Abstract Every noise channel acting on a quantum neural network leaves a structural signature in the way it decays the coherences of the circuit, and this paper shows that the signature can be read before any training takes place. For the noise families that dominate practical device models, the decay rates within a charge sector of an equivariant network assemble into a weighted graph with closed-form edge weights, and the trainability behaviour of the circuit is organised by this coherence-rate graph. We prove a sufficient protection theorem on the graph, certified by a purely combinatorial construction from the circuit connectivity, we validate a predictor of the depth at which correlated noise degrades each trainable parameter, and we derive exact product laws for the measured degradation. We also refute the natural conjecture that operator-level decay bounds the measured decay, exhibiting explicit counterexamples with large gaps. A deterministic population benchmark, cross-validated to ten digits against an independent implementation, supplies exact targets for the statistical analysis and clarifies the finite-ensemble variation in the deep-circuit comparisons.
Abstract Every noise channel acting on a quantum neural network leaves a structural signature in the way it decays the coherences of the circuit, and this paper shows that the signature can be read before any training takes place. For the noise families that dominate practical device models, the decay rates within a charge sector of an equivariant network assemble into a weighted graph with closed-form edge weights, and the trainability behaviour of the circuit is organised by this coherence-rate graph. We prove a sufficient protection theorem on the graph, certified by a purely combinatorial construction from the circuit connectivity, we validate a predictor of the depth at which correlated noise degrades each trainable parameter, and we derive exact product laws for the measured degradation. We also refute the natural conjecture that operator-level decay bounds the measured decay, exhibiting explicit counterexamples with large gaps. A deterministic population benchmark, cross-validated to ten digits against an independent implementation, supplies exact targets for the statistical analysis and clarifies the finite-ensemble variation in the deep-circuit comparisons.
Symmetry provides a quantum neural network structure, but on its own it does not keep the network trainable once noise is present. We ask which physical quantity decides whether the gradients of an equivariant circuit survive decoherence, and we answer with a compact training law. Working with U(1)-equivariant brickwork circuits that conserve a charge, we find that two distinct effects govern a trainable gradient. Causality fixes where the gradient can live, confining it to the backward light cone of the readout inside the active charge sector. Coherence then determines how fast it decays through the contraction of the off-diagonal sector modes that the projected readout can actually observe. We prove a light-cone reduction that pins the noiseless gradient to the sector-restricted cone with a lower bound independent of the total qubit number, and we define a readout-visible aligned coherence rate as a Rayleigh quotient of the noise generator along the gradient-carrying mode. A perturbative open-system analysis turns this rate into a leading-order training law. Density-matrix simulations then confirm that the finite-noise degradation follows a single accumulated variable built from noise depth and coherence contraction, with a coefficient of determination of 0.979. The sharpest test comes from a correlated-dephasing channel that has a large worst-case rate but a near-zero aligned rate. The law predicts no gradient loss for this channel, and none is seen. Sector coherence outperforms every standard channel diagnostic we compare it against, and the analysis identifies readout-visible sector coherence as the quantity that links equivariant architecture, open-system dynamics and noisy trainability.
Quantum Neural Networks (QNNs) with random structures have poor trainability due to the exponentially vanishing gradient as the circuit depth and the qubit number increase. This result leads to a general belief that a deep QNN will not be feasible. In this work, we provide the first viable solution to the vanishing gradient problem for deep QNNs with theoretical guarantees. Specifically, we prove that for circuits with controlled-layer architectures, the expectation of the gradient norm can be lower bounded by a value that is independent of the qubit number and the circuit depth. Our results follow from a careful analysis of the gradient behaviour on parameter space consisting of rotation angles, as employed in almost any QNNs, instead of relying on impractical 2-design assumptions. We explicitly construct examples where only our QNNs are trainable and converge, while others in comparison cannot.
Graphs provide a natural language for relational data in chemistry, biology and optimisation. Graph neural networks (GNNs) have driven much of the recent progress in learning from such data through message passing, a single primitive that generalises convolution and attention. Quantum counterparts have been proposed, but with limited connection to message passing and few guarantees on performance or scalability. More broadly, the trainability of variational quantum circuits is a recognised bottleneck for their wide applicability, and pre-training has emerged as one way to address it. Yet for a quantum model to be useful, it must offer expressivity guarantees along with demonstrable scalability. Here we show how a quantum graph neural network can be built to perform message passing, to be permutation equivariant, and to sit at a chosen level of the Weisfeiler-Leman hierarchy, the standard measure of how finely a model can tell graphs apart. We show that, as for classical GNNs, the training can be done first on small graph instances, allowing for a pre-training that can mitigate usual training issues, and its output can be read out at a cost that stays low as the graph grows. We validate the framework in large-scale simulations of up to 56 qubits across three datasets, on synthetic graphs that ordinary message passing cannot separate, on molecular property prediction, and on the travelling salesperson problem. Our framework opens a path for near-term quantum algorithms with theoretical guarantees and practical scalability, bringing the principles of graph learning into quantum circuit design.
Quantum neural networks learn by following gradients, and on real hardware those gradients are washed out by noise. Predicting how quickly this happens, before any finite noise experiment or circuit sweep is run, remains an open problem. We present CohAlign, an open source toolkit that computes the first order noise susceptibility of the ensemble second moment of a specified parameter gradient in a charge conserving brickwork quantum neural network. The computation needs only a classical description of the circuit, the noise model, the prepared state, the readout and the initialisation prior. The predicted quantity is distinct from training convergence or final accuracy. The key observation is that only the part of the noise that a measurement can actually see matters. A purposefully constructed control channel makes this clear, damaging the operator carrying a gradient at nearly the maximal rate while the measured gradient remains exactly as predicted and untouched. We validate the predictions against separately evaluated finite difference slopes across the benchmark suite, including a signed coherent case, in exact simulations at eight and ten qubits and on two circuit topologies.
A central model in quantum machine learning is the quantum neural network (QNN), whose design requires balancing expressivity and trainability. Technically, expressivity is studied through circuit-function analysis, such as quantum signal processing, while trainability is analyzed using dynamical-Lie-algebra (DLA) methods. To support certified QNN design, we formalize these major components of QNN theory in a connected lean 4 development checked by a proof kernel, where every analytic input is either proved or exposed as a named hypothesis. On the expressivity side, we prove exact if-and-only-if characterizations of single-qubit QNNs, a resource-counted quantum phase processing theorem, and an overparameterization ceiling that bounds the quantum Fisher information rank by the DLA dimension. On the trainability side, we derive the direct-sum loss-variance law through a de-circularized second-moment interface. A parameterized Casimir-uniqueness engine discharges the required inputs for fully controllable, orthogonal, and matchgate circuit families, while single-qubit and product-Clifford ensembles close the two-design assumptions directly. A capstone theorem pairs the conditional variance law with exact loss reconstruction in DLA coordinates. The development record identifies eight corrections and clarifications that were not explicit in the informal arguments. We expect this work to provide a machine-checkable foundation for QNN theory and a step toward AI-assisted or automated design of quantum machine learning algorithms.
Abstract Quantum neural network (QNN), or equivalently, the variational quantum circuits with a gradient-based classical optimizer, has been broadly applied to many experimental proposals for noisy intermediate scale quantum (NISQ) devices. However, the learning capability of QNN remains largely unknown due to the non-convex optimization landscape, the measurement error, and the unavoidable gate noise introduced by NISQ machines. In this study, we theoretically explore the learnability of QNN from the perspective of the trainability and generalization. Particularly, we derive the convergence performance of QNN under the NISQ setting, and identify classes of computationally hard concepts that can be efficiently learned by QNN. Our results demonstrate that large gate noise, few quantum measurements, and deep circuit depth will lead to poor convergence rates of QNN towards the empirical risk minimization. Moreover, we prove that any concept class, which is efficiently learnable by a restricted quantum statistical query (QSQ) learning model, can also be efficiently learned by QNN. Since the restricted QSQ learning model can tackle certain problems such as parity learning with a runtime speedup, our result suggests that QNN established on NISQ devices will retain the quantum advantage. Our work provides the theoretical guidance for developing advanced QNNs and opens up avenues for exploring quantum advantages using NISQ devices.