CVBM: Continuous-Variable Quantum Model

• CVBM is a quantum generative model that employs continuous-variable photonic circuits and both Gaussian and non-Gaussian gates to directly model real-valued distributions.
• It trains by minimizing statistical distances like the squared Maximum Mean Discrepancy between generated and target distributions using parameter-shift gradients and stochastic gradient descent.
• Its design overcomes exponential resource overheads of discrete models, enabling scalable simulation and efficient learning of multimode continuous data.

A Continuous-Variable Born Machine (CVBM) is a quantum generative model that parameterizes continuous-valued probability distributions using multimode photonic quantum circuits. By exploiting the native infinite-dimensional Hilbert space and measurement protocols of continuous-variable (CV) quantum systems—such as modes of light—CVBMs model, generate, and learn continuous probability densities over $\mathbb{R}^d$ directly, without recourse to discretization. This architecture stands in contrast to discrete-variable Born machines, which are inherently limited to modeling distributions over bitstrings and suffer exponential resource overheads when approximating real-valued targets. CVBMs employ parameterized quantum circuits composed of Gaussian and, where necessary, non-Gaussian gates and are trained by minimizing a statistical distance (commonly the squared Maximum Mean Discrepancy, MMD) between the generated and target distributions, typically via stochastic gradient descent that leverages parameter-shift rules or automatic differentiation of Gaussian moments (Čepaitė et al., 2020, Kolarovszki et al., 11 Mar 2026, Kolarovszki et al., 2024).

1. Mathematical Structure and Probability Model

The core element of a CVBM is a family of multimode photonic quantum states

$$\rho(\theta) = U(\theta)|0\rangle^{\otimes d}\langle 0| U(\theta)^\dagger$$

where $U(\theta)$ is a parameterized CV quantum circuit and $d$ is the mode (feature) count. Sampling is performed via homodyne detection in the quadrature basis, typically at phase $\phi=0$, yielding continuous real vectors $x=(x_1,\ldots,x_d)\in\mathbb{R}^d$ according to the Born rule:

$$p(x|\theta) = \mathrm{Tr}[|x\rangle\langle x|\,\rho(\theta)]$$

with $$|x\rangle = |x_1\rangle\otimes\ldots\otimes|x_d\rangle$$ and $|x_j\rangle$ the eigenstate of $\hat{x}_j$ (Čepaitė et al., 2020, Kolarovszki et al., 2024).

The position-basis wavefunction's overlap with Fock states is given by

$$\langle n|x\rangle = \psi_n(x) = \frac{e^{-x^2/2}}{\pi^{1/4}\sqrt{2^n n!}} H_n(x)$$

where $H_n(x)$ are the physicists' Hermite polynomials, facilitating explicit evaluation and classical simulation in truncated Fock spaces (Kolarovszki et al., 2024).

2. Quantum Circuit Architecture

CVBMs are realized with architectures comprising alternating layers of Gaussian and non-Gaussian gates. The parameterized circuit can be captured by:

$$U(\theta) = \prod_{\ell=1}^L \left[ U^{(NG)}(\gamma^{(\ell)},\kappa^{(\ell)}) U^{(G)}(\theta^{(\ell)},\phi^{(\ell)},r^{(\ell)},\alpha^{(\ell)}) \right]$$

where:

• $U^{(G)}$ includes Gaussian gates:
  • Displacement: $$D(\alpha) = \exp(\alpha \hat{a}^\dagger - \alpha^* \hat{a})$$
  • Squeezing: $$S(r) = \exp[\frac{1}{2}(r^*\hat{a}^2 - r\hat{a}^{\dagger 2})]$$
  • Phase rotation: $R(\varphi) = \exp(i\varphi \hat{a}^\dagger\hat{a})$
  • Beam splitter: $$BS(\theta,\phi) = \exp\left(\theta \left( e^{i\phi} \hat{a}_j\hat{a}_k^\dagger - h.c. \right)\right)$$
• $U^{(NG)}$ incorporates non-Gaussian elements:
  • Cubic-phase: $V(\gamma) = \exp(i\gamma \hat{x}^3/6)$
  • Cross-Kerr: $$CK_{jk}(\kappa) = \exp(i\kappa \hat{n}_j \hat{n}_k)$$
  • Kerr: $K(\kappa) = \exp(i\kappa \hat{n}^2)$

The full variational parameter set $\theta$ collects all gate arguments across layers and modes (Čepaitė et al., 2020, Kolarovszki et al., 2024).

For circuits restricted to Gaussian gates, the resulting states are pure Gaussian and completely specified by their mean vector $\mu(\theta)\in\mathbb{R}^{2d}$ and covariance matrix $\Sigma(\theta)\in\mathbb{R}^{2d\times 2d}$, with successive layers updating $\mu$ and $\Sigma$ via real-symplectic affine maps (Kolarovszki et al., 11 Mar 2026).

3. Training Procedures: Loss Functions and Gradient Estimation

Training aims to match the generated distribution $p(x|\theta)$ to a target $Q(x)$ using the squared Maximum Mean Discrepancy (MMD):

$$\mathrm{MMD}^2(P_\theta, Q) = \mathbb{E}_{x,x'\sim P_\theta}[k(x,x')] + \mathbb{E}_{y,y'\sim Q}[k(y,y')] - 2\mathbb{E}_{x\sim P_\theta, y\sim Q}[k(x,y)]$$

where $k$ is a positive-definite kernel, typically a Gaussian $k(x,y) = \exp(-\|x-y\|^2/2\sigma^2)$ (Čepaitė et al., 2020, Kolarovszki et al., 2024, Kolarovszki et al., 11 Mar 2026).

Gradients of the loss with respect to circuit parameters are computed via the quantum parameter-shift rule for those parameters (notably Gaussian gates):

$$\frac{\partial}{\partial\theta_k} p(x;\theta) = \frac{1}{2}(p_{\theta_k^+}(x) - p_{\theta_k^-}(x))$$

where $\theta_k^\pm = \theta_k \pm s$, with gate-dependent shifts $s$ and scaling. For Gaussian gates, analytic shift rules are available; for non-Gaussian gates, finite-difference methods are used—an approach that is markedly sample-inefficient (Čepaitė et al., 2020, Kolarovszki et al., 2024).

Stochastic gradient descent (SGD) is employed in practice, with gradients estimated from finite samples (“shots”) drawn from parameter-shifted circuits. Empirically, $10^3$ shots per gradient estimate suffice for reliable optimization using Adam (Kolarovszki et al., 2024).

4. Classical Simulation and Resource Scaling

Classical simulation of CVBM sampling proceeds by representing $\rho(\theta)$ in a truncated multimode Fock basis (cutoff $c$), enabling joint position probabilities to be constructed as

$$p(x) = \sum_{\vec{n},\vec{m}} \rho_{\vec{n},\vec{m}} \prod_{i=1}^d \psi_{n_i}(x_i)\psi_{m_i}(x_i)$$

Sampling is performed sequentially by conditioning modes; marginal densities for each mode are polynomials times $e^{-x^2}$, enabling inverse-transform sampling (Kolarovszki et al., 2024).

The computational complexity of this procedure grows exponentially with the number of modes $d$, with the observed empirical scaling $\approx\exp(1.3d)$ (Kolarovszki et al., 2024). On physical quantum hardware, this cost is absent, but practical quantum computation remains limited by decoherence, shot noise, and currently available non-Gaussian gate fidelities.

For circuit resources, a single CV mode encodes an infinite-dimensional variable; as a result, CVBMs avoid the exponential scaling associated with discretizing continuous variables in discrete-variable circuits. For $n$-mode, $L$-layer circuits, the total gate and parameter counts scale as $O(Ln)$, with gradient estimation requiring $M+2(R+S)\cdot$(number of parameters) measurement shots per training step (Čepaitė et al., 2020).

In the special case of purely Gaussian CVBMs realized as Gaussian Boson Sampling (GBS) circuits, the state evolution and loss/gradient evaluation reduce to linear algebra on $(\mu,\Sigma)$, allowing $O(Ld^3)$ scaling and enabling classical training for hundreds of modes and $10^6+$ parameters (Kolarovszki et al., 11 Mar 2026).

5. Empirical Results and Training Performance

Numerical experiments demonstrate the capacity of CVBMs to learn both classical and quantum continuous distributions (Čepaitė et al., 2020, Kolarovszki et al., 2024). Key findings include:

• Learning a 1D classical Gaussian: a single-mode CVBM with displacement and squeezing converges within $\lesssim20$ iterations, MMD $\sim10^{-3}$, closely matching the target. Discrete-variable Born machines with six qubits plateau in accuracy due to coarse discretization (Čepaitė et al., 2020).
• Learning multimode targets: circuits with $L$ alternations of non-Gaussian and Gaussian blocks were trained using up to four modes and $1500$ shots per gradient. Training converged in $100$–$600$ iterations, final joint distributions matching target densities (Kolarovszki et al., 2024).
• GBS-based CVBMs (“parity-GBBM”): demonstrated classical trainability with JAX on up to $805$-mode circuits, $1.3\times10^6$ parameters, and datasets including genomic and image data. Model performance exceeded classic RBMs and Chow-Liu trees in matching reference statistics (Kolarovszki et al., 11 Mar 2026).

SGD with stochastic parameter-shift gradients was shown to be robust to gradient noise, converging reliably even with $O(1)$ estimator fluctuations (Kolarovszki et al., 2024).

6. Noise and Robustness

The dominant noise mechanism on CV hardware is modeled as photon loss, represented as a beamsplitter of transmissivity $T$. Numerical analyses show:

• Gaussian targets are robust to $T \sim 0.8$–$0.9$; learning remains effective for moderate loss.
• Non-Gaussian targets are more sensitive but show partial learnability under realistic noise (Čepaitė et al., 2020).
• In GBS-based architectures, classical training is unaffected by hardware noise, but quantum sampling for inference retains hardware-induced imperfections (Kolarovszki et al., 11 Mar 2026).

In practice, transmissivity may be calibrated directly by fitting the model to experimental data (Čepaitė et al., 2020).

7. Limitations, Variants, and Outlook

Several limitations and open questions are evident:

• Parameter-shift rules for non-Gaussian gates are absent, necessitating inefficient finite-difference gradients for these directions (Kolarovszki et al., 2024).
• The convergence of CVBM training depends critically on the chosen circuit ansatz, kernel, learning rate, and initialization. Optimal architectures for given data distributions are undetermined (Kolarovszki et al., 2024).
• For large mode count, classical simulation becomes infeasible; quantum-gradient-based training is expected to scale favorably with access to photonic hardware (Kolarovszki et al., 2024).
• The regime of shot budgets that ensure convergence is not yet quantitatively characterized; rigorous sample complexity analysis remains open (Kolarovszki et al., 2024).

Empirical evidence suggests CVBMs can be trained with practical shot counts using quantum stochastic gradient descent and can leverage hybrid approaches for model classes—such as GBS circuits with classical differentiability—to enable classically scalable training while sampling remains quantum (Kolarovszki et al., 11 Mar 2026).

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