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Gaussian Boson Sampling

The term Gaussian Boson Sampling (GBS) essentially refers to photon number detection in Gaussian states. The most prominent case is the particle-resolved GBS, where the concrete particle numbers are observed during the measurement via a particle number-resolving photon detector. There is theoretical evidence that it is intractable to simulate this sampling problem on a classical computer, making it a promising quantum advantage scheme with several experimental demonstrations in recent years. Through the hafnian, GBS is connected to certain graph-theoretical problems.

For a Gaussian input state \(\rho\) without displacement (i.e., \(\mu \equiv 0\)), the probability of an detecting a sample \(T = (T_1, \dots, T_d)\) during particle number resolved photon detection is given by [1]

\[p(T) = \bra{T} \rho \ket{T} = \frac{1}{\sqrt{~\det Q}} \frac{ \operatorname{haf} \left( \operatorname{br}_{T} A \right) }{T_1! \dots T_d!}.\]

In this formula, \(\operatorname{br}_{T}\) is the block reduction according to Definition~:nbsphinx-math:ref{def:block_reduction} from Appendix~:nbsphinx-math:ref{app:red} and

\[\begin{split}A &= \begin{bmatrix} & I \\ I & \\ \end{bmatrix} \left(I - Q^{-1}\right), \\ Q &= \Sigma + \frac{1}{2} I, \\ \Sigma &= W \sigma W^\dagger,\end{split}\]

where \(W\) is the coordinate transformation from between the ladder and quadrature operators:

\[\mathbf{\xi} = W \mathbf{\chi},\]

where the operators are collected in a vector as

\[\begin{split}\mathbf{\xi} &= [ a_1, \dots, a_d, a_1^\dagger, \dots, a_d^\dagger ]^T, \\ \mathbf{\chi} &= \left [ \hat{x}_1, \dots, \hat{x}_d, \hat{p}_1, \dots, \hat{p}_d \right]^T, \\ W &= \frac{1}{\sqrt{2}} \begin{bmatrix} I & iI \\ I & -iI \end{bmatrix}.\end{split}\]

Furthermore, the hafnian of a matrix is defined by

\[\operatorname{haf}(A) = \sum_{M \in \operatorname{PMP}(n)} \prod_{(i,j) \in M} A_{i, j},\]

where \(\operatorname{PMP}(n)\) is the set of perfect matching permutations of \(n\) even elements, such that \(\sigma(2i-1)<\sigma(2i)\) and \(\sigma(2i-1)<\sigma(2i+1)\), where \(\sigma\) is a permutation.

For displaced Gaussian states, one needs to use a different formula accounting for the non-zero displacement vector:

\[p(T) = \bra{T} \rho \ket{T} = \frac{\exp\left( -\frac{1}{2} \mathbf{\gamma}^T \mathbf{\alpha} \right)}{\sqrt{\operatorname{det} Q}} \frac{ \operatorname{lhaf} \left( \operatorname{vid}(\operatorname{br}_{T} A, \operatorname{br}_{T} \mathbf{\gamma}) \right) }{T_1! \dots T_d!},\]

where \(\operatorname{vid}\) (vector in diagonal) denotes putting the second (vector) argument in the diagonal of the first (matrix) argument, \(\mathbf{\alpha} = W \mu\) and \(\mathbf{\gamma}^T = \mathbf{\alpha}^\dagger \Sigma^{-1}\). The loop hafnian \(\operatorname{lhaf}\) is defined similarly as the hafnian, but the concept of single pair matchings (SPM) is used instead of perfect matching permutations:

\[\operatorname{lhaf}(A) = \sum_{M \in \operatorname{SPM}(n)} \prod_{(i,j) \in M} A_{i, j}.\]

Here, the definition of single-pair matchings is almost the same as the definition of perfect matching permutations, the difference is that when enumerating single-pair matchings, single vertices are also allowed to be paired up with themselves.

Although particle-resolved GBS is the most typical variant, it is often difficult to realize experimentally due to detector inefficiencies [2]. One can easily specialize towards a coarser detection scheme by using threshold detectors instead, which do not resolve exactly how many particles are detected, and only signals detection or no detection. In particular, this variant of GBS, called threshold GBS, was used in several quantum advantage experiments in recent years [3] [4].

More specifically, let output \(1\) correspond to “click”, and \(0\) correspond to “no click”. In the case of threshold measurement of a non-displaced (\(\mu \equiv 0\)) Gaussian state, the probability of an output \(T = (T_1, \dots, T_d) \in {\{0, 1 \}}^d\) during detection is given by [5]

\[p(T) = \frac{ \operatorname{tor} ( \operatorname{br}_{T} A ) }{T_1! \dots T_d! \sqrt{\det Q}},\]

where \(A\) is defined as

\[A = I - Q^{-1},\]

and the torontonian \(\operatorname{tor}\) of a matrix \(M \in \mathbb{C}^{2n \times 2n}\) is

\[\operatorname{tor}(M) = \sum_{X \in \{ 0, 1 \}^{n} } \frac{ {(-1)}^{n - \sum_{i=1}^n X_i } }{ \sqrt{ |\det(I-\operatorname{br}_{X} M)| } }.\]

When the Gaussian state is displaced with complex displacement vector \(\mathbf{\alpha}\), the probability distribution is modified as [6]

\[p(T) = \frac{\exp(-\frac{1}{2}\mathbf{\alpha}^\dagger\Sigma^{-1}\mathbf{\alpha} )}{\sqrt{\operatorname{det}Q}} \operatorname{ltor}( \operatorname{br}_T A, \operatorname{br}_T \mathbf{\gamma}),\]

where $:nbsphinx-math:mathbf{gamma} = \overline{(\Sigma^{-1} \mathbf{\alpha})} $ and \(\operatorname{ltor}\) is the loop torontonian defined by

\[\operatorname{ltor}(M, \mathbf{\gamma}) = \sum_{X \in \{ 0, 1 \}^{n} } \frac{ (-1)^{ n -\sum_{i=1}^n X_i } }{ \sqrt{ |\operatorname{det}(I-\operatorname{br}_X M)|} } \exp( \frac{1}{2} \mathbf{\gamma}^T (I - \operatorname{br}_X M)^{-1} \overline{\mathbf{\gamma}} ).\]

Particle-resolved Gaussian Boson Sampling can be performed using Piquasso as shown by the following example:

import piquasso as pq
import numpy as np

from scipy.stats import unitary_group


shots = 1000
squeezings = np.concatenate([0.25 * np.ones(5), np.zeros(5)])
d = len(squeezings)
unitary = unitary_group.rvs(d)

with pq.Program() as program:
    for i in range(d):
        pq.Q(i) | pq.Squeezing(squeezings[i])

    pq.Q() | pq.Interferometer(unitary)
    pq.Q() | pq.ParticleNumberMeasurement()

simulator = pq.GaussianSimulator(d=d)

result = simulator.execute(program, shots=shots)

print(result.samples)

[[0 0 0 ... 0 0 0]
 [0 0 0 ... 0 0 0]
 [0 0 0 ... 0 0 0]
 ...
 [0 0 0 ... 0 0 0]
 [0 0 0 ... 0 0 0]
 [0 0 0 ... 0 0 0]]

Graph adjacency matrices can also be embedded:

A = np.array(
    [
        [0, 1, 0, 1, 1],
        [1, 0, 0, 0, 1],
        [0, 0, 0, 1, 0],
        [1, 0, 1, 0, 1],
        [1, 1, 0, 1, 0],
    ]
)

with pq.Program() as program:
    pq.Q(all) | pq.Graph(A, mean_photon_number=1.0)

    pq.Q(all) | pq.ParticleNumberMeasurement()


simulator = pq.GaussianSimulator(d=len(A))

result = simulator.execute(program, shots=shots)

print("Samples:", result.samples)

Samples: [[1 0 0 2 3]
 [1 1 0 1 1]
 [2 0 0 2 2]
 ...
 [0 0 0 0 0]
 [3 1 0 1 1]
 [0 0 0 0 0]]

The generated samples could be used to acquire some denser subgraphs of the generated graph with

print("Subgraphs:", result.to_subgraph_nodes())

Subgraphs: [[0, 3, 3, 4, 4, 4], [0, 1, 3, 4], [0, 0, 3, 3, 4, 4], [0, 0, 0, 1, 3, 4, 4, 4], [], [0, 2, 2, 3, 3, 3], [0, 0, 0, 3, 4, 4], [0, 4], [3, 4], [0, 3, 3, 4], [0, 3, 4, 4], [0, 1, 3, 3, 4, 4], [], [2, 3], [], [0, 0, 1, 4, 4, 4], [], [0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4], [0, 4], [], [0, 0, 0, 1, 1, 1, 3, 4], [0, 0, 0, 3, 3, 4], [0, 0, 0, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4], [2, 3], [0, 0, 0, 0, 1, 1, 3, 4, 4, 4], [], [], [3, 4], [0, 0, 0, 0, 1, 2, 3, 3, 3, 4], [0, 0, 1, 1, 1, 1, 4, 4, 4, 4], [0, 0, 0, 2, 3, 3, 3, 4], [], [], [], [0, 3, 4, 4], [0, 3, 3, 4, 4, 4], [0, 1], [0, 0, 0, 1, 1, 1, 1, 3, 3, 4, 4, 4], [0, 0, 0, 1, 2, 3, 3, 3, 3, 4], [], [2, 3], [], [0, 0, 0, 1, 1, 2, 3, 3, 3, 3, 4, 4, 4, 4], [0, 4], [], [], [0, 1], [], [], [0, 2, 3, 3, 3, 4], [0, 0, 0, 0, 1, 1, 1, 3, 4, 4, 4, 4], [1, 4], [], [2, 3], [0, 2, 3, 4], [], [0, 0, 0, 1, 3, 4], [0, 0, 0, 3, 3, 4, 4, 4], [0, 0, 0, 0, 1, 1, 1, 1, 3, 4], [], [0, 0, 0, 3, 4, 4], [0, 0, 0, 0, 1, 3, 3, 3, 3, 4, 4, 4], [3, 4], [0, 1, 3, 4], [3, 4], [0, 3], [0, 0, 0, 0, 1, 3, 3, 4, 4, 4], [0, 3], [0, 3], [0, 0, 0, 1, 1, 3, 4, 4, 4, 4], [0, 1, 3, 4], [0, 0, 0, 0, 1, 1, 2, 3, 3, 3, 3, 4], [], [0, 0, 0, 1, 1, 4], [], [1, 1, 2, 3, 3, 4, 4, 4], [0, 2, 3, 3], [], [0, 0, 0, 1, 1, 2, 3, 3, 3, 3, 4, 4], [0, 0, 0, 1, 1, 1, 3, 4, 4, 4], [0, 0, 0, 0, 1, 1, 1, 2, 3, 4], [0, 0, 0, 3, 3, 4], [0, 0, 0, 1, 1, 2, 3, 3, 3, 3, 4, 4, 4, 4], [], [0, 0, 0, 2, 2, 3, 3, 3, 3, 4, 4, 4], [0, 0, 3, 4], [0, 0, 1, 3, 3, 4, 4, 4], [0, 0, 0, 0, 1, 1, 1, 1, 3, 3, 3, 4, 4, 4], [], [], [0, 0, 0, 3, 3, 4, 4, 4], [], [], [0, 1], [0, 0, 0, 1, 1, 3, 3, 4, 4, 4], [0, 0, 0, 2, 3, 3, 4, 4, 4, 4], [], [0, 4], [0, 0, 0, 0, 1, 1, 1, 3, 3, 3, 4, 4, 4, 4], [0, 1, 3, 4], [0, 3, 4, 4], [0, 0, 0, 1, 1, 1, 1, 3, 4, 4, 4, 4], [], [0, 0, 3, 4, 4, 4], [0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 3], [], [0, 3, 3, 4, 4, 4], [], [], [], [0, 1, 1, 3, 4, 4], [0, 0, 1, 2, 3, 3, 4, 4], [], [0, 1, 3, 4], [0, 0, 1, 3, 3, 4, 4, 4], [], [1, 4], [], [], [3, 3, 4, 4], [0, 0, 0, 1, 3, 3, 4, 4], [0, 0, 0, 1, 1, 1], [], [3, 4], [0, 1], [], [0, 0, 0, 1, 3, 3, 3, 4], [2, 3], [1, 3, 4, 4], [2, 3], [0, 0, 0, 1, 1, 3, 4, 4], [], [2, 3], [0, 0, 1, 3, 4, 4], [], [], [], [], [0, 0, 2, 2, 2, 3, 3, 3, 3, 4], [1, 2, 3, 4], [0, 1], [1, 1, 4, 4], [3, 4], [0, 0, 1, 4], [], [2, 3], [0, 2, 2, 3, 3, 3], [0, 0, 2, 3, 3, 3], [0, 0, 0, 1, 1, 2, 3, 3, 3, 4, 4, 4], [0, 0, 1, 3], [], [0, 0, 0, 1, 2, 3, 3, 3, 3, 4], [0, 2, 2, 3, 3, 3, 3, 4], [2, 2, 3, 3, 3, 4], [0, 1, 3, 4], [], [], [0, 1, 3, 4, 4, 4], [0, 4], [], [0, 0, 0, 1, 3, 3, 4, 4], [0, 0, 1, 3, 3, 4], [0, 2, 3, 4], [0, 0, 1, 2, 3, 3, 4, 4, 4, 4], [1, 2, 3, 4], [], [0, 0, 1, 3, 3, 4], [0, 4], [2, 3], [], [0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4], [], [], [0, 1], [2, 3], [0, 0, 1, 1, 2, 3, 3, 4], [3, 4], [], [0, 0, 3, 4], [], [], [0, 0, 2, 3, 3, 3], [0, 3], [0, 3], [0, 2, 3, 3], [3, 4], [0, 4], [0, 0, 0, 2, 2, 3, 3, 3, 4, 4], [2, 3], [0, 3], [0, 1, 1, 3, 3, 4, 4, 4], [1, 4], [0, 4], [0, 0, 0, 0, 1, 2, 3, 3, 3, 3, 4, 4], [0, 1, 2, 3], [0, 0, 0, 0, 1, 2, 3, 3, 3, 4, 4, 4], [0, 0, 0, 0, 1, 3, 3, 3, 4, 4, 4, 4], [0, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4], [3, 3, 4, 4], [1, 2, 3, 3, 4, 4], [0, 0, 0, 3, 3, 4], [0, 0, 0, 0, 1, 1, 1, 3, 3, 3, 4, 4, 4, 4], [], [], [], [0, 0, 0, 0, 1, 2, 3, 3, 3, 3, 4, 4], [0, 0, 0, 0, 1, 1, 3, 3, 3, 4, 4, 4], [], [0, 0, 0, 3, 3, 4, 4, 4], [0, 0, 3, 4], [0, 1], [], [0, 0, 1, 3, 3, 3, 3, 4, 4, 4], [0, 0, 0, 1, 1, 3, 4, 4], [], [], [], [3, 4], [0, 0, 1, 3], [0, 1, 1, 2, 3, 3, 3, 3, 4, 4, 4, 4], [0, 0, 1, 1, 3, 4], [], [1, 3, 4, 4], [0, 0, 0, 1, 2, 3, 3, 3, 4, 4], [0, 0, 1, 4, 4, 4], [0, 3, 3, 4], [0, 4], [], [0, 2, 3, 3, 4, 4], [0, 0, 1, 3], [0, 0, 0, 3, 3, 3, 4, 4], [0, 0, 1, 1, 4, 4], [0, 0, 1, 1, 1, 2, 3, 3, 3, 4, 4, 4], [1, 2, 3, 4], [0, 2, 3, 3], [], [], [0, 0, 0, 2, 2, 3, 3, 3, 3, 4, 4, 4], [0, 0, 1, 3], [0, 4], [0, 4], [0, 0, 4, 4], [0, 0, 0, 1, 1, 4, 4, 4], [0, 4], [0, 3, 3, 3, 4, 4], [0, 0, 0, 0, 1, 1, 2, 3, 3, 3, 4, 4], [0, 0, 3, 4], [0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 4, 4, 4], [0, 0, 1, 1, 3, 3, 4, 4], [2, 3, 3, 4], [], [0, 0, 1, 2, 3, 4], [], [0, 0, 1, 3], [0, 0, 0, 0, 1, 3, 4, 4], [], [0, 0, 3, 4, 4, 4], [3, 3, 4, 4], [], [0, 2, 3, 3, 3, 4, 4, 4], [2, 3], [0, 0, 4, 4], [], [1, 4], [], [0, 1], [0, 0, 1, 1, 2, 3, 3, 3, 3, 4, 4, 4], [], [3, 4], [], [], [0, 0, 0, 1, 3, 3, 3, 4, 4, 4], [], [], [0, 0, 3, 4], [], [0, 0, 1, 3, 4, 4], [0, 0, 1, 4], [], [], [0, 0, 0, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4], [0, 0, 1, 2, 2, 3, 3, 3, 4, 4], [0, 1, 1, 3, 4, 4, 4, 4], [], [1, 4], [2, 2, 2, 3, 3, 3], [0, 1, 1, 4], [0, 1, 2, 3, 3, 3, 4, 4], [0, 3], [2, 3], [3, 4], [0, 0, 1, 1, 1, 4, 4, 4], [0, 1, 4, 4], [1, 3, 4, 4], [0, 0, 1, 1, 3, 4, 4, 4], [0, 0, 1, 1], [0, 2, 3, 3], [0, 1], [0, 2, 2, 2, 3, 3, 3, 3], [0, 0, 2, 3, 3, 3, 4, 4], [0, 3], [], [0, 3], [0, 1, 3, 4, 4, 4], [], [0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4], [0, 0, 0, 1, 1, 1, 4, 4], [0, 0, 3, 3, 3, 4], [0, 0, 1, 4, 4, 4], [0, 0, 0, 0, 1, 1, 1, 3, 3, 3, 4, 4, 4, 4], [0, 3], [3, 4], [], [], [], [], [0, 1, 4, 4], [3, 4], [0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 4, 4, 4], [0, 1, 2, 2, 3, 3, 3, 4], [], [], [], [0, 4], [0, 1], [], [], [3, 4], [0, 0, 0, 0, 1, 1, 3, 3, 4, 4], [], [0, 3], [0, 0, 0, 3, 4, 4], [], [0, 0, 1, 1, 2, 3, 3, 4], [2, 3, 3, 4], [0, 3], [0, 3], [0, 0, 0, 1, 1, 1, 4, 4], [], [0, 0, 0, 1, 1, 1, 2, 3, 3, 3, 4, 4, 4, 4], [], [0, 4], [0, 0, 0, 1, 3, 3, 3, 4, 4, 4], [2, 3, 3, 4], [0, 1], [0, 4], [1, 4], [], [0, 0, 0, 1, 2, 3, 3, 3, 4, 4], [], [], [0, 0, 1, 3, 3, 4, 4, 4], [3, 4], [], [0, 1, 2, 3, 3, 4], [2, 3], [], [], [1, 4], [0, 0, 1, 1], [], [], [0, 4], [3, 4], [0, 0, 1, 3], [0, 2, 3, 3, 4, 4], [0, 4], [0, 0, 0, 1, 1, 3, 4, 4], [], [0, 0, 0, 0, 1, 3, 3, 3, 4, 4, 4, 4], [], [3, 3, 4, 4], [], [0, 0, 1, 1, 3, 3, 3, 3, 4, 4, 4, 4], [0, 0, 2, 3, 4, 4], [], [0, 0, 0, 1, 3, 3], [0, 0, 0, 2, 3, 3, 3, 4, 4, 4], [], [], [0, 0, 0, 1, 3, 3, 3, 3, 4, 4], [1, 1, 4, 4], [], [], [0, 2, 2, 2, 3, 3, 3, 3], [0, 1, 1, 3, 4, 4], [0, 2, 3, 3], [0, 0, 3, 4], [0, 0, 1, 3, 3, 4], [0, 0, 0, 1, 1, 1, 3, 3, 4, 4, 4, 4], [], [0, 1, 3, 4], [0, 1, 3, 4], [], [], [3, 4], [], [1, 4], [2, 3, 3, 4], [0, 0, 1, 1, 2, 2, 3, 3, 3, 4], [0, 4], [], [0, 0, 1, 4], [0, 0, 0, 1, 4, 4], [], [0, 3], [0, 0, 0, 0, 1, 1, 1, 3], [], [2, 3], [0, 0, 1, 1, 3, 4], [0, 0, 1, 1, 3, 3, 4, 4], [0, 0, 1, 4], [], [0, 2, 3, 4], [0, 1, 3, 4], [0, 0, 2, 2, 2, 3, 3, 3, 3, 4], [], [], [], [2, 3], [3, 3, 4, 4], [0, 4], [2, 2, 3, 3], [0, 2, 3, 3], [3, 4], [0, 3], [1, 2, 2, 3, 3, 3, 3, 4, 4, 4], [], [0, 0, 1, 1, 3, 4], [0, 0, 3, 3, 4, 4], [], [0, 0, 1, 1, 1, 3, 3, 3, 4, 4, 4, 4], [], [], [0, 1, 1, 1, 1, 3, 4, 4, 4, 4], [0, 0, 0, 1, 3, 4], [0, 0, 0, 0, 1, 1, 1, 1, 3, 3, 4, 4, 4, 4], [], [2, 3], [3, 4], [0, 0, 0, 0, 1, 1, 2, 3, 3, 3, 3, 4, 4, 4], [0, 0, 1, 3, 4, 4], [], [0, 0, 0, 1, 3, 4, 4, 4], [0, 1], [0, 0, 3, 3, 3, 4, 4, 4], [1, 1, 3, 3, 4, 4, 4, 4], [0, 4], [0, 1], [], [2, 3], [], [], [0, 3], [], [], [], [], [0, 0, 1, 2, 2, 3, 3, 3, 3, 4], [], [], [], [0, 0, 3, 4], [], [2, 3], [2, 3], [0, 1, 1, 3, 4, 4], [2, 3], [0, 4], [], [0, 1, 4, 4], [], [0, 0, 0, 0, 1, 1, 3, 3, 3, 4, 4, 4], [0, 0, 0, 0, 1, 3, 3, 3, 4, 4], [0, 4], [2, 2, 3, 3], [], [], [0, 0, 0, 1, 1, 3, 3, 3, 4, 4, 4, 4], [], [1, 1, 4, 4], [], [], [0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4], [0, 0, 1, 3, 3, 4, 4, 4], [2, 3, 3, 4], [0, 1, 3, 4], [0, 0, 0, 1, 3, 3, 4, 4], [], [0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4], [], [3, 4], [0, 0, 0, 1, 3, 3], [0, 1, 1, 4], [], [0, 1, 3, 3, 3, 4, 4, 4], [0, 0, 1, 1, 1, 2, 3, 3, 4, 4, 4, 4], [0, 0, 3, 4], [2, 3, 3, 4], [0, 0, 0, 1, 1, 2, 3, 3, 3, 4, 4, 4], [3, 3, 4, 4], [0, 0, 0, 1, 2, 3, 3, 3, 3, 4, 4, 4], [0, 0, 0, 1, 3, 3], [0, 0, 0, 0, 1, 1, 3, 3, 3, 3, 4, 4, 4, 4], [], [], [], [], [3, 3, 4, 4], [], [], [], [0, 4], [], [0, 0, 1, 1, 1, 3, 4, 4], [0, 0, 0, 0, 1, 1, 1, 1, 3, 4], [0, 0, 3, 3], [0, 0, 0, 0, 1, 1, 1, 3, 3, 4], [0, 0, 3, 4, 4, 4], [1, 4], [], [2, 3], [], [0, 0, 0, 1, 2, 3, 3, 3, 4, 4, 4, 4], [0, 1, 4, 4], [], [1, 3, 3, 4, 4, 4], [0, 1, 2, 2, 3, 3, 3, 4], [], [0, 0, 0, 1, 1, 3, 4, 4], [2, 3, 3, 3, 3, 4, 4, 4], [2, 3, 3, 4], [], [0, 3, 4, 4], [], [], [0, 0, 0, 0, 1, 1, 3, 3, 3, 4, 4, 4], [0, 0, 0, 3, 3, 4], [], [1, 4], [0, 4], [0, 0, 0, 1, 3, 3, 3, 4, 4, 4], [0, 0, 2, 3, 3, 4], [0, 0, 3, 3, 4, 4], [], [], [0, 0, 1, 1, 1, 1, 4, 4], [], [0, 0, 1, 2, 3, 3, 3, 3, 4, 4, 4, 4], [2, 3], [], [1, 4], [0, 3, 3, 4, 4, 4], [], [], [0, 0, 0, 0, 1, 1, 1, 1, 3, 4], [0, 3, 4, 4], [], [0, 1, 4, 4], [0, 1, 1, 4], [0, 0, 1, 1, 3, 3, 4, 4, 4, 4], [0, 3], [2, 2, 2, 3, 3, 3], [], [1, 1, 4, 4], [0, 3, 4, 4], [0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 4, 4, 4, 4], [0, 3, 4, 4], [], [], [], [], [0, 1], [0, 0, 0, 1, 2, 3, 3, 4, 4, 4], [2, 3], [], [0, 0, 0, 2, 3, 3, 4, 4], [], [], [0, 1, 3, 4], [0, 3], [1, 2, 3, 3, 4, 4], [], [0, 0, 0, 0, 1, 2, 3, 3, 3, 3, 4, 4], [], [], [2, 3], [], [0, 4], [], [0, 0, 1, 3, 3, 4, 4, 4], [3, 4], [0, 0, 0, 1, 3, 3, 3, 3, 4, 4, 4, 4], [3, 4], [], [], [3, 4], [0, 0, 0, 0, 1, 2, 3, 3, 3, 4, 4, 4], [], [], [0, 1], [], [0, 0, 0, 1, 1, 1, 3, 3, 3, 4, 4, 4], [], [0, 4], [], [0, 0, 0, 0, 2, 3, 3, 3, 4, 4, 4, 4], [0, 3, 4, 4], [3, 4], [2, 3], [0, 1], [], [0, 1, 1, 2, 3, 4], [0, 0, 0, 0, 1, 1, 1, 2, 3, 3, 3, 3, 4, 4], [0, 1, 3, 4], [0, 0, 0, 1, 1, 1, 1, 3, 3, 4, 4, 4], [], [0, 0, 0, 1, 1, 3], [], [0, 1, 3, 4, 4, 4], [0, 0, 0, 1, 3, 3, 4, 4], [], [0, 0, 0, 1, 2, 3, 3, 3, 4, 4], [1, 4], [1, 4], [], [2, 3], [0, 0, 1, 2, 3, 3, 4, 4], [1, 4], [0, 1, 1, 2, 3, 3, 3, 4, 4, 4], [], [], [], [0, 3], [], [0, 1, 3, 4], [], [0, 0, 0, 0, 1, 3, 3, 4], [0, 0, 1, 1], [0, 1], [], [], [0, 4], [1, 3, 4, 4], [], [2, 3], [0, 0, 1, 3, 4, 4, 4, 4], [0, 4], [0, 2, 3, 3, 3, 4], [1, 2, 3, 4], [0, 0, 3, 3, 4, 4], [], [0, 4], [0, 1, 1, 3, 4, 4], [], [], [0, 0, 0, 0, 1, 1, 1, 2, 3, 3, 3, 4, 4, 4], [], [0, 0, 1, 3, 4, 4], [], [0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4], [], [0, 0, 1, 1, 3, 4], [0, 0, 0, 1, 1, 1, 4, 4], [0, 1], [0, 3, 3, 4], [3, 4], [0, 1, 3, 4], [], [1, 4], [1, 4], [], [0, 0, 0, 0, 1, 2, 3, 3, 3, 3, 4, 4], [], [0, 0, 0, 1, 1, 1, 1, 3, 3, 4, 4, 4], [], [0, 0, 0, 1, 3, 4, 4, 4], [], [3, 3, 4, 4], [1, 4], [0, 0, 1, 1, 1, 3, 4, 4, 4, 4], [0, 0, 0, 1, 1, 3, 4, 4], [3, 4], [0, 0, 1, 1], [0, 1, 3, 4], [0, 3], [], [], [], [], [], [0, 3, 4, 4], [0, 3, 4, 4], [0, 0, 0, 4, 4, 4], [0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 3, 4, 4, 4], [0, 4], [], [0, 0, 0, 0, 1, 1, 2, 3, 3, 3], [0, 0, 1, 2, 3, 3, 3, 4, 4, 4], [], [0, 0, 1, 3, 4, 4], [], [0, 0, 0, 0, 1, 1, 1, 1, 3, 3, 3, 3, 4, 4, 4, 4], [2, 2, 3, 3], [0, 1, 4, 4], [0, 0, 0, 1, 1, 1, 2, 3, 3, 4], [0, 0, 0, 1, 1, 1, 3, 4, 4, 4], [0, 0, 0, 1, 1, 1, 3, 4, 4, 4], [0, 3], [0, 0, 0, 1, 3, 4, 4, 4], [1, 3, 4, 4], [0, 3], [], [], [], [1, 3, 3, 3, 4, 4, 4, 4], [0, 1], [0, 0, 4, 4], [], [], [2, 3], [0, 3], [0, 1], [2, 3], [], [], [1, 3, 4, 4], [0, 1, 2, 2, 2, 3, 3, 3, 3, 4], [3, 4], [], [0, 1, 3, 4], [0, 0, 0, 0, 1, 3, 3, 4, 4, 4], [0, 0, 1, 1, 1, 2, 3, 3, 4, 4], [], [], [0, 0, 0, 2, 3, 3, 4, 4], [0, 0, 0, 0, 1, 1, 1, 3, 4, 4], [], [], [1, 4], [], [0, 4], [0, 1, 2, 3, 3, 4], [], [0, 1], [], [0, 0, 0, 0, 1, 1, 3, 4], [0, 0, 1, 3, 3, 4], [0, 0, 1, 4], [2, 3], [0, 3], [0, 0, 1, 4], [2, 3], [0, 1, 2, 3, 3, 3, 4, 4], [], [], [0, 1], [0, 0, 1, 1, 3, 4], [], [0, 4], [], [], [0, 2, 2, 3, 3, 3], [], [0, 1, 3, 4, 4, 4], [2, 3], [2, 2, 3, 3], [], [3, 4], [1, 4], [], [0, 4], [1, 4], [], [1, 4], [0, 1, 1, 4, 4, 4], [], [0, 1, 3, 3, 4, 4], [3, 4], [0, 4], [0, 3, 3, 4], [0, 3], [0, 0, 0, 0, 1, 3, 3, 4], [0, 1, 3, 4], [0, 0, 0, 0, 3, 3, 3, 3, 4, 4], [0, 2, 3, 3, 3, 3, 4, 4], [0, 0, 0, 0, 1, 1, 1, 2, 3, 3, 3, 4, 4, 4], [2, 2, 3, 3], [0, 0, 0, 0, 1, 1, 1, 3, 4, 4], [0, 1, 2, 3, 3, 3, 4, 4], [1, 4], [0, 0, 0, 1, 1, 1, 1, 3, 3, 3, 4, 4, 4, 4], [], [0, 1], [], [], [0, 1], [], [], [], [0, 3], [2, 3], [], [], [0, 4], [], [0, 1, 2, 3, 3, 4, 4, 4], [0, 1, 2, 3], [0, 0, 1, 4], [0, 1, 1, 4], [0, 3], [], [0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4], [0, 0, 0, 0, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4], [0, 0, 1, 3, 3, 4, 4, 4], [0, 1, 1, 3, 4, 4], [0, 2, 3, 4], [2, 3], [0, 1, 3, 3, 4, 4], [0, 1, 3, 3, 3, 3, 4, 4, 4, 4], [], [], [], [0, 0, 3, 3, 3, 4, 4, 4], [0, 0, 0, 1, 1, 3, 3, 4, 4, 4], [], [0, 0, 1, 1], [0, 4], [0, 0, 3, 4], [0, 0, 0, 0, 3, 4, 4, 4], [], [0, 1], [0, 0, 1, 2, 3, 3], [0, 0, 4, 4], [0, 0, 0, 1, 1, 4], [0, 4], [3, 4], [], [2, 3], [0, 3], [0, 1], [0, 0, 0, 0, 1, 1, 2, 3, 3, 3, 4, 4, 4, 4], [0, 0, 0, 1, 3, 3, 4, 4], [0, 4], [0, 3], [0, 0, 1, 3], [3, 3, 4, 4], [0, 0, 3, 4], [0, 0, 0, 1, 1, 3, 4, 4], [0, 0, 0, 0, 3, 3, 4, 4], [0, 1, 4, 4], [0, 0, 0, 2, 3, 3, 3, 3, 4, 4], [0, 1, 1, 2, 3, 4], [0, 0, 0, 0, 1, 3, 3, 3, 4, 4, 4, 4], [], [0, 1], [0, 0, 1, 2, 3, 4], [2, 3, 3, 3, 4, 4], [1, 2, 3, 4], [0, 0, 0, 1, 3, 3], [0, 0, 0, 1, 1, 2, 3, 3, 3, 4, 4, 4], [0, 0, 0, 0, 2, 2, 3, 3, 3, 3, 4, 4], [0, 0, 0, 0, 1, 1, 3, 4], [2, 2, 3, 3], [], [], [0, 0, 1, 1, 3, 4, 4, 4], [2, 2, 3, 3], [2, 3], [0, 0, 0, 3, 3, 4], [0, 0, 0, 2, 3, 3, 4, 4], [], [0, 0, 1, 1, 4, 4], [0, 0, 0, 1, 3, 3, 4, 4], [0, 1, 3, 3, 4, 4], [1, 3, 4, 4], [0, 4], [], [], [0, 0, 0, 1, 2, 3, 3, 4, 4, 4], [], [0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4], [0, 0, 0, 0, 1, 1, 1, 4, 4, 4], [0, 3], [], [0, 1, 3, 4, 4, 4], [0, 0, 2, 2, 3, 3, 3, 3], [], [], [], [2, 3], [], [1, 4], [0, 4], [0, 0, 1, 1], [0, 3, 4, 4], [0, 1, 3, 3, 4, 4], [1, 4], [0, 0, 1, 1, 3, 3, 3, 4, 4, 4], [], [0, 0, 1, 3, 3, 4, 4, 4], [2, 3, 3, 4], [0, 0, 0, 1, 1, 1, 3, 3, 4, 4, 4, 4], [], [1, 4], [1, 4], [], [0, 0, 1, 3], [0, 0, 0, 1, 1, 1, 3, 4, 4, 4], [], [], [0, 0, 3, 3], [3, 4], [0, 0, 0, 0, 1, 3, 4, 4, 4, 4], [], [0, 0, 0, 1, 3, 3, 3, 4], [0, 0, 0, 1, 1, 3, 3, 3, 4, 4, 4, 4], [0, 0, 1, 3, 3, 4], [], [], [0, 0, 0, 0, 1, 1, 1, 3, 4, 4], [2, 3], [0, 1, 2, 2, 3, 3], [0, 1], [], [0, 0, 0, 2, 3, 3, 3, 3, 4, 4], [], [0, 3], [0, 0, 3, 3, 4, 4], [0, 0, 0, 1, 3, 4, 4, 4], [], [0, 2, 2, 3, 3, 3, 3, 4], [2, 3], [], [0, 0, 0, 0, 1, 1, 1, 2, 3, 3, 3, 3, 4, 4, 4, 4], [0, 0, 1, 3, 4, 4], [], [], [2, 3, 3, 4], [3, 4], [], [0, 3], [0, 1], [1, 4], [], [], [0, 1, 1, 3, 4, 4], [], [], [0, 3, 4, 4], [0, 1, 1, 2, 3, 3, 4, 4], [], [], [0, 0, 0, 1, 2, 3, 3, 4], [0, 1, 3, 4], [0, 0, 1, 1, 1, 3, 4, 4], [0, 2, 3, 3], [0, 0, 1, 1, 2, 3, 3, 3, 4, 4, 4, 4], [0, 0, 1, 2, 3, 3], [2, 3, 3, 4], [], [0, 1, 3, 3, 4, 4], [], [0, 4], [], [0, 0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4], [1, 4], [3, 4], [0, 3], [0, 0, 0, 0, 1, 3, 3, 4, 4, 4], [0, 0, 1, 1, 3, 3, 4, 4], [1, 4], [0, 0, 1, 1, 3, 4], [3, 4], [0, 3], [1, 4], [0, 1], [], [3, 4], [0, 0, 0, 0, 1, 1, 3, 3, 4, 4], [], [0, 4], [0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 4], [], [], [0, 1, 3, 4], [0, 1, 3, 4], [0, 3, 3, 4], [3, 4], [0, 3, 3, 4], [], [], [0, 3, 4, 4], [0, 1], [2, 3], [0, 0, 1, 1, 4, 4], [0, 0, 0, 1, 1, 1, 3, 3, 3, 4, 4, 4], [], [0, 0, 0, 1, 3, 4], []]

One can also implement threshold GBS with ThresholdMeasurement:

with pq.Program() as program:
    for i in range(d):
        pq.Q(i) | pq.Squeezing(squeezings[i])

    pq.Q() | pq.Interferometer(unitary)
    pq.Q() | pq.ThresholdMeasurement()

simulator = pq.GaussianSimulator(d=d)

result = simulator.execute(program, shots=shots)

print(result.samples)

[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 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