Instantaneous Quantum Polynomial Circuit Born Machines (IQP-QCBMs) have been proposed as quantum generative models that combine a classically tractable training objective - based on the maximum mean discrepancy (MMD) - with a potential quantum advantage motivated by sampling-complexity arguments. While recent works have explored this model across various application domains, fundamental questions remain: does the model suffer from exponentially vanishing loss gradients, known as the barren plateau problem - a pervasive obstacle in quantum machine learning - and how do regimes of trainability relate to regimes of possible quantum advantage? Here, we address both questions analytically. To study trainability, we derive closed-form expressions for the variances of the partial derivatives of the MMD loss function and establish general upper and lower bounds. We explicitly characterize how trainability depends on the generator set and the spectrum of the chosen kernel, identifying regimes in which low Hamming-weight kernels avoid exponential gradient suppression under structured topologies. Regarding potential quantum advantage, we reformulate the anti-concentration property in terms of the same generator-set quantities that govern trainability, enabling a unified analysis. We show that sparse IQP architectures can produce classically intractable output distributions while simultaneously remaining trainable, at least at lower-weight frequencies. Our analytical results corroborate the numerical observations of prior work and provide principled guidelines for designing scalable IQP-QCBM architectures.
We present a unified quantum sensitivity sampling framework that streamlines coreset construction for many classical approximation problems, significantly speeding up clustering, regression, and low-rank approximation. We assume $n,d\geq\mathrm{poly}(k/\epsilon)$ and, for regression, $n\geq\mathrm{poly}(d)$: - $k$-median/$k$-means: For $n$ points in $\mathbb{R}^d$, we build an $\epsilon$-coreset of input points of size $\tilde O(k^{4/3}\epsilon^{-2})$ (resp. $\tilde O(k^{3/2}\epsilon^{-2})$) in time $\tilde O(\epsilon^{-1}n^{0.5}dk^{1.67})$ (resp. $\tilde O(\epsilon^{-1}n^{0.5}dk^{1.75})$). Our better $d$- and linear $1/\epsilon$-dependence improve on [Xue, Chen, Li and Jiang, ICML'23] for $d\geq\Omega(k^{7/3})$ (resp. $\Omega(k^{2.5})$). - $\ell_p$ regression: For $\min_x\|Ax-b\|_p$, $A\in\mathbb{R}^{n\times d}$, we build an $\epsilon$-coreset of size $\tilde O_p(d^{\max\{1,p/2\}}\epsilon^{-2})$ in time $\tilde O_p(n^{0.5}d^{1.5}\epsilon^{-1})$ for $p\leq 2$ and $\tilde O_p(n^{0.5}d^{p/4+1}(\epsilon^{-1}+d^{0.5}))$ for $p>2$, matching [Liu and Ji, ICML'26] for $p\leq 2$ and improving on the prior best quantum sampling approach [Apers and Gribling, SICOMP'26] for all $p>2$. - Low-rank approximation: We give the first quantum sublinear-time algorithm for Frobenius-norm rank-$k$ approximation of $A\in\mathbb{R}^{d\times n}$ free of data-dependent parameters, running in $\tilde O(n^{0.5}dk^{0.5}\epsilon^{-0.5})$ time with factored output (plus $\tilde O(nk^2\epsilon^{-2})$ to materialize them). We also give sublinear algorithms for kernel and tensor variants. - Lower bound: For generic sensitivity sampling with total sensitivity $S$, any quantum algorithm outputting an $\epsilon$-coreset of size $o(\sqrt{nS})$ needs $\Omega(\sqrt{nS})$ queries, so our query complexity is optimal up to $\mathrm{poly}(\epsilon^{-1},\log n)$ and VC-dimension factors.
We consider the problem of synthesizing Clifford quantum circuits for devices with all-to-all qubit connectivity. We approach this task as a reinforcement learning problem in which an agent learns to discover a sequence of elementary Clifford gates that reduces a given symplectic matrix representation of a Clifford circuit to the identity. This formulation permits a simple learning curriculum based on random walks from the identity. We introduce a novel neural network architecture that is equivariant to qubit relabelings of the symplectic matrix representation, and which is size-agnostic, allowing a single learned policy to be applied across different qubit counts without circuit splicing or network reparameterization. On six-qubit Clifford circuits, the largest regime for which optimal references are available, our agent finds circuits within one two-qubit gate of optimality in milliseconds per instance, and finds optimal circuits in 99.2% of instances within seconds per instance. After continued training on ten-qubit instances, the agent scales to unseen Clifford tableaus with up to thirty qubits, including targets generated from circuits with over a thousand Clifford gates, where it achieves lower average two-qubit gate counts than Qiskit's Aaronson-Gottesman and greedy Clifford synthesizers.
Quantum circuit Born machines (QCBMs) can learn distributions that are hard to sample from classically, but their value depends on generalizing from finite training data. Here, we study how hardware noise affects this generalization. We show that, for any noise process, the generalization gap of a QCBM trained with the maximum mean discrepancy is bounded by the complexity of the distributions the noisy model can reach. Scrambled local Pauli noise contracts this complexity but biases the model towards the uniform distribution, so residual noise acts as a regularizer whose optimal strength can be intermediate. We propose GENIE, a framework that realizes a chosen noise level with capped error mitigation and selects it during training from held-out data within a fixed execution budget, with provably bounded regret. In simulations on synthetic and census data, GENIE stays close to the best of four noise-control strategies, whereas every fixed noise level falls short in many settings.
When an image classifier decides, we can usually only estimate why. For quantum and all-optical classifiers that act linearly on amplitude-encoded pictures up to the measurement, with no trained classical network before or after them, each class's Born probability above its no-information share, $P_g - b_g$, splits exactly into contributions of the encoded amplitudes, shown as Born evidence maps (BEM). The split itself follows from known results on quadratic forms: with the transmitted light held fixed, it is what Integrated Gradients from a zero baseline and the Shapley value give. What we add is a way to measure it on the hardware that classifies, from class probabilities on a quantum chip or, in simulation, from camera frames on a lens bench, with no gradient circuit and no light sent backwards; a $g\times g$ map takes $1 + 2g^2$ runs. Deleting the 5% of pixels with the highest evidence drops accuracy from 73% to 9% (one seed), as occlusion and gradient $\times$ input do, while plain gradients remove much less. The classifiers share one architecture, a Fourier-plane mask followed by a brickwall circuit on qubits or by phase plates on a lens bench, read by a fixed Born headroom readout with no trained parameter; centring each picture on its own mean lets dark features count as much as bright ones. In simulation, four phase plates at one wavelength reach 72.3% on Caltech-10 colour photographs. On IBM's ibm_kingston, with the mask computed classically before approximate loading, 300 test pictures were classified at 37.7% against 10% chance (56.3% in exact simulation), and $2\times2$ maps of the filtered input of 30 test pictures drawn at random in advance, measured on the chip, agreed with the exact ones ($r = 0.88$; standard estimator 0.82). The maps also revealed a shortcut in the data; as the trim that removed it was chosen on test pictures, the Caltech-10 results are exploratory.
Quantum low-density parity-check codes promise low-overhead error correction, but high-performing constructions remain hard to find. Here we introduce structured concept evolution, in which a large language model (LLM) evolves construction programs and their internal search strategies by modifying the finite group, protograph shapes or entry-assignment rule. We search for codes with fewer than $600$ data qubits and check weight at most $12$, optimizing $kd^2/n$ or the circuit-level logical error rate. Our method improves on published lifted-product benchmarks in selected parameter regimes, with exactly certified distances for the leading results. Circuit-level optimization typically yields lower logical error rates than $kd^2/n$ optimization, often despite smaller code distances. With the same evolution framework and step budget, the best LLM outperforms all random-mutation and hill-climbing runs on both objectives. Our results point to LLM-driven program search as an approach to designing general quantum LDPC codes.
Trainable-frequency quantum models train the prefactors of their encoding gates so that the circuit's frequencies can move onto the target's. From the unit start of prior work, training rarely reaches a target more than one resolution step away. We show that the target-driven part of the prefactor gradient is a weighted sum of slopes of the overlap $\sin u/u$, where $u$ is the model-target frequency distance scaled by the window half-width. This gives each frequency a reach: under gradient flow, an isolated real frequency whose coefficient adapts optimally reaches its target from within half a step, for every target at least one step from zero frequency; with a fixed coefficient, a phase-dependent barrier can stop it. A ternary prefactor grid places a model frequency within half a step of every band frequency, with logarithmically many encoding gates. Single-frequency circuits trained with Adam and jointly trained amplitudes behave as predicted: inside the first side lobe, a frequency starts toward its target exactly when in phase with it in 98% of runs, and only 11 of 768 frequencies starting beyond the lobe reached their target. With several frequencies, the grid start raised successful fits on synthetic targets from 69% to 84% against the best random start. With many target frequencies, least squares on the fixed grid fits near-perfectly, but the circuit does not reach the coefficients; training the prefactors narrows this shortfall, but with 8 and 24 target frequencies, training the fixed grid ten times longer fitted better. On real series, training the frequencies improved held-out $R^2$ at equal budget by 0.010--0.045 on four of seven series, but a classical screen did not separate these from the others. A classical model with independently trainable frequencies shows the same dependence on the start; our results describe how to train such models, not an advantage of the circuit.
Many quantum software development kits provide a suite of circuit optimisation passes. These passes have been highly optimised and tested in isolation. However, the order in which they are applied is left to the user, or else defined in general-purpose default pass sequences. While general-purpose sequences miss opportunities for optimisation which are particular to individual circuits, designing pass sequences bespoke to particular circuits requires exceptional knowledge about quantum circuit design and optimisation. Here we propose and demonstrate training a reinforcement learning agent to compose optimisation-pass sequences. In particular the agent's action space consists of passes for two-qubit gate count reduction used in default PyTKET pass sequences. For the circuits in our diverse test set, the (mean, median) fraction of two-qubit gates removed by the agent is $(57.7\%, \ 56.7 \%)$, compared to $(41.8 \%, \ 50.0 \%)$ for the next best default pass sequence. In doing so our model demonstrates the ability to forgo reward to improve the final two-qubit gate count. Our approach clearly separates the problem of designing circuit optimisation passes from the problem of using them optimally in practice. We expect this separation of concerns to facilitate innovations in both areas.
The manifold hypothesis suggests that data lies on manifolds with smaller intrinsic dimension (ID) than their ambient dimension. However there is no empirical agreement on the estimates for ID from different estimators for realistic datasets. Thus it is important to test ID estimators (IDEs) with targeted stressors. In this work, we consider the role of anisotropy. To this end, we propose HomID, a collection of homogeneous spaces with anisotropic embedding, for benchmarking ID estimators. We observe that methods that perform well on standard benchmarks systematically degrade on HomID under identical resource allocation. We further observe that anisotropic distortion of such benchmarks also results in performance degradation. Finally, we demonstrate that controlled anisotropic distortions induce systematic shifts in the distributions on which these methods rely, providing a concrete mechanism for the resulting estimation errors in two particular IDEs. The code for the project is available at https://github.com/a-writes-code/homid
Choosing a quantum error-correcting code for a real device means scoring candidates with circuit-level simulations so slow that only a few hundred can be run. So every evaluation counts, and a search is judged by how many good codes it finds per paid evaluation. Quantum LDPC codes built from a few group elements protect many logical qubits, but good ones are rare in a huge space. We introduce GFlowQEC, to our knowledge the first generative flow network (GFlowNet) for quantum code design. It builds a code one group element at a time and learns to propose codes in proportion to a power of their score. We train it once on scores from earlier searches, freeze it, and let it propose lifted-product codes with 180 physical qubits, scored by their distances. Ranking ten of its draws per paid call, it finds more than twice as many new top-scoring codes (inequivalent to any it was trained on) as tuned parallel tempering, even when tempering starts from GFlowQEC's training codes. It finds more than three and a half times as many as a supervised screen ranking ten random candidates per call. When both rank a hundred, the screen beats GFlowQEC, and does best ranking GFlowQEC's draws. Soft Q-learning, which shares the GFlowNet's optimal policy, does as well at one noise bias and better at the other, so the gain does not depend on the GFlowNet's training loss. Trained proposers can make a budget of expensive evaluations go further when good codes are rare.
Deep eutectic solvents (DESs) offer a tunable platform for lignocellulose refining, but their solvation behavior depends on multicomponent hydrogen-bond networks rather than individual constituent properties. We develop TiNN, a thermodynamics-informed neural network that hierarchically represents HBA-HBD association and solute accommodation through a learned free-energy pathway. Trained on 938 records, TiNN achieves a test-set $R^2$ of 0.917 and generalizes to new HBA-HBD combinations and sparse conditions. Ablation identifies solute-specific stabilization as the largest contributor to prediction accuracy, while the importance of hydrogen-bond compatibility depends on the dissolution mechanism. Experimental measurements of glucose, xylan, and Kraft lignin in acidic, basic, and neutral DESs show that TiNN reduces the geometric mean solubility error from more than 23-fold with COSMO-RS to approximately 1.5-fold. Without training on fractionation data, TiNN also distinguishes DES formulations with high and low lignin and hemicellulose removal from corn stover. These results support chemically grounded solvent screening for lignocellulose fractionation.
Instantaneous quantum polynomial-time (IQP) Born machines are attractive quantum generative models, as certain IQP circuit families are conjectured to be hard to sample, yet they can be trained classically. Previously proposed conditional IQP Born machines make the parameters of the diagonal layer condition dependent. We introduce conditional IQP Born machines, in which the condition is encoded in the input state and a single IQP unitary is shared across conditions. Efficient sampling and amplitude-ratio evaluation for the reference state entering the diagonal layer suffice to obtain unbiased classical estimates of a conditional maximum mean discrepancy objective and its gradients. We show that input conditioning can represent conditional families that no diagonally conditioned IQP model with the same number of qubits can reproduce. For the sixteen-qubit Curie-Weiss spin model, a single conditional model captures the magnetization transition at held-out temperatures. Its mean total variation distance is six times lower than that of diagonal conditioning under the same protocol and comparable to that of classical generative models with up to seven times as many parameters. We also train conditional models classically with up to 2048 qubits, which reproduce the mean squared magnetization of the target away from the crossover. These results extend classically trained IQP Born machines to conditional generation.
This work studies hybrid quantum-classical neural networks for learning routing heuristics. Specifically, this paper asks whether small quantum neural networks can replace parameter-heavy modules inside a competitive attention-based routing model while maintaining solution quality. For the capacitated vehicle routing problem, encoder feed-forward replacement emerges as the most promising design: it reduces the number of model parameters by 56.6% while keeping the hybrid model close to the classical neural baseline at small and medium instance sizes, although the gap grows for larger instances. This work also compares to classical routing algorithms, which remain highly competitive and often superior on the fixed Euclidean test sets. Our results therefore do not indicate quantum advantage or solver dominance, but identify encoder feed-forward replacement as a viable hybrid-module compression strategy for neural combinatorial optimization.
We investigate whether a multiscale tensor-network architecture provides a useful inductive bias for reconstruction-based anomaly detection in collider jets. Jets are produced by a branching cascade, so their internal structure is naturally organised across angular and momentum scales. This motivates an autoencoder that compresses information hierarchically and can reorganise short-range correlations before coarse-graining. Guided by this picture, we formulate a Multiscale Entanglement Renormalization Ansatz (MERA)-inspired autoencoder acting on ordered jet constituents. To our knowledge, a MERA-inspired autoencoder has not previously been proposed, and this architecture has not been explored in collider anomaly detection. We compare this architecture to a dense autoencoder, the corresponding tree-tensor-network limit, and classical baselines within a background-only reconstruction framework. The paper is organised around two main questions: whether locality-aware hierarchical compression is genuinely supported by the data, and whether the disentangling layers of MERA contribute beyond a simpler tree hierarchy. To address these questions, we combine benchmark comparisons with a training-free local-compressibility diagnostic and a direct identity-disentangler ablation. In the nominal benchmark, the MERA-inspired model attains a higher mean AUC than the dense autoencoder at all three latent dimensions while using roughly one-third of its trainable parameters. The locality-preserving multiscale structure is well matched to jet data, and, at the strongest compression bottleneck, the trained disentanglers move substantially away from the identity and are accompanied by a positive mean performance shift. Overall, the study supports locality-aware hierarchical compression as a useful and parameter-efficient inductive bias for reconstruction-based jet anomaly detection in this benchmark setting.
Choosing a variational quantum circuit (VQC) usually means searching over ansatze and depths, and training each candidate. We give three rules that fix these choices from the data and the gate set alone, before any training. First, a circuit without entangling gates has a rank-one Fourier coefficient tensor on every subset of qubits. A singular value decomposition of a dataset's Fourier coefficients therefore indicates whether entanglement is needed, which qubits it connects, and a lower bound on the error of omitting it. Second, the rank of the quantum Fisher information matrix grows by $p_0$ per layer, after a first-layer deficit $d_0$, until it reaches a ceiling set by the dynamical Lie algebra. This law fixes the depth. Third, reading out all $2^N - 1$ Pauli-Z products from one batch of shots resolves half of the state's directions, the most any single-basis readout can. A single-qubit readout resolves an exponentially small fraction. For the Madelon dataset, the rules prescribe 7 blocks of 3 qubits at depth 3: a 239-parameter classifier with 78.5% test accuracy, found without a grid search. Ablations confirm three predicted effects: shallower circuits, smaller blocks and dropping reuploading all lower accuracy.
This work investigates the application of Quantum Recurrent Neural Networks to the binary classification of sleep-disordered breathing events in time-series data, aiming to distinguish between apneic and normal breathing episodes. A variational classical--quantum architecture is proposed and evaluated against classical Recurrent Neural Network models. Experiments were conducted on multiple real-world datasets with varying levels of complexity to assess the strengths and limitations of the quantum approach. In the comparative analysis, both classical and quantum models showed comparable performance. The quantum model achieved superior performance on the smallest dataset, while yielding slightly weaker results in the most complex setting. However, the quantum approach exhibited notable strengths, such as competitive performance despite a drastically reduced parameter count and low-qubit regime, as well as good generalization capabilities. These results suggest that, despite current hardware and algorithmic limitations, variational quantum recurrent models represent a promising alternative for time-series classification as Quantum Computing technologies continue to mature.
Implicit neural representations (INRs) model signals as continuous functions from coordinates to values, but their performance depends strongly on how spectral capacity is allocated. Yet most Quantum INR (QINR) architectures treat spectral components as independent outputs of separate quantum blocks. We introduce the Spectral Quantum Memory Model (SQMM), a memory coupled quantum INR in which sequential hybrid data reuploading layers (bands) communicate through a shared coherent quantum memory register. Each band reads the memory, computes on its own coordinate upload, and writes part of its state back, so later bands can build on the computation of earlier ones. When memory read and write layers are input independent, we prove that the memory state forms an operator valued Fourier series whose support is contained in the Minkowski sum of the initial memory spectrum and the local band spectra. Downstream bands then condition their Fourier coefficients on the spectral content stored by earlier bands, and the memory coupling creates no new primitive frequencies. This theory motivates a spectral memory alignment loss that encourages memory to encode the spectrum expressed by the deployed prefix of the model. Across audio and image reconstruction benchmarks, SQMM improves over compact QINR and INR baselines.
We introduce the Pauli-Correlator Autoencoder (PCAE), a hybrid quantum--classical architecture for image reconstruction. PCAE splits an image into patches and compresses each patch as a matrix product state (MPS). A small quantum circuit turns each compressed patch into a few measured numbers, and a classical neural network rebuilds the patch from these numbers and the patch position. A learnable energy term, taken from physics, is added to training and tracked as training proceeds. Two versions place the six qubits on a line (PCAE1D) or on a grid (PCAE2D). An averaging step makes the PCAE2D features follow rotations and reflections of the image exactly. Trained on one image at a time, PCAE2D rebuilds images from six public datasets with a peak signal-to-noise ratio of 25.8 to 41.6 decibels. On CIFAR-10 images not used in training, a formula-based stand-in for the PCAE2D features performs within 1 decibel of several simpler classical methods. Five trained models also run on real IBM quantum computers without retraining, and rebuild images at 25.90 to 31.52 decibels. A short argument and a synthetic test show that, with a linear final layer, measuring qubits in pairs is needed for targets built from products of inputs. These results show that PCAE works as a hybrid model that rebuilds varied images and runs on today's quantum hardware at the six-qubit scale.
Quantum reservoir computing (QRC) processes temporal information through quantum dynamics, but it remains unclear whether small QRC models can forecast real data competitively with classical models and whether their predictions survive the transfer from ideal simulation to noisy hardware. We address both questions for a six-qubit gate-based reservoir that forecasts next-day motor-insurance claim counts. Five reservoir candidates were selected and frozen using hardware-native interaction graphs, Trotterized Ising dynamics, memory and measurement-robustness criteria, circuit cost, and calibrated-noise screening. On 232 held-out test days, the ideal candidates reached root-mean-square errors (RMSEs) of 5.66-6.12 claims/day, compared with 5.36 for ridge regression. For four of the five, the difference from ridge was not significant under a 95% block bootstrap, and their accuracy was comparable with classical recurrent and reservoir baselines. On the 156-qubit IBM Kingston processor, we ran 63 evaluations in 21 paired rounds, comparing hardware-aware, random-native, and Qiskit optimization level 1 layouts. Hardware-aware placement gave lower RMSE than Qiskit in every round, with a maximum transfer distortion of 7.59 claims/day (11.84 for Qiskit), and avoided the severe transfer failures seen for some random layouts. Most of the hardware distortion appeared as a systematic forecast shift, which a short hardware probe on training inputs detected before the test runs. Longer replay windows raised circuit cost without a consistent gain in forecasting accuracy or hardware transfer. Small QRC models can thus reach competitive accuracy on this task, and hardware-aware layout selection makes their transfer to IBM Kingston more consistent.
Feedback-based adaptive quantum optimization (FALQON) is a promising approach for combinatorial optimization on noisy intermediate-scale quantum devices, but prior studies report that standard fixed-parameter schedules can require relatively deep circuits to reach strong solution quality. We study Optimal FALQON, a layer-wise formulation that treats the time-step $\delta_k$ and scaling parameter $w_k$ as classical decision variables optimized per layer. The benchmark uses all 94 non-isomorphic 3-regular graphs with 12 vertices under 8192-shot simulation and compares standard FALQON, Optimal FALQON, and QAOA/QAOA-MA Powell runs warm-started from FALQON-derived schedules. Across approximation ratio, success probability, evaluation-normalized efficiency, and layer-normalized efficiency, Optimal FALQON substantially improves on standard FALQON, and Optimal-FALQON warm starts substantially improve on standard-FALQON warm starts.