
QuadPairBasis(2)

julia> _test_twomode_squeeze(basis,r,theta)[1]
4-element Vector{Float64}:
 0.0
 0.0
 0.0
 0.0

julia> _test_twomode_squeeze(basis,r,theta)[2]
4×4 Matrix{Float64}:
  0.468149   0.0       0.603379  0.0
  0.0        0.802684  0.0       1.03455
 -0.603379   0.0       0.468149  0.0
  0.0       -1.03455   0.0       0.802684

julia> twosqueeze(basis, r, theta)
GaussianUnitary for 2 modes.
  symplectic basis: QuadPairBasis
displacement: 4-element Vector{Float64}:
 0.0
 0.0
 0.0
 0.0
symplectic: 4×4 Matrix{Float64}:
  1.03656    0.0       -0.252686  -0.102981
  0.0        1.03656   -0.102981   0.252686
 -0.252686  -0.102981   1.03656    0.0
 -0.102981   0.252686   0.0        1.03656

The current approach appears to be incorrect. I initially thought we could use the Bloch-Messiah decomposition for the two-sum gate and then leverage it to implement the gate. 😅

[codecov]

Codecov Report

Attention: Patch coverage is 36.66667% with 19 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/unitaries.jl	36.66%	19 Missing ⚠️

Files with missing lines	Coverage Δ
src/Gabs.jl	100.00% <ø> (ø)
src/unitaries.jl	95.51% <36.66%> (-4.25%)	⬇️

[Fe-r-oz]

Hi, Andrew! This paper provides (equation 14) the symplectic matrix of SUM gate https://arxiv.org/pdf/2502.07670 which in Gabs convention is in QuadBlockBasis :) I hope this is more relevant than my previous message 😅

The symplectic matrix of the SUM gate in the quadrature basis $(\hat{q}_1,, \hat{q}_2,, \hat{p}_1,, \hat{p})$

$$\mathrm{SUM} = \begin{pmatrix} 1 & 0 & 0 & 0\\[1mm] 1 & 1 & 0 & 0\\[1mm] 0 & 0 & 1 & -1\\[1mm] 0 & 0 & 0 & 1 \end{pmatrix}\, \tag{14}$$

The aforementioned paper cited this paper when providing the symplectic representation in quadrature basis.