Parametrized Gaussian Boson Sampling

• Parametrized Gaussian Boson Sampling is a family of photonic sampling models that employ explicit control over Gaussian states, circuits, and measurement maps to modify output distributions.
• It offers versatility through multiple parameterizations—such as WAW, BipartiteGBS, and threshold-detection—that influence computational complexity via hafnian, loop hafnian, or Torontonian functions.
• Its programmable and trainable frameworks facilitate applications in optimization, generative modeling, and simulation of quantum phenomena with scalable, near-term photonic devices.

Parametrized Gaussian Boson Sampling denotes a family of photonic sampling models in which the Gaussian state, Gaussian circuit, or measurement map is controlled by explicit parameters rather than treated as a fixed boson-sampling primitive. In the standard formulation, Gaussian Boson Sampling (GBS) uses squeezed Gaussian inputs, linear optics, and photon-number measurements, with output probabilities governed by hafnians of matrices derived from the covariance structure of the state (Hamilton et al., 2016). Later work extends this baseline in several directions: programmable transition-matrix embeddings, variational reweightings of GBS distributions, displacement-controlled circuits, threshold-detection models, and photonic generative models with layered Gaussian ansätze (Grier et al., 2021, Banchi et al., 2020, Thekkadath et al., 2022, Cazalis et al., 2024, Kolarovszki et al., 11 Mar 2026). The term is therefore not a single universally fixed abstraction. One paper states this explicitly for its own setting, noting that it does not introduce a free-standing “parametrized GBS” model but rather a programmable generalization, BipartiteGBS (Grier et al., 2021).

1. Standard GBS as the reference model

Standard GBS is the point of departure for all parametrized variants. It replaces the single-photon inputs of Aaronson–Arkhipov boson sampling with Gaussian inputs, especially single-mode squeezed states, followed by a linear interferometer and Fock-basis detection. For a zero-displacement Gaussian state, the output probabilities are expressed through the hafnian; for single-mode squeezed inputs with squeezing parameters $r_j$, the matrix entering the probability law is

$B = T \left(\oplus_{j=1}^{M}\tanh r_j \right )T^t,$

so the sampled matrix depends jointly on the interferometer $T$ and the squeezing profile (Hamilton et al., 2016).

This baseline already contains the main control parameters that later literature treats more explicitly: the number of active squeezed sources $K$, the squeezing strengths, and the interferometer size $M$. In the early Gaussian-state boson-sampling precursor, the analogous control variable was the squeezing parameter $\chi$ of two-mode squeezed vacuum sources, together with the interferometer $\hat U$ and the heralding or output pattern. That work framed Gaussian-state sampling as a generalized boson-sampling problem and identified $\chi$ as the parameter controlling the rate of valid heralded events (Lund et al., 2013).

A second reference point is the broader Gaussian formulation that includes squeezed and displaced Gaussian states. In that setting the state is characterized by its covariance matrix and, when present, its displacement vector. This establishes the later parametrized literature’s central theme: changing state parameters changes not merely rates or calibration constants, but the matrix function governing the output law itself (Kruse et al., 2018).

2. Parameter spaces and mathematical representations

Parametrized GBS appears in the literature through several inequivalent representations of the Gaussian state or circuit. Some papers parameterize the sampling law directly; others parameterize the underlying Gaussian transformation.

Framework	Parameters	Role
Standard GBS	$r_j$, $K$, $B = T \left(\oplus_{j=1}^{M}\tanh r_j \right )T^t,$0, interferometer $B = T \left(\oplus_{j=1}^{M}\tanh r_j \right )T^t,$1	Fixes the matrix $B = T \left(\oplus_{j=1}^{M}\tanh r_j \right )T^t,$2 and photon statistics
WAW parametrization	$B = T \left(\oplus_{j=1}^{M}\tanh r_j \right )T^t,$3, weights $B = T \left(\oplus_{j=1}^{M}\tanh r_j \right )T^t,$4	Makes the trainable part explicit in the distribution
BipartiteGBS	$B = T \left(\oplus_{j=1}^{M}\tanh r_j \right )T^t,$5, scale $B = T \left(\oplus_{j=1}^{M}\tanh r_j \right )T^t,$6	Programs arbitrary transition matrices and collision regime
GBBM	$B = T \left(\oplus_{j=1}^{M}\tanh r_j \right )T^t,$7	Layered parametrized Gaussian circuit for generative modeling
Threshold-VQE ansatz	$B = T \left(\oplus_{j=1}^{M}\tanh r_j \right )T^t,$8 or symmetric $B = T \left(\oplus_{j=1}^{M}\tanh r_j \right )T^t,$9 with $T$0	Variational family for binary optimization

In the WAW construction for trainable GBS distributions, a fixed symmetric matrix $T$1 is dressed by a diagonal matrix $T$2, so that $T$3. The resulting probability law isolates the trainable weights in a multiplicative factor while leaving the difficult hafnian term independent of them, which is the key reason gradients become tractable (Banchi et al., 2020).

In the threshold-detector variational ansatz, the same pure Gaussian state may be parameterized either by circuit data $T$4 or by a Bargmann matrix $T$5 with $T$6, related by

$T$7

This duality between optical and algebraic parameterizations is recurrent in the parametrized GBS literature (Cazalis et al., 2024).

A further distinction concerns the output primitive. Zero-displacement photon-number GBS is governed by the hafnian. For displaced pure Gaussian states, the relevant quantity is the loop hafnian,

$T$8

while threshold-detection GBS replaces the hafnian by the Torontonian (Quesada et al., 2020, Quesada et al., 2018). This is not merely a change of notation: the choice of parameters and detector model changes the underlying combinatorial object.

3. Programmability and physical realizations

One major branch of parametrized GBS is explicitly programmable rather than merely tunable. The clearest example is BipartiteGBS, where an arbitrary $T$9 transition matrix $K$0 is encoded into a $K$1-mode Gaussian state by preparing two-mode squeezed states on pairs $K$2, then applying one interferometer $K$3 to the first half of the modes and another interferometer $K$4 to the second half. The programmed matrix is

$K$5

and the output distribution becomes

$K$6

This construction generalizes Scattershot BosonSampling and Twofold Scattershot BosonSampling, and its scaling parameter $K$7 tunes the mean photon number relative to the number of modes (Grier et al., 2021).

A physically different notion of parametrization appears in the dynamical Casimir effect proposal. There the effective mirror motion

$K$8

is controlled through SQUID flux modulation. The pump frequency selects whether the resonant process is beam splitting or two-mode squeezing, while the pump amplitude, duration, and phase set the coupling strength and phase shifts. In this sense the DCE platform is a programmable Gaussian sampler in which the Gaussian network is parameterized by experimentally controlled drives rather than by an abstract matrix alone (Peropadre et al., 2016).

A broader generalization is Vibronic Boson Sampling, motivated by molecular vibronic spectra at finite temperature. There the input may be any Gaussian state

$K$9

including correlated thermal Gaussian states. The paper shows that any GBS instance with initial correlations can be simulated by a GBS instance without initial correlations, with only polynomial overhead, by enlarging the mode space. This places correlated thermal Gaussian inputs within the same generalized parametrized landscape (Huh et al., 2016).

4. Trainable and variational parametrized models

A second branch of parametrized GBS treats the Gaussian sampler as a trainable model. In the WAW framework, analytic gradients are derived for GBS probabilities, and for the trainable weights one obtains

$M$0

This yields sample-based gradient estimators drawn from the same GBS device being optimized. For Kullback–Leibler divergence and log-likelihood objectives, the gradients simplify further and can be computed classically from the difference between model and data mean photon numbers, which the authors state can be done in $M$1 time. The resulting framework turns GBS from a fixed sampler into a variational family and supports applications such as a variational Ising solver (Banchi et al., 2020).

A closely related but formally distinct direction represents the Gaussian-state characteristic function as a neural network. Gaussian transformations are modeled as linear layers through a pullback mechanism, and boson pattern probabilities are computed by automatic differentiation of the resulting computational graph. In the optimization example, the trainable object is an additional interferometer inserted before a fixed Haar-random interferometer, and the loss

$M$2

is used to enhance correlated photon-pair events in selected modes (Conti, 2021).

More recent work defines a Gaussian Bosonic Born Machine as a parametrized GBS circuit used for generative modeling. Each layer is parameterized by displacement, two interferometers, and squeezing values,

$M$3

and training is based on the maximum mean discrepancy loss. Because the state remains Gaussian, training can be carried out classically by propagating only first and second moments and evaluating local parity-string expectations in closed form. Numerical experiments reach up to 805 modes and over a million trainable parameters, which the paper presents as evidence of scalability for near-term photonic devices (Kolarovszki et al., 11 Mar 2026).

A variational threshold-detector formulation adapts parametrized GBS to binary optimization. There the ansatz is a pure zero-displacement Gaussian state measured with threshold detectors, and the cost is the Conditional Value-at-Risk of a diagonal PUBO or QUBO Hamiltonian. In the standard VQE limit $M$4, the paper states that the cost function becomes analytical and can be computed efficiently, along with its gradient, for low-degree polynomials using only classical computing resources. Numerical experiments on 3-SAT and Graph Partitioning show significant performance gains over random guessing (Cazalis et al., 2024).

5. Displacement, detector models, and state reconstruction

Displacement is one of the most important control parameters in parametrized GBS because it changes both the photonic state and the associated classical simulation problem. In displaced GBS, coherent light is added to the squeezed Gaussian resource, and the output probabilities are governed by loop hafnians rather than ordinary hafnians. The experimental demonstration of displaced GBS implements this by injecting a laser beam alongside a two-mode squeezed vacuum state into a 15-mode interferometer. The paper emphasizes two consequences: displacement is needed for applications such as molecular vibronic spectra, graph similarity or graph classification, and quantum state engineering; and increasing the coherent component can push the output distribution toward a more semiclassical regime (Thekkadath et al., 2022).

The same experiment introduces an in situ reconstruction method for the full multimode Gaussian output state using only three measurement settings regardless of dimension. The comparison between reconstructed and theoretical photon statistics yields total variation distances $M$5 for the direct reconstruction and $M$6 for the indirect reconstruction, which the paper uses to validate the method (Thekkadath et al., 2022).

Detector choice is another parameterization axis. Threshold GBS replaces photon-number-resolving outcomes by binary click patterns and is governed by the Torontonian rather than the hafnian. The model is physically attractive because threshold detectors implement only the two outcomes “no click” and “click,” but mathematically the resulting click probabilities remain nontrivial and can still support hardness arguments when collisions are sufficiently rare (Quesada et al., 2018).

This suggests a useful conceptual partition. Some parametrized GBS models change the state while keeping the measurement fixed, as in displaced GBS. Others keep the Gaussian state family but change the readout map, as in threshold-detector formulations. In both cases the parameterization acts at the level of the sampled distribution rather than only at the level of calibration.

6. Complexity, classical simulation, and conceptual boundaries

The computational significance of parametrized GBS is inseparable from the hardness structure inherited from standard GBS. The foundational hardness intuition comes from the fact that GBS probabilities are governed by the hafnian, a $M$7-complete function closely related to the permanent through

$M$8

so efficient classical simulation would entail consequences analogous to those in boson sampling (Hamilton et al., 2016).

At the same time, parametrization can alter classical simulability. An exact classical simulation algorithm for GBS based on auxiliary heterodyne variables reduces global sampling to a sequence of conditional pure-state probability evaluations. Its runtime scales as

$M$9

improving on the earlier

$\chi$0

scaling. The paper interprets this as a quadratic speedup because, for squeezing-generated GBS, complexity is governed by the number of photon pairs rather than the total photon number. It also reports an implementation in The Walrus with about a $\chi$1 speedup over an earlier diagonalization-based routine and a relative-error improvement of about three orders of magnitude, reaching about one part in $\chi$2 accuracy for loop hafnians of size $\chi$3 (Quesada et al., 2020).

For programmable BipartiteGBS, hardness is placed on a more explicit complexity-theoretic footing. Under the Permanent-of-Gaussians and Permanent Anti-Concentration conjectures, approximate classical sampling in the dilute regime $\chi$4 would imply a collapse of the polynomial hierarchy. The same work also extends the analysis to the constant-collision regime by reductions involving permanents with repeated rows and columns (Grier et al., 2021).

A recurring misconception is that classical trainability implies classical easiness. The recent generative-model literature rejects that equivalence: training may be classically efficient because Gaussian states are tracked through first and second moments, while inference remains tied to a sampling problem believed to be classically hard in appropriate regimes (Kolarovszki et al., 11 Mar 2026). Conversely, another misconception is that adding more parameters always increases hardness. The displaced-GBS literature states the opposite possibility: when a significant fraction of detected light is classical, approximate semi-classical models become accurate and computational cost drops (Thekkadath et al., 2022).

Taken together, these results indicate that parametrized GBS is best understood as a controlled family of Gaussian sampling problems rather than as a single model. The parameters may encode a target matrix, a trainable variational ansatz, a phase-space displacement, a detector coarse-graining rule, or an operating regime such as dilute, constant-collision, or semiclassical. What remains stable across these variants is the central role of Gaussian-state structure and the persistence of nontrivial matrix functions—hafnians, loop hafnians, Torontonians, or programmed permanents—as the objects governing the output distribution.