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arXiv:2607.24014

Scalable Quantum Machine Learning: Trainability, Expressivity and Efficiency

Iordanis Kerenidis

quant-phSubmitted Jul 27, 2026• Announced Jul 28, 2026• 0 citations
v1 · Jul 27, 2026v2 · Aug 20, 2026v3 · Oct 27, 2026
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Abstract

Designing scalable parameterized quantum circuits for machine learning faces three fundamental obstacles: barren plateaus that prevent gradient-based training, the absence of provable guarantees that the learned function class is classically hard, and prohibitive circuit evaluations per gradient step. We propose the unitary brick-wall: a $k$-particle fermionic architecture for nearest-neighbor hardware, combining Reconfigurable Beam Splitter gates with interleaved single-qubit phase gates and a non-Gaussian magic-state encoding, where the particle number $k$ is a tunable dial trading classical simulation hardness against training cost. Trainable. The brick-wall has dynamical Lie algebra $\mathfrak{u}(n)$ and directly parametrizes $U(n)$ via Givens rotations, enabling Haar initialization. Two-body correlator readouts achieve gradient variance $Θ(k^2/n^5)$. Expressive. Classical hardness is controlled by the particle number $k$: best-known classical algorithms for sampling and for two-body expectation values run in time $2^{Θ(k)}\mathrm{poly}(n)$, worst-case #P-hardness holds at $k = n^ε$, and average-case hardness applies at $k = Θ(n)$. Efficient. A multi-layer parallel parameter-shift rule computes all $O(n^2)$ gradients from $k(8n+4)$ circuit evaluations per gradient step, a factor $3n/(8k)$ reduction over the $3n^2$ evaluations required by the standard parameter-shift rule. The unitary butterfly variant targets all-to-all hardware, with depth $2\log n$ and $n\log n$ parameters. It achieves similar hardness guarantees at $8k\log n$ evaluations per gradient step, the same factor $3n/(8k)$ reduction. Its trainability is established at two levels: the absence of exponential barren plateaus is unconditional, whereas the sharp $Θ(k^2/n^5)$ rate holds under a two-particle approximate-2-design conjecture.

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Abstract Barren plateaus are the main obstacle to training quantum machine-learning models. On a barren plateau, the cost-function gradient shrinks exponentially as the circuit grows, and training stalls. Today the only reliable test is to measure it. That means sampling many gradients for every candidate circuit, which is expensive. We ask a different question. Can a classical model predict trainability from the circuit architecture alone? The architecture here means qubit count, depth, ansatz type, entanglement pattern, entangler gate, and cost-observable locality. We test this directly. We use an exact statevector simulator to build a dataset of \Nrows{} random variational circuits. For each circuit we estimate the gradient variance $\mathrm{Var}[\partial C/\partial\theta]$ from \Samples{} random parameter vectors, using the parameter-shift rule. This variance is the label. A gradient-boosted regressor then predicts $\log_{10}\mathrm{Var}$ with $R^2 = \RsqHold{}$ on held-out architectures. A classifier separates barren from trainable circuits with an area under the ROC curve of \AucHold{}. We also test transfer to larger circuits. We train only on small circuits and test on the two largest unseen sizes ($n=11,12$). The model still reaches $R^2 = \RsqExtrap{}$. Permutation importance then recovers established theory. Cost-observable locality and entanglement structure matter most. We make no quantum-advantage claim. This is classical machine learning about quantum circuits. It offers a cheap way to screen ans\"atze, and it shows which design choices cause untrainability.

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We propose SUN-VQC, a variational-circuit architecture whose elementary layers are single exponentials of a symmetry-restricted Lie subgroup, $\mathrm{SU}(2^{k}) \subset \mathrm{SU}(2^{n})$ with $k \ll n$. Confining the evolution to this compact subspace reduces the dynamical Lie-algebra dimension from $\mathcal{O}(4^{n})$ to $\mathcal{O}(4^{k})$, ensuring only polynomial suppression of gradient variance and circumventing barren plateaus that plague hardware-efficient ansätze. Exact, hardware-compatible gradients are obtained using a generalized parameter-shift rule, avoiding ancillary qubits and finite-difference bias. Numerical experiments on quantum auto-encoding and classification show that SUN-VQCs sustain order-of-magnitude larger gradient signals, converge 2--3$\times$ faster, and reach higher final fidelities than depth-matched Pauli-rotation or hardware-efficient circuits. These results demonstrate that Lie-subalgebra engineering provides a principled, scalable route to barren-plateau-resilient VQAs compatible with near-term quantum processors.

arXiv:2507.05535•IEEE International Conference on Quantum Artificial Intelligence (QAI)•Nov 2, 2025•0 citations
Designing scalable parameterized quantum circuits for machine learning faces three fundamental obstacles: barren plateaus that prevent gradient-based training, the absence of provable guarantees that the learned function class is classically hard, and prohibitive circuit evaluations per gradient step. We propose the unitary brick-wall: a kkk-particle fermionic architecture for nearest-neighbor hardware, combining Reconfigurable Beam Splitter gates with interleaved single-qubit phase gates and a non-Gaussian magic-state encoding, where the particle number kkk is a tunable dial trading classical simulation hardness against training cost.
Trainable. The brick-wall has dynamical Lie algebra u(n)\mathfrak{u}(n)u(n) and directly parametrizes U(n)U(n)U(n) via Givens rotations, enabling Haar initialization. Two-body correlator readouts achieve gradient variance Θ(k2/n5)Θ(k^2/n^5)Θ(k2/n5).
Expressive. Classical hardness is controlled by the particle number kkk: best-known classical algorithms for sampling and for two-body expectation values run in time 2Θ(k)poly(n)2^{Θ(k)}\mathrm{poly}(n)2Θ(k)poly(n), worst-case #P-hardness holds at k=nεk = n^εk=nε, and average-case hardness applies at k=Θ(n)k = Θ(n)k=Θ(n).
Efficient. A multi-layer parallel parameter-shift rule computes all O(n2)O(n^2)O(n2) gradients from k(8n+4)k(8n+4)k(8n+4) circuit evaluations per gradient step, a factor 3n/(8k)3n/(8k)3n/(8k) reduction over the 3n23n^23n2 evaluations required by the standard parameter-shift rule.
The unitary butterfly variant targets all-to-all hardware, with depth 2log⁡n2\log n2logn and nlog⁡nn\log nnlogn parameters. It achieves similar hardness guarantees at 8klog⁡n8k\log n8klogn evaluations per gradient step, the same factor 3n/(8k)3n/(8k)3n/(8k) reduction. Its trainability is established at two levels: the absence of exponential barren plateaus is unconditional, whereas the sharp Θ(k2/n5)Θ(k^2/n^5)Θ(k2/n5) rate holds under a two-particle approximate-2-design conjecture.
Abstract Barren plateaus are the main obstacle to training quantum machine-learning models. On a barren plateau, the cost-function gradient shrinks exponentially as the circuit grows, and training stalls. Today the only reliable test is to measure it. That means sampling many gradients for every candidate circuit, which is expensive. We ask a different question. Can a classical model predict trainability from the circuit architecture alone? The architecture here means qubit count, depth, ansatz type, entanglement pattern, entangler gate, and cost-observable locality. We test this directly. We use an exact statevector simulator to build a dataset of \Nrows{} random variational circuits. For each circuit we estimate the gradient variance Var[∂C/∂θ]\mathrm{Var}[\partial C/\partial\theta]Var[∂C/∂θ] from \Samples{} random parameter vectors, using the parameter-shift rule. This variance is the label. A gradient-boosted regressor then predicts log⁡10Var\log_{10}\mathrm{Var}log10​Var with $R^2 = \RsqHold{}$ on held-out architectures. A classifier separates barren from trainable circuits with an area under the ROC curve of \AucHold{}. We also test transfer to larger circuits. We train only on small circuits and test on the two largest unseen sizes (n=11,12n=11,12n=11,12). The model still reaches $R^2 = \RsqExtrap{}$. Permutation importance then recovers established theory. Cost-observable locality and entanglement structure matter most. We make no quantum-advantage claim. This is classical machine learning about quantum circuits. It offers a cheap way to screen ans\"atze, and it shows which design choices cause untrainability.
We propose an approach to generative quantum machine learning that overcomes the fundamental scaling issues of variational quantum circuits. The core idea is to use a class of generative models based on instantaneous quantum polynomial circuits, which we show can be trained efficiently on classical hardware. Although training is classically efficient, sampling from these circuits is widely believed to be classically hard, and so computational advantages are possible when sampling from the trained model on quantum hardware. By combining our approach with a data-dependent parameter initialisation strategy, we do not encounter issues of barren plateaus and successfully circumvent the poor scaling of gradient estimation that plagues traditional approaches to quantum circuit optimisation. We investigate and evaluate our approach on a number of real and synthetic datasets, training models with up to one thousand qubits and hundreds of thousands of parameters. We find that the quantum models can successfully learn from high dimensional data, and perform surprisingly well compared to simple energy-based classical generative models trained with a similar amount of hyperparameter optimisation. Overall, our work demonstrates that a path to scalable quantum generative machine learning exists and can be investigated today at large scales.
Abstract—Quantum Machine Learning (QML) operating on Noisy Intermediate-Scale Quantum (NISQ) hardware represents an emerging computational paradigm to address high-dimensional optimization problems that challenge classical Von Neumann supercomputing architectures. By mapping complex feature vectors into exponentially large 2^N Hilbert state spaces via Parameterized Quantum Circuits (PQCs) and Variational Quantum Algorithms (VQAs), QML targets computational advantages in combinatorial optimization, high-dimensional classification, and quantum-enhanced molecular modeling. However, the theoretical promise of NISQ-era variational computing is severely constrained by three fundamental bottlenecks: (1) Barren Plateaus, characterized by the exponential concentration of measure and vanishing variance of cost function gradients; (2) Hardware Decoherece Noise and readout error rates that corrupt quantum state fidelity; and (3) Classical-to-Quantum State Preparation bottlenecks that can erase theoretical computational speedups. This paper delivers an exhaustive, mathematically formal technical investigation and empirical benchmark of QML architectures for high-dimensional optimization. We formalize the Haar-distributed geometric mechanics of barren plateaus under 2-design unitary assumptions, evaluate barren plateau mitigation through shallow local cost operators and layer-by-layer parameter initialization, and contrast Variational Quantum Eigensolvers (VQE) with Quantum Kernel Support Vector Machines (Q-SVM). Furthermore, we construct an empirical benchmarking testbed executing across both state-vector simulators and superconducting quantum backends (IBM Quantum Heron / Eagle architectures), evaluating 127-qubit Hamiltonian optimization and high-dimensional financial portfolio allocation. Our findings demonstrate that while global cost functions suffer exponential gradient decay Var[grad] ~ O(2^-N) across circuit depth d >= 4, local Hamiltonian decomposition maintains trainable variance, achieving a 98.2% ground-state fidelity under zero-noise extrapolation (ZNE) error mitigation. Finally, we formulate open challenges in quantum error mitigation scaling, non-perturbative barren plateau escapes, and quantum data loading efficiency. Index Terms—Quantum Machine Learning (QML), Parameterized Quantum Circuits (PQCs), Variational Quantum Eigensolver (VQE), Barren Plateaus, Quantum Kernel Methods, NISQ Computing, Error Mitigation, Hilbert Space.
Whether parameterized quantum circuits (PQCs) can be systematically constructed to be both trainable and expressive remains an open question. Highly expressive PQCs often exhibit barren plateaus, while several trainable alternatives admit efficient classical simulation. We address this question by deriving a finite-sample, dimension-independent concentration bound for estimating the variance of a PQC cost function, yielding explicit trainability guarantees. Across commonly used ansätze, we observe an anticorrelation between trainability and expressibility, consistent with theoretical insights. Building on this observation, we propose a property-based ansatz-search framework for identifying circuits that combine trainability and expressibility. We demonstrate its practical viability on a real quantum computer and apply it to variational quantum algorithms. We identify quantum neural network ansätze with improved effective dimension using over 6×6 \times6× fewer parameters, and for VQE on H2\mathrm{H}_2H2​ we achieve UCCSD-like accuracy at substantially reduced circuit complexity.
Variational quantum circuits (VQCs) face an expressivity-trainability dilemma and scalability challenges. We propose Multi-Layer Fully-Connected Variational Quantum Circuits (FC-VQC), a general-purpose quantum machine learning framework that connects local VQC blocks through measurement, deterministic parameter-free routing, and re-encoding. All trainable model parameters reside within the quantum blocks, without trainable classical neural components. We study fully connected, sliding-window, and parallel block mixing and analyze computational costs, conditional error propagation, and block information exchange. Using classical simulation, we evaluate tabular regression and classification, with our main comparison addressing spatio-temporal function approximation for the Black--Scholes, Burgers, and time-dependent oscillatory PDEs with up to 144144144 spatial dimensions. Comparisons include neural-network, explicit angle-feature, tensor-network, and gradient-boosted-tree baselines. The results show that FC-VQC achieves the lowest trajectory relative MAE for all PDEs at the higher dimensions d∈{81,144}d\in\{81,144\}d∈{81,144}. Gradient-dynamics and depolarizing-noise experiments provide complementary empirical diagnostics of trainability and preliminary noise sensitivity.
We propose SUN-VQC, a variational-circuit architecture whose elementary layers are single exponentials of a symmetry-restricted Lie subgroup, SU(2k)⊂SU(2n)\mathrm{SU}(2^{k}) \subset \mathrm{SU}(2^{n})SU(2k)⊂SU(2n) with k≪nk \ll nk≪n. Confining the evolution to this compact subspace reduces the dynamical Lie-algebra dimension from O(4n)\mathcal{O}(4^{n})O(4n) to O(4k)\mathcal{O}(4^{k})O(4k), ensuring only polynomial suppression of gradient variance and circumventing barren plateaus that plague hardware-efficient ansätze. Exact, hardware-compatible gradients are obtained using a generalized parameter-shift rule, avoiding ancillary qubits and finite-difference bias. Numerical experiments on quantum auto-encoding and classification show that SUN-VQCs sustain order-of-magnitude larger gradient signals, converge 2--3×\times× faster, and reach higher final fidelities than depth-matched Pauli-rotation or hardware-efficient circuits. These results demonstrate that Lie-subalgebra engineering provides a principled, scalable route to barren-plateau-resilient VQAs compatible with near-term quantum processors.