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

## What the paper contributes (one paragraph)
The paper identifies the Quantum Circuit Born Machine (QCBM) as a quantum Fourier model whose output distribution is a Fourier expansion of the Born rule in terms of Pauli-Z correlators. This lets the authors import known dequantization conditions and state when a classically trained surrogate, deployed on quantum hardware, can or cannot match the ideal quantum model. They prove a dequantization theorem for shift-invariant kernels and a bound that decomposes the classical-versus-quantum deployment discrepancy into a feature-cardinality gap and a surrogate-training mismatch. To make correlators classically computable they study two concrete surrogates, tensor networks (controlled by bond dimension) and Pauli propagation (controlled by weight or flip truncation), deriving a correlator-variance result for tensor networks and a closed-form Pauli-propagation expression for IQP circuits. As supporting characterisation, they derive the previously unknown dynamical Lie algebra of the Haldane chain. The work is theoretical and numerical, on small systems, and is about mapping the classical limit rather than demonstrating advantage.

## Techniques offered
- **Classical surrogation dequantization conditions (Deployment-Dequantization theorems)** - states conditions under which a polynomial-resource classical surrogate of a deployed QCBM reproduces the ideal quantum distribution, and bounds the residual classical-versus-quantum discrepancy.
  - guarantee: high-probability excess-risk upper bound under efficient sampling, polynomial concentration across the frequency support, and coefficient alignment, plus a triangle-inequality bound separating the feature-cardinality gap from the surrogate-training mismatch
  - quote: "Under these assumptions, the distribution $\boldsymbol c_{\mathcal{C}\!\ell}$ enables efficient classical dequantization for shift-invariant kernel methods."
- **Tensor-network correlator surrogate** - approximates the QCBM correlators with a matrix product state whose bond dimension sets the classical-versus-quantum gap, giving a classical simulation route for the truncated Born distribution.
  - guarantee: closed-form correlator-variance expression showing approximated correlators can still vanish as their complexity grows; accuracy controlled by the bond dimension as a complexity cutoff
  - quote: "Tensor Networks have been used for sampling problems~\cite{ferris_perfect_2012} as well as for the Born Machine~\cite{rudolph_synergy_2023} where it is numerically proven that they provide useful initialisations."
- **Pauli-propagation surrogate with closed form for IQP circuits** - estimates QCBM correlators by Heisenberg-picture Pauli propagation with weight or flip truncation, and supplies an explicit closed-form expression for IQP circuits using their commuting-gate symmetry.
  - guarantee: closed-form truncated-probability expression for Z-like IQP circuits, with a flip-Hamming-weight truncation reducing the term count from exponential to polynomial; approximate rather than exact for general circuits
  - quote: "However, in doing so, we present a closed form for the Pauli propagation for the IQP circuit where we leverage the symmetries within the circuit structure, which may be of independent interest."
- **Dynamical Lie algebra of the Haldane chain** - derives the previously unknown dynamical Lie algebra of the one-dimensional Haldane chain Hamiltonian, characterising the expressivity and trainability of a Haldane-inspired QCBM ansatz.
  - guarantee: explicit algebra (stated as a proposition, verified computationally up to a finite system size) with a proven exponential dimension via the Cartan decomposition
  - quote: "However, unlike the Heisenberg chain, the algebra for the Haldane chain was previously unknown, to the best of our knowledge."

## Where it could apply
- Target primitive(s): sampling (generative modelling via the Born rule); classical surrogate training and quantum deployment of parameterised quantum circuits
- Target application group(s): quantum_ml, platforms_tooling, classical_simulation
- Code family / hardware assumptions: no error-correcting code; gate-model QCBM ansatze (IQP circuits, matchcircuits, Heisenberg-chain and Haldane-chain circuits); tensor-network (MPS) and Pauli-propagation classical surrogates; small-scale numerics (6 to 20 qubits)

## Caveats
This is a dequantization and classical-simulation analysis, so the techniques it supplies are limits and surrogates rather than performance-enhancing primitives for an external quantum result. The dequantization theorem assumes access to the ideal quantum minimiser, which the paper notes is not achievable in practice at scale because of vanishing gradients; results are demonstrated on small systems (6 to 20 qubits); the Pauli-propagation surrogate is approximate for general circuits and sensitive to hyperparameters; and the Haldane Lie algebra is a proposition verified only up to a finite size. No hardware accreditation, error correction, or verification protocol is offered.