The Born Ultimatum: Conditions for Classical Surrogation of Quantum Generative Models with Correlators

TL;DR

The paper treats Quantum Circuit Born Machines (QCBMs) as quantum Fourier models whose distributions admit a correlator-based Fourier decomposition and develops Deployment-Dequantization conditions to determine when classical surrogates can match or supplant quantum inference. It analyzes several ansatz families (IQP, matchcircuits, Heisenberg, Haldane) and two surrogate classes (Tensor Networks and Pauli Propagation) to quantify discrepancies between classically trained parameters and quantum deployments, especially under k-order truncation of the Fourier expansion. The authors derive closed-form results for Pauli propagation in IQP circuits, establish variance and generalization analyses for truncated distributions (including RMPS-based tensor networks), and provide numerical results across TFIM, Haldane, and scRNA-seq datasets to illustrate when classical surrogates suffice and where quantum advantages may persist. Overall, the work offers a principled framework to benchmark train-on-classical, deploy-on-quantum strategies, clarifying how correlator significance and inductive biases shape the attainable quantum advantage and guiding practical choices for surrogate methods and circuit architectures.

Abstract

Quantum Circuit Born Machines (QCBMs) are powerful quantum generative models that sample according to the Born rule, with complexity-theoretic evidence suggesting potential quantum advantages for generative tasks. Here, we identify QCBMs as a quantum Fourier model independently of the loss function. This allows us to apply known dequantization conditions when the optimal quantum distribution is available. However, realizing this distribution is hindered by trainability issues such as vanishing gradients on quantum hardware. Recent train-classical, deploy-quantum approaches propose training classical surrogates of QCBMs and using quantum devices only for inference. We analyze the limitations of these methods arising from deployment discrepancies between classically trained and quantumly deployed parameters. Using the Fourier decomposition of the Born rule in terms of correlators, we quantify this discrepancy analytically. Approximating the decomposition via distribution truncation and classical surrogation provides concrete examples of such discrepancies, which we demonstrate numerically. We study this effect using tensor-networks and Pauli-propagation-based classical surrogates. Our study examines the use of IQP circuits, matchcircuits, Heisenberg-chain circuits, and Haldane-chain circuits for the QCBM ansatz. In doing so, we derive closed-form expressions for Pauli propagation in IQP circuits and the dynamical Lie algebra of the Haldane chain, which may be of independent interest.

Figures (17)

• Figure 1: a) We adopt a train-on-classical, deploy-on-quantumrecio-armengol_train_2025 approach for quantum circuit Born machines. Training is performed using the Fourier decomposition in terms of correlators of the QCBM’s probability distribution. $\boldsymbol{c}_X(\boldsymbol{\theta}^*_Y)$ are the set of correlators produced by model $X \in \{\mathcal{C}\!\ell, \mathcal{Q}\}$ using optimal parameters ($\boldsymbol{\theta}^*_Y$) generated by training model, $Y \in \{\mathcal{C}\!\ell, \mathcal{Q}\}$. Classical surrogates, $\boldsymbol{c}_{\mathcal{C}\!\ell}$, are used to approximate a 'constrained' subset of correlators within a truncated version of the Fourier decomposition to train the model. Once the optimal classical parameters $\boldsymbol \theta^*_{\mathcal{C}\!\ell}$ are obtained, they are deployed on the quantum model, $\boldsymbol c_{\mathcal{Q}}({\boldsymbol{\theta}}^*_{\mathcal{C}\!\ell})$. A quantum advantage arises when the distribution induced by the quantum deployment approximates the underlying target distribution better than the classical counterpart and we provide conditions under such an advantage holds. b) Deployment-Dequantization conditions: When quantum training is possible to find ${\boldsymbol{\theta}}^*_{\mathcal{Q}}$and if the learned quantum distribution $\boldsymbol c_{\mathcal{Q}}({\boldsymbol{\theta}}^*_{\mathcal{Q}})$: b.1) has significant weight across the full frequency support, this prevents alignment with any classical counterpart, $c_{\mathcal{C}\!\ell}$; or b.2) is less expressive and concentrates only on a restricted support, the classical distribution can align with it. c) Sources of discrepancy in deploying the ideal quantum and classical parameters, ${\boldsymbol{\theta}}^*_{\mathcal{Q}}, {\boldsymbol{\theta}}^*_{\mathcal{C}\!\ell}$ respectively: c.1) the quantum model, $\boldsymbol c_{\mathcal{Q}}({\boldsymbol{\theta}}^*_{\mathcal{Q}})$, can represent a broader range of frequencies, even if not all are necessarily useful (cf. b.2); c.2) using optimal classical - ${\boldsymbol{\theta}}^*_{\mathcal{C}\!\ell}$ - rather than quantum parameters - ${\boldsymbol{\theta}}^*_{\mathcal{Q}}$ - decreases the overall optimality, with the extent of this reduction determined by the inductive bias of the ansatz. We elabotate on this and give precise statements in Sec. \ref{['sec:comparison']}.
• Figure 2: The Quantum Circuit Born Machine (QCBM). In its original version the PQC circuit is sampled to obtain the discrete binary bitsring $\boldsymbol{x}$. After sampling sufficiently many times one can build an estimate of $\operatorname{Pr}_{{\boldsymbol{\theta}}}$. The same is done for the Hamiltonain $\mathcal{H}$ and $\operatorname{Pr}_{\mathcal{H}}$ or a different data source. Both probabilities are compared via the gradient of the loss function of choice $\mathcal{L}$ which informs parameter updates. This procedure is repeated until $\mathcal{L}$ reaches a predefined convergence criterion, such as a small gradient norm, a plateau in loss values, or a target accuracy threshold.
• Figure 3: Approximations summary. In the first approximation we select the set of constrained correlators to train. In the second approximation we choose the correlator surrogate. The restricted landscape to the correlator $\mathcal{L}(\langle Z_{\boldsymbol{i}}\rangle)$ changes with respect to the original quantum model one. When deploying, after the classical training, the previous optimum value in the landscape is no longer optimum for the constrained surrogated correlators. Furthermore, new values arise from the unconstrained correlators.
• Figure 4: $Z$-basis $k$-order correlations for ground state of $n=6$ qubits in: a) the one-dimensional Ising Hamiltonian, with $J=0.7$ and $h=0.33$; b) the Alternating Heisenberg Hamiltonian with $J_{\rm even}=1.4$ and $J_{\rm odd} = 0.6$; then for the correlations of the Haldane Hamiltonian with parameters $J=1$, $h_1=0.7$ and $h_2=0.33$ we have c) the one-dimensional chain with $n=6$; and d) the rectangular $y$-periodic lattice with $n_x=3$ and $n_y=2$.
• Figure 5: $k$-order correlations in the scRNA-seq dataset derived from the data probability distribution amongst $n=6$ genes with activated and not activated encoding.

Theorems & Definitions (21)

• Lemma 1.1
• Lemma 1.2
• Definition 1.1
• Definition 1.2
• Proposition 1.1: Dynamical Lie algebra for the Haldane chain Hamiltonian (Eq. \ref{['eq:haldane_1d']})
• Definition 2.1: RFC Dequantization
• Theorem 2.1: Fourier Distribution Dequantization for Shift-Invariant Kernels sweke_potential_2025sahebi_dequantization_2025
• Theorem 2.2: Discrepancy Sources between Classical and Quantum Deployed Distributions
• Lemma 3.1: Variance of Truncated Probability for Matchcircuits
• Definition 3.1: Random Matrix Product State (RMPS) haferkamp_emergent_2021