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.
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.
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.
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.
A quantum neural network (QNN) built from gates that conserve the Hamming weight keeps its data in one weight sector, so the shots that left the sector can be discarded at readout. This is the qg filter of the qang library [1, 2]. We show that, with equal amplitude damping (T1) on every qubit, the filtered readout of such a network is exactly the noiseless one in any weight sector. The kept fraction of shots is (1 − γ)k d for weight k and depth d, and training under T1 with the filter gives exactly the parameters of noiseless training. We then measure what this is worth on four small classification datasets. Every result is reported with the filter and without it, and every prediction was committed before its run, failures included. Over 60 runs per model, a network trained on a simulator and run under T1 gains +2.5 points from the filter at weight 1 (95% CI [0.8, 4.1]) and +16.5 points at weight 2 ([11.8, 21.2]), recovering its noiseless accuracy exactly. A network trained under the calibrated noise without the filter catches up (diflerences of 0.1 points), and the filter alone corrects neither dephasing nor unequal T1 across qubits, although it loses less than one point up to a ±80% spread of 1/T1. Unequal T1 is corrected by training UÏtl the filter under the calibrated decay rates: the kept state depends only on their ratios, so the model lands within 0.5 points of noiseless at a ±100% spread, ignores a 10% calibration error and, at weight 2, loses 0.7 points under a 1.5-fold drift of all T1 that costs the raw noise-aware model 10. With noise-aware training and a richer readout, the model without the filter was 1.2 points better in one study, but over 60 new runs that gap shrank to +0.5 points and was not significant. The weight-2 deficit comes from the pair-product encoding, not from the readout; with an encoding that fills the whole sector and a full-rank readout, weight 2 reaches the weight-1 accuracy. On noise models of IBM and IonQ devices the filter cuts the error of the measured qgZ by 2.7-3.6×; on the IonQ Forte-1 noise model over 189 inputs of four datasets the filtered model is 1.1 points more accurate than the raw one, and calibrating the errors the filter keeps with five echo circuits per model halves the remaining error of the decision value: decisions that difler from the noiseless model fall from 15 (raw) and 7 (filter) to 3 over 945 readings on five device noise models. With a faithful weight-2 encoding the advantage of the unfiltered model under noise-aware training disappears, and a classifier given both the filtered and the raw features gains nothing over the filter alone. Over 513 readout configurations, including kept fractions down to 2.6 ×10−7, unequal T1 and dephasing, the filter has the lower readout error whenever about 20 shots are kept, and a closed-form rule predicts the winner. The same holds beyond the first model: at 8 qubits a weight-2 network keeps its noiseless 0.965 with the filter and drops to 0.705 without it; a five-class classifier (one class per qubit) keeps its noiseless accuracy under T1 with the filter and loses 15-18 points without it on five-class digits; training from 100 or 1000 shots per step with the filter is 1.0-1.5 points ahead of training without it; and the filtered gradient equals the noiseless one, while the raw one is K times smaller at weight 1. On five device noise models the echo calibration turns class confusions back: flipped decisions of the five-class model fall from 51 (raw) and 24 (filter) to 15 over 500 readings. The echo helped only where the filtered error was well above shot noise, and it still removed a third of the filtered error with a 28 × 28 matrix at 8 qubits. These models are classically simulable and no quantum advantage is claimed. The filter is
Optimisation of training still faces challenges due to an ill-conditioned loss surface and noisy gradient signals; at the same time, limited computational resources also restrict training options. Quantum Gradient Descent as a Hybrid Optimisation Strategy for Neural Network Training. The proposed noise-resilient quantum gradient descent framework maps selected gradient components to parameterised quantum circuits, estimates directional derivatives via parameter-shift measurements, and returns stable updates to the classical optimizer. It is not directly substituted for backpropagation; instead, it is a type of controllable gradient module that can be added to sensitive layers or low-dimensional projection heads. Add adaptive shot scheduling, damped momentum and layer-wise trust constraints to reduce estimator variance and prevent unstable quantum updates. Representative classification tasks are selected as the subjects of experiments for convergence speed, validation accuracy, gradient fidelity, measurement cost and noise robustness. Based on the above analysis, the new optimizer outperforms standard SGD in terms of final accuracy by 1.6% - 3.2% when using the same number of epochs; at the same time, it reduces the median gradient variance by 28.4% and maintains stable convergence even with an increase in simulated depolarizing noise from 0.001 to 0.015. Based on the above, it can be seen that Quantum Gradient Descent is not an all-purpose optimizer but rather a target-specific training accelerator for gradient-sensitive modules that can trade off sampling cost for more informative descent directions in measurement-aware update design.
Training quantum neural networks (QNNs) on near-term hardware remains hampered by two compounding difficulties: the exponential vanishing of gradient variance known as the barren plateau, and the O(L⋅2n) time and memory cost of differentiating through an n-qubit, L-layer circuit. We propose RLQ-Grad, a reinforcement-learning-based optimizer in which a classical policy πφ (a spectrally-normalized PPO agent) learns to propose parameter updates directly, conditioned on the QNN's current parameters, loss, accuracy, and previous update. Because the surrogate gradient is emitted by a classical network rather than obtained by differentiating through the unitary U(θ), its variance is not constrained by the barren plateau concentration bound, and its cost scales with the number of trainable parameters rather than the Hilbert-space dimension. We prove these properties formally and verify them on a hardware-efficient ansatz across four supervised benchmarks with up to n=20 qubits. RLQ-Grad preserves a near-flat gradient-variance curve where backpropagation, parameter-shift, and adjoint differentiation decay by 1 to 2 orders of magnitude. Accounting for the full training pipeline (PPO rollouts, actor-critic updates, and optimizer states), RLQ-Grad needs under 2 MB of memory and runs 2490×, 7876×, and 673× faster per iteration than these three methods at n=20. It improves top-1 accuracy by up to +10% over gradient-based baselines on circuits of up to 12 qubits, and matches dedicated barren plateau mitigation methods on CIFAR-10 at 14 to 20 qubits, where evolutionary and gradient-free optimizers collapse to chance.
A quantum neural network (QNN) built from gates that conserve the Hamming weight keeps its data in one weight sector, so the shots that left the sector can be discarded at readout. This is the qg filter of the qang library [1, 2]. We show that, with equal amplitude damping (T1) on every qubit, the filtered readout of such a network is exactly the noiseless one in any weight sector. The kept fraction of shots is (1 − γ)k d for weight k and depth d, and training under T1 with the filter gives exactly the parameters of noiseless training. We then measure what this is worth on four small classification datasets. Every result is reported with the filter and without it, and every prediction was committed before its run, failures included. Over 60 runs per model, a network trained on a simulator and run under T1 gains +2.5 points from the filter at weight 1 (95% CI [0.8, 4.1]) and +16.5 points at weight 2 ([11.8, 21.2]), recovering its noiseless accuracy exactly. A network trained under the calibrated noise without the filter catches up (diflerences of 0.1 points), and the filter corrects neither dephasing nor unequal T1 across qubits. These models are classically simulable and no quantum advantage is claimed. The filter is a way to make noise—free training valid under relaxation, at a shot cost known in advance. The models are released as qang. qml (version 0.6.0, pip install qang).