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arXiv:2605.27735

An IQP Born Machine for Calorimeter Image Generation at 64 Qubits with Compiled-IQP Deployment

Jamal Slim, Saverio Monaco, Florian Rehm, Dirk Krücker, Kerstin Borras

quant-phSubmitted May 26, 2026• Announced May 28, 2026• 0 citations
v1 · May 26, 2026v2 · Jun 5, 2026v3 · Aug 7, 2026
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Abstract

We train an instantaneous quantum polynomial-time (IQP) Born machine on real high-energy-physics calorimeter shower images at $64$ qubits and compile the trained model into a single sampling-hard IQP circuit for quantum deployment. The pipeline has three components: a Mixture-of-IQP (\moiqp{}) architecture, whose Walsh-diagonal MMD$^{2}$ loss is classically trainable by Van den Nest Fourier Monte Carlo; the Pearson-Stabilized Correlation Kernel (\psck{}), a positive-definite MMD kernel that biases descent toward correlation-sensitive directions through a data-evaluated Jacobian of the empirical Pearson matrix; and an exact deferred-measurement compilation of \moiqp{} into a single IQP circuit on $\nfeat + \lceil \log_2 \Lcomp \rceil$ qubits (\ciqp{}). Across five seeds at $\Lcomp = 8$, $1500$ epochs, the model reaches $\maerho = 0.069 \pm 0.008$ against a $0.052$ encoding-fidelity floor on the training split and $0.071 \pm 0.008$ on a held-out test split, versus a Liu--Wang baseline at $\maerho = 0.100$. The compiled \ciqp{} reproduces the \moiqp{} marginal to $0.591 \pm 0.012$ times the Monte Carlo noise floor.

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We train an instantaneous quantum polynomial-time (IQP) Born machine on real high-energy-physics calorimeter shower images at 646464 qubits and compile the trained model into a single sampling-hard IQP circuit for quantum deployment. The pipeline has three components: a Mixture-of-IQP (\moiqp{}) architecture, whose Walsh-diagonal MMD2^{2}2 loss is classically trainable by Van den Nest Fourier Monte Carlo; the Pearson-Stabilized Correlation Kernel (\psck{}), a positive-definite MMD kernel that biases descent toward correlation-sensitive directions through a data-evaluated Jacobian of the empirical Pearson matrix; and an exact deferred-measurement compilation of \moiqp{} into a single IQP circuit on \nfeat+⌈log⁡2\Lcomp⌉\nfeat + \lceil \log_2 \Lcomp \rceil\nfeat+⌈log2​\Lcomp⌉ qubits (\ciqp{}). Across five seeds at \Lcomp=8\Lcomp = 8\Lcomp=8, 150015001500 epochs, the model reaches \maerho=0.069±0.008\maerho = 0.069 \pm 0.008\maerho=0.069±0.008 against a 0.0520.0520.052 encoding-fidelity floor on the training split and 0.071±0.0080.071 \pm 0.0080.071±0.008 on a held-out test split, versus a Liu--Wang baseline at \maerho=0.100\maerho = 0.100\maerho=0.100. The compiled \ciqp{} reproduces the \moiqp{} marginal to 0.591±0.0120.591 \pm 0.0120.591±0.012 times the Monte Carlo noise floor.
Quantum circuit Born machines (QCBMs) built with instantaneous quantum polynomial (IQP) circuits have loss functions that can be efficiently estimated classically, but require quantum computers during inference. Thus, classical computers can train large IQP QCBMs that exceed the qubit numbers of available quantum hardware. This motivates research into how qubit-limited quantum computers can be used to sample from larger IQP QCBMs. Leveraging insights from classical simulation of noisy IQP circuits, we propose a deployment protocol for IQP QCBMs that injects noise at the logical level to probabilistically decompose the original circuit into smaller clusters that can be sampled on qubit-limited quantum processors. We also apply lightweight resampling of known noisy qubits to mitigate the impact of the introduced error. Numerical simulations of our protocol applied to shallow IQP QCBMs and IBM Heron experiments deploying a 316-qubit IQP circuit onto 156-qubit hardware show that mitigation improves the similarity of the empirical distribution to the target distribution. The proposed deployment protocol also crucially preserves correlation structures more accurately than a classical baseline that has accurate single-qubit marginals.
Simulating the showers that particles deposit in a calorimeter, the detector that records their energy across many cells, is one of the largest computing costs in high-energy physics, and fast generative surrogates are needed. Quantum circuits have been proposed as such surrogates, but demonstrations on calorimeter data have either required a register that grows with the image, roughly one qubit per cell, or have passed the information between parts of the image through a classical channel. We remove both constraints for this problem, using a construction drawn from the sequential-generation and hidden-quantum-Markov-model literature. A shower image is generated block by block on a fixed register of six qubits. Three of them, the memory, are never measured. They hold what the circuit knows about the blocks already generated as a quantum state, so the correlations between distant parts of the image cross each block boundary as unmeasured amplitudes. The other three are measured and reset once per block, and each outcome selects the energy pattern of one block. The register size is set by the block, so adding cells to the image adds steps to the sequence, not qubits. We call the model a coherent-memory Born machine (CoMB), a Born machine being a circuit whose measurement statistics are the generated distribution. Its training loss grows only linearly with the number of blocks and never requires enumerating the image distribution, and a depth-two instance runs on an IBM superconducting processor. On a twelve-cell benchmark the model reproduces the energy distribution of every cell, the correlation matrix between cells, and the total-energy spectrum. Removing the three memory qubits, with everything else unchanged, produces independent blocks. Every correlation between blocks is carried by the unmeasured qubits and by nothing else.
Generative modeling is one of the most promising applications of quantum machine learning, yet training and deploying Quantum Generative Models (QGMs) on near-term hardware remains effectively intractable due to prohibitive gradient estimation and implementation costs. We propose a resource-efficient approach based on shallow Instantaneous Quantum Polynomial-time (IQP) circuits that circumvents these bottlenecks by leveraging efficient classical training while retaining the guarantee of sampling hardness. To validate this approach, we formalize graph generation as a hierarchy of physical correlations, allowing us to map abstract data features, such as edge density and bipartiteness, directly to the quantum observables required to learn them. We validate our protocol through demonstrations both on real hardware (from 282828 to 153153153 qubits) and simulations (282828 qubits). Results show that while global structural features exhibit significant degradation beyond 919191 qubits, our models achieve high-precision reproduction of local correlations, even up to 153153153 qubits. These findings establish shallow IQP circuits as a robust, scalable candidate for generative tasks on current Noisy Intermediate-Scale Quantum (NISQ) devices.
Quantum circuit Born machines based on instantaneous quantum polynomial-time (IQP) circuits are natural candidates for quantum generative modeling, both because of their probabilistic structure and because IQP sampling is provably classically hard in certain regimes. Recent proposals focus on training IQP-QCBMs using Maximum Mean Discrepancy (MMD) losses built from low-body Pauli-ZZZ correlators, but the effect of initialization on the resulting optimization landscape remains poorly understood. In this work, we address this by first proving that the MMD loss landscape suffers from barren plateaus for random full-angle-range initializations of IQP circuits. We then establish lower bounds on the loss variance for identity and an unbiased data-agnostic initialization. We then additionally consider a data-dependent initialization that is better aligned with the target distribution and, under suitable assumptions, yields provable gradients and generally converges quicker to a good minimum (as indicated by our training of circuits with 150 qubits on genomic data). Finally, as a by-product, the developed variance lower bound framework is applicable to a general class of non-linear losses, offering a broader toolset for analyzing warm-starts in quantum machine learning.
Instantaneous quantum polynomial quantum circuit Born machines (IQP-QCBMs) have been proposed as quantum generative models with a classically tractable training objective based on the maximum mean discrepancy (MMD) and a potential quantum advantage motivated by sampling-complexity arguments, making them an exciting model worth deeper investigation. While recent works have further proven the universality of a (slightly generalized) model, the next immediate question pertains to its trainability, i.e., whether it suffers from the exponentially vanishing loss gradients, known as the barren plateau issue, preventing effective use, and how regimes of trainability overlap with regimes of possible quantum advantage. Here, we provide significant strides in these directions. To study the trainability at initialization, we analytically derive closed-form expressions for the variances of the partial derivatives of the MMD loss function and provide general upper and lower bounds. With uniform initialization, we show that barren plateaus depend on the generator set and the spectrum of the chosen kernel. We identify regimes in which low-weight-biased kernels avoid exponential gradient suppression in structured topologies. Also, we prove that a small-variance Gaussian initialization ensures polynomial scaling for the gradient under mild conditions. As for the potential quantum advantage, we further argue, based on previous complexity-theoretic arguments, that sparse IQP families can output a probability distribution family that is classically intractable, and that this distribution remains trainable at initialization at least at lower-weight frequencies.
Quantum Circuit Born Machines (QCBMs) offer a natural approach to generative machine learning by leveraging the Born rule. Recent work has provided a method to classically train QCBMs with Instantaneous Quantum Polynomial (IQP) circuits via the Maximum Mean Discrepancy (MMD) loss. Despite the assumed intractability of sampling from IQP circuits classically, their expectation values can be computed classically, enabling training of these IQP QCBMs. However, quantum machine learning (QML) models have various other challenges, including trainability issues caused by exponential concentration or barren plateaus. While these issues have been explored for parameters sampled from a uniform distribution, little work has been done to rigorously treat the use of arbitrary Gaussian initialization schemes. This work leverages Stein's lemma and Lipschitz concentration bounds for Gaussian random variables to provide an analytical lower bound of the variance of the gradient and a probabilistic concentration bound of the deviation of the gradient from its mean. It discusses strategies to either avoid or encourage exponential concentration, as well as the conditions under which barren plateaus are more likely to occur.