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Lie-theoretic structure of parameter gradients in variational quantum circuits

María-Luisa Pérez-Delgado, Roberto Casado Vara

Published Oct 13, 2026• 0 citations

Abstract

Abstract Variational quantum circuits are a central paradigm of quantum machine learning and near-term quantum computing, yet their practical optimization is often limited by strongly suppressed parameter gradients. Dynamical Lie algebras provide a rigorous description of the infinitesimal directions accessible to a variational architecture, but dynamical accessibility alone does not determine the gradient signal exposed by an actual circuit parametrization. In this work, we develop a Lie-theoretic description of parameter-gradient structure that separates four successive mechanisms: dynamical accessibility, observable visibility, state-dependent activation, and local parameter realization. The optimization observable determines which accessible directions can influence the objective and how strongly they couple to it, while the quantum state selects which of these response modes are locally activated. This information is then pulled back through the circuit parametrization to recover the actual parameter gradients. The resulting construction yields an exact modal decomposition of the accessible gradient signal, a hierarchy of structural bounds, and a natural description of state averaging. An analytical single-qubit example and controlled numerical experiments validate the successive levels of the hierarchy and show that accessible Lie dimension alone is insufficient to characterize the parameter-gradient scale. The results support a mechanism-based view of variational trainability in which parameter gradients emerge from the interaction between accessible dynamics, observable coupling, quantum-state activation, and circuit parametrization.

DOI: 10.1088/2058-9565/aeb3aa

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Abstract Variational quantum circuits are a central paradigm of quantum machine learning and near-term quantum computing, yet their practical optimization is often limited by strongly suppressed parameter gradients. Dynamical Lie algebras provide a rigorous description of the infinitesimal directions accessible to a variational architecture, but dynamical accessibility alone does not determine the gradient signal exposed by an actual circuit parametrization. In this work, we develop a Lie-theoretic description of parameter-gradient structure that separates four successive mechanisms: dynamical accessibility, observable visibility, state-dependent activation, and local parameter realization. The optimization observable determines which accessible directions can influence the objective and how strongly they couple to it, while the quantum state selects which of these response modes are locally activated. This information is then pulled back through the circuit parametrization to recover the actual parameter gradients. The resulting construction yields an exact modal decomposition of the accessible gradient signal, a hierarchy of structural bounds, and a natural description of state averaging. An analytical single-qubit example and controlled numerical experiments validate the successive levels of the hierarchy and show that accessible Lie dimension alone is insufficient to characterize the parameter-gradient scale. The results support a mechanism-based view of variational trainability in which parameter gradients emerge from the interaction between accessible dynamics, observable coupling, quantum-state activation, and circuit parametrization.

Variational quantum algorithms (VQAs) are commonly developed under the assumption that a parameterized circuit corresponds to a single, well-defined physical realization. In practice, this assumption is not satisfied. Differences in compilation, gate ordering, and hardware-level perturbations lead to multiple admissible circuit realizations, each associated with a distinct objective over the same parameter space. The resulting optimization problem is therefore not defined by a single loss function, but by a family of configuration-dependent objectives whose aggregate behaviour must be considered. This study formulates optimization explicitly at the level of such configuration families. A consistent extension of natural-gradient methods is developed in which both the descent direction and the local geometric structure are obtained through averaging across configurations. The resulting update preserves the structure of quantum natural gradient methods while ensuring alignment with the family-defined objective. The approach is evaluated using exact statevector simulations on transverse-field Ising and Heisenberg Hamiltonians under controlled variability. Across all tested regimes, the method produces lower average loss, substantially reduced dispersion across configurations, and improved worst-case performance relative to both configuration-specific optimization and gradient-averaged baselines. These results show that when the objective depends on a set of admissible circuit realizations, optimization must be constructed at the same level to obtain stable and reliable solutions.

Parameterized Quantum Circuits have emerged as a central framework for variational quantum algorithms in the NISQ regime, with major applications in quantum simulation, combinatorial optimization, and, more recently, quantum machine learning (QML). By developing a new approach for computing dynamical Lie algebras associated with quantum circuits combining parametrized gates and non-parametrized entangling gates, we aim to extend the applicability of recent algebraic results to architectures that are actually used in quantum machine learning.

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.

Abstract Ensuring data privacy in machine learning models is critical, especially in distributed settings where model gradients are shared among multiple parties for collaborative learning. Motivated by the increasing success of recovering input data from the gradients of classical models, this study investigates the analogous challenge for variational quantum circuits (VQC) as quantum machine learning models. We highlight the crucial role of the dynamical Lie algebra (DLA) in determining privacy vulnerabilities. While the DLA has been linked to the trainability and simulatability of VQC models, we establish its connection to privacy for the first time. We show that properties conducive to VQC trainability, such as a polynomial-sized DLA, also facilitate extracting detailed snapshots of the input, posing a weak privacy breach. We further investigate conditions for a strong privacy breach, where original input data can be recovered from snapshots by classical or quantum-assisted methods. We establish properties of the encoding map, such as classical simulatability, overlap with DLA basis, and its Fourier frequency characteristics that enable such a privacy breach of VQC models. Our framework thus guides the design of quantum machine learning models, balancing trainability and robust privacy protection.

An important application for near-term quantum computing lies in optimization tasks, with applications ranging from quantum chemistry and drug discovery to machine learning. In many settings, most prominently in so-called parametrized or variational algorithms, the objective function is a result of hybrid quantum-classical processing. To optimize the objective, it is useful to have access to exact gradients of quantum circuits with respect to gate parameters. This paper shows how gradients of expectation values of quantum measurements can be estimated using the same, or almost the same, architecture that executes the original circuit. It generalizes previous results for qubit-based platforms, and proposes recipes for the computation of gradients of continuous-variable circuits. Interestingly, in many important instances it is sufficient to run the original quantum circuit twice while shifting a single gate parameter to obtain the corresponding component of the gradient. More general cases can be solved by conditioning a single gate on an ancilla.

Understanding how the performance of parameterized quantum circuits scales with available resources is important for characterizing their trainability and effective model capacity. In this study, we numerically investigate data scaling, model scaling, and compute scaling in parameterized quantum circuits and examine Lie-algebraic quantities as alternative measures of model size. In addition to the number of circuit parameters, we consider the dimension of the dynamical Lie algebra, the observable-orbit dimension, and a Jacobian effective dimension defined as the rank of the Jacobian of the parameterized observable orbit. Using a regression task with randomly generated Pauli-string generators, we observe decreasing loss with increasing training dataset size, parameter size, and number of optimization iterations over the ranges investigated. For model scaling, the dynamical Lie algebra and observable orbit dimensions rapidly saturate as the parameter size increases, whereas the Jacobian effective dimension remains strongly correlated with the parameter size and exhibits comparable scaling behavior. These results suggest that the Jacobian effective dimension provides a geometry-aware measure of the locally accessible observable degrees of freedom of finite-depth parameterized quantum circuits and may serve as a useful scaling parameter beyond the nominal parameter size.