Continuous-Variable Quantum Neural Networks

• CV-QNNs are layered continuous-variable circuits that utilize Gaussian operations for affine transformations and non-Gaussian gates for activation functions.
• They mimic classical neural networks by implementing mode mixing, squeezing, and displacement through optical components like interferometers.
• CV-QNNs are applied in tasks such as classification, regression, and state restoration within hybrid quantum-classical frameworks.

Continuous-variable quantum neural networks (CV-QNNs) are variational quantum circuits defined on continuous-variable quantum hardware, where information is carried by bosonic modes rather than qubits and processed through layered Gaussian and non-Gaussian operations. In the canonical formulation, a CV-QNN layer is built to mirror the classical map $\mathcal{L}(x)=\varphi(Wx+\mathbf{b})$: Gaussian gates realize an affine transformation in phase space, while non-Gaussian gates supply the activation-like component, yielding a layered architecture that is also universal for continuous-variable quantum computation (Killoran et al., 2018).

1. Foundational formulation

The foundational CV-QNN construction defines a single layer on $N$ modes as

$$\mathcal{L}:= \Phi \circ \mathcal{D} \circ \mathcal{U}_2 \circ \mathcal{S} \circ \mathcal{U}_1,$$

where $\mathcal{U}_1$ and $\mathcal{U}_2$ are interferometers, $\mathcal{S}$ is a modewise squeezing layer, $\mathcal{D}$ is a displacement layer, and $\Phi$ is a non-Gaussian transformation such as a Kerr or cubic-phase operation. In phase space, the Gaussian subsequence implements

$$\begin{bmatrix} \mathbf{x}\ \mathbf{p} \end{bmatrix} \mapsto M \begin{bmatrix} \mathbf{x}\ \mathbf{p} \end{bmatrix} + \begin{bmatrix} \boldsymbol{\alpha}_r\ \boldsymbol{\alpha}_i \end{bmatrix},$$

with $M$ symplectic, so the layer can be written in neural-network-like form as $N$0 (Killoran et al., 2018).

A closely related presentation makes the classical analogy explicit at the level of encoded states: $N$1 There the linear map $N$2 is motivated through singular value decomposition,

$N$3

and realized as two interferometers around diagonal squeezing, while displacement implements the bias term $N$4 (Choe, 2022).

The original work also showed that a classical feedforward network can be embedded in the CV formalism by identifying classical inputs with quadrature eigenstates and constructing unitary dilations of activation functions. This is why CV-QNNs are often presented not merely as quantum-inspired analogues of neural networks, but as a quantum-native generalization of the affine-plus-activation paradigm itself (Killoran et al., 2018).

2. Gaussian affine structure and non-Gaussian activation

The Gaussian part of a CV-QNN layer comprises rotations, beamsplitters, squeezing, and displacement. The rotation, displacement, squeezing, and beamsplitter gates are given in phase-space form in the foundational construction, and together realize general Gaussian affine transformations compatible with symplectic structure. The Bloch–Messiah decomposition underlies this choice: a general symplectic transformation factors into orthogonal interferometers around diagonal squeezing, after which displacement supplies translation (Killoran et al., 2018).

Within the explicit neural-network analogy, squeezing acts as trainable scaling, interferometers implement mode mixing, and displacement acts as additive bias. One application-oriented exposition states this directly through

$N$5

with the bias correspondence chosen by $N$6, while the affine part is summarized as

$N$7

In the same presentation, Kerr gates are treated as the source of the layer’s activation-like behavior (Choe, 2022).

Non-Gaussianity is structurally central. The original CV-QNN paper states that Gaussian operations enact affine transformations and non-Gaussian gates enact nonlinear activation functions; it also states that the non-Gaussian gates provide both the nonlinearity and the universality of the model (Killoran et al., 2018). A plausible implication is that, in this literature, “activation” refers to nonlinear action on encoded variables or phase-space structure rather than a violation of linear unitary evolution on Hilbert space.

Later controlled comparisons sharpened this point by showing that truncation can materially alter the meaning of a nominally non-Gaussian gate. In a wafer-map study using cutoff $N$8, the Kerr action

$N$9

reduces on $$\mathcal{L}:= \Phi \circ \mathcal{D} \circ \mathcal{U}_2 \circ \mathcal{S} \circ \mathcal{U}_1,$$0 to $$\mathcal{L}:= \Phi \circ \mathcal{D} \circ \mathcal{U}_2 \circ \mathcal{S} \circ \mathcal{U}_1,$$1, so the observed advantage of the CV head in that regime was attributed not to larger Hilbert-space dimensionality or genuine non-Gaussianity, but to two other properties: a structured neural-network-analogue layer and continuous phase-space encoding (Kim et al., 1 Jul 2026).

3. Encoding, readout, and hybridization

CV-QNNs appear in both fully quantum and hybrid quantum-classical pipelines. In the original demonstrations, classical inputs were encoded by displacement of vacuum, outputs were obtained by homodyne, heterodyne, or photon counting, and training was performed in a hybrid loop using Strawberry Fields with automatic differentiation (Killoran et al., 2018). Subsequent application papers retained this hybrid character but diversified both the front end and the readout.

Several works compress classical features before entering the quantum circuit. In the forecasting model QNNet10, one or two classical features are loaded through displacement-based encoding into a two-mode, single-layer CV circuit, and the scalar forecast is read out through

$$\mathcal{L}:= \Phi \circ \mathcal{D} \circ \mathcal{U}_2 \circ \mathcal{S} \circ \mathcal{U}_1,$$2

with the model using 2 modes, 1 layer, 10 quantum gates, and 8 trainable parameters (Abdulrahman, 4 Mar 2025). In the biomedical imaging feasibility study, $$\mathcal{L}:= \Phi \circ \mathcal{D} \circ \mathcal{U}_2 \circ \mathcal{S} \circ \mathcal{U}_1,$$3 images are flattened, reduced by PCA to 4 principal components, encoded into 4 qumodes via displacement, processed by a depth-2 Gaussian circuit, and measured through

$$\mathcal{L}:= \Phi \circ \mathcal{D} \circ \mathcal{U}_2 \circ \mathcal{S} \circ \mathcal{U}_1,$$4

after which a classical linear head performs classification (Lopez et al., 3 Nov 2025).

Other models use full probability vectors as quantum features. In the controlled CV-versus-DV wafer-map comparison, the CV head outputs the Fock-basis probability distribution of dimension

$$\mathcal{L}:= \Phi \circ \mathcal{D} \circ \mathcal{U}_2 \circ \mathcal{S} \circ \mathcal{U}_1,$$5

which is then passed to a final dense layer. This made the 4-qumode, cutoff-2 CV head directly comparable to a 4-qubit DV head with a 16-dimensional computational-basis output (Kim et al., 1 Jul 2026). In a different direction, the entanglement-witness architecture concatenates probability vectors from multiple informationally complete measurement settings and feeds them into a small classical MLP to produce a scalar witness value (Shokou et al., 7 Sep 2025).

A more elaborate use of hybridization appears in the continuous-variable diffusion framework. There, a CVQNN is not a one-shot classifier or regressor but a timestep-conditioned denoiser $$\mathcal{L}:= \Phi \circ \mathcal{D} \circ \mathcal{U}_2 \circ \mathcal{S} \circ \mathcal{U}_1,$$6 acting jointly on a noisy state and a quantum time-embedding qumode, with non-unitarity realized by tracing out one subsystem after a two-mode unitary (Huang et al., 24 Jun 2025). This suggests that CV-QNNs are increasingly used not only as static ansätze but also as trainable quantum channels inside larger generative or restoration schemes.

4. Representative tasks and empirical behavior

Application work on CV-QNNs spans regression, classification, state generation, restoration, and quantum-state characterization. The numerical evidence is heterogeneous: some papers emphasize conceptual architecture, some emphasize simulator-based benchmarks, and some stress controlled paradigm comparisons.

A recurring pattern is that CV-QNNs are competitive as small hybrid heads or small photonic models, while their strongest claims often concern structural suitability rather than unqualified superiority over classical baselines. For example, the “structured classical information” paper reports 94.9% MNIST classification accuracy but also separately reports fidelities greater than 99.98%; those are distinct quantities, with the latter referring to state-overlap objectives rather than classification accuracy (Basani et al., 2020). The controlled wafer-map study, by contrast, isolates the quantum head as the sole structural variable and reports a clear CV-over-DV gap while still finding that the classical baseline remains best overall (Kim et al., 1 Jul 2026).

Task	Representative result	Paper
Fraud detection	AUC $$\mathcal{L}:= \Phi \circ \mathcal{D} \circ \mathcal{U}_2 \circ \mathcal{S} \circ \mathcal{U}_1,$$7 for a hybrid classifier	(Killoran et al., 2018)
MNIST / structured classical information	94.9% classification accuracy; fidelities greater than 99.98% reported separately	(Basani et al., 2020)
Experimentally realizable hybrid CV-QNN	99.9% single-photon fidelity, 99.8% cat fidelity, 93.9% GKP fidelity, and more than 95% classification accuracy	(Bangar et al., 2023)
State generation and restoration	Final generation fidelities under pure loss of 99.95% for $$\mathcal{L}:= \Phi \circ \mathcal{D} \circ \mathcal{U}_2 \circ \mathcal{S} \circ \mathcal{U}_1,$$8, 99.56% for $$\mathcal{L}:= \Phi \circ \mathcal{D} \circ \mathcal{U}_2 \circ \mathcal{S} \circ \mathcal{U}_1,$$9, 99.85% for $\mathcal{U}_1$0, and 99.61% for $\mathcal{U}_1$1	(Huang et al., 24 Jun 2025)
Controlled CV-versus-DV classifier head	4-qumode CV head reaches $\mathcal{U}_1$2 accuracy versus $\mathcal{U}_1$3 for a matched 4-qubit DV head	(Kim et al., 1 Jul 2026)

Beyond these representative points, the biomedical imaging study reports that a 4-qumode, depth-2 Gaussian-only hybrid CV-QNN achieves ACC 0.8429 on PneumoniaMNIST, ACC 0.7564 on BreastMNIST, and ACC 0.4563 on OrganAMNIST, typically outperforming the matched DV model on the harder multiclass task but not surpassing the classical baseline (Lopez et al., 3 Nov 2025). The entanglement-witness work reports about 99% accuracy on both 2-mode and 3-mode benchmarks, together with a large gap over strong classical baselines in the 3-mode case (Shokou et al., 7 Sep 2025).

5. Hardware-oriented realizations

A major line of work has tried to make CV-QNNs experimentally realistic by replacing idealized nonlinear gates with components closer to current photonic practice. One approach uses only Gaussian gates together with ancillary qumodes and repeat-until-success measurements. In that protocol, ancilla-assisted controlled displacement followed by photon detection induces an effective nonlinear transformation on the primary mode; for small $\mathcal{U}_1$4 and $\mathcal{U}_1$5, a successful click implements

$\mathcal{U}_1$6

so the nonlinearity is measurement-induced rather than supplied by an explicit Kerr gate. This Gaussian-plus-ancilla architecture was used for state preparation, curve fitting, and classification, with reported fidelities of 99.9% for single-photon states, 99.8% for cat states, and 93.9% for approximate GKP states, together with more than 95% classification accuracy (Bangar et al., 2023).

A second hardware-oriented formulation replaces Kerr-style activation with heralded photon subtraction after a general Gaussian block. The proposed quantum optical neural network takes coherent-state inputs, applies

$\mathcal{U}_1$7

then performs photon subtraction on selected modes and reads out homodyne expectations. In the single-mode analysis, the effective activation becomes

$\mathcal{U}_1$8

which the authors interpret as an adaptive activation generated directly by photonic hardware. That work further states that the proposed design satisfies the Universal Approximation Theorem within a single layer and introduces the QuaNNTO simulator to evaluate exact expectation values of these non-Gaussian states without truncating the infinite-dimensional Hilbert space (Krasimirov-Ivanov et al., 4 Dec 2025).

These hardware-oriented proposals do not remove all implementation difficulties. The experimentally realizable protocol is probabilistic because the nonlinear step is heralded, and the photon-subtraction architecture makes success probability a resource issue when many modes are subtracted simultaneously. Nonetheless, both works recast the central CV-QNN idea—Gaussian affine structure plus non-Gaussian activation—into forms that are substantially closer to established optical components than a deterministic Kerr-heavy circuit (Bangar et al., 2023).

6. Boundaries of the concept and open technical questions

The CV-QNN label is used unevenly across adjacent areas, and several neighboring literatures must be distinguished carefully. A paper on “continuous variable prediction” is mostly about a qubit-based hybrid regressor for a continuous-valued target; its genuinely continuous-variable model is the photonic Strawberry Fields baseline using DisplacementEmbedding and CVNeuralNetLayers, and that component remained heavily constrained by simulation cost (Jain et al., 2022). By contrast, continuous-variable neural-network quantum states for the quantum rotor model and for CV optimization are classical variational wavefunction ansätze trained by variational Monte Carlo rather than gate-based photonic CV-QNNs proper (Stokes et al., 2021). Likewise, continuous-variable quantum kernel theory analyzes fixed CV feature maps and shows that finite-stellar-rank kernels take the form “Gaussian times algebraic,” which is highly relevant to expressivity but is not itself a trainable CV-QNN architecture (Henderson et al., 2024).

Within gate-based CV-QNNs themselves, several technical issues remain recurrent. Classical simulation cost grows rapidly with cutoff and qumode count; some application papers therefore use only very small mode numbers or Gaussian-state backends, while others stress that Fock-space scaling is a major bottleneck (Jain et al., 2022). Readout design remains task-dependent: some models use quadrature expectations, some use Fock-basis probabilities, and some rely on informationally complete measurement plus a classical head (Shokou et al., 7 Sep 2025). A plausible implication is that “CV-QNN” now denotes a family of architectures sharing a gate-level language rather than a single settled protocol.

Several papers also identify substantive open questions. The controlled CV-versus-DV wafer-map study asks how much of the observed advantage comes from continuous encoding versus structured layer design, whether the advantage persists at higher cutoff where genuine non-Gaussianity enters, and how sensitive results are to interferometer parameterization and truncation (Kim et al., 1 Jul 2026). The biomedical imaging study points to larger qumode counts, deeper circuits, and non-Gaussian gates as the next steps beyond small Gaussian-only feasibility demonstrations (Lopez et al., 3 Nov 2025). The diffusion model notes that larger-support states, broader noise models, and hardware-level realization of Kerr nonlinearity remain unresolved (Huang et al., 24 Jun 2025). The entanglement-witness work proves a dense approximation result for the hybrid model under informationally complete measurement on compact finite-dimensional state sets, but that theorem addresses approximation capacity rather than training efficiency or hardware feasibility (Shokou et al., 7 Sep 2025).

Taken together, these results delimit the current meaning of CV-QNNs. At minimum, the term refers to layered continuous-variable circuits in which Gaussian operations implement trainable affine transformations on bosonic modes and a non-Gaussian or measurement-induced mechanism supplies activation-like structure. Around that core, the literature now includes hybrid classifier heads, generative denoisers, hardware-inspired photonic neurons, and data-driven quantum-state functionals, while still contending with the field’s central constraints: non-Gaussian resource realism, measurement design, cutoff dependence, and simulator-limited scaling.