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arXiv:2604.01975

Efficient generation and explicit dimensionality of Lie group-equivariant and permutation-invariant bases

Eloïse Barthelemy, Geneviève Dusson, Camille Hernandez, Liwei Zhang

math.NAquant-phSubmitted Apr 2, 2026• Announced Apr 3, 2026
v1 · Apr 2, 2026v2 · May 22, 2026
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Abstract

In this article, we propose a practical construction of Lie group-equivariant and permutation-invariant functions of $N$ variables from the knowledge of a one-particle basis that is stable with respect to the group action. The construction is generic for any linear Lie group and relies on building a matrix constructed from the Lie algebra whose kernel is spanned by a group-equivariant and permutation-invariant basis. In particular, this construction does not require the knowledge of Clebsch--Gordan coefficients and instead directly builds generalized Clebsch--Gordan coefficients. For specific groups such as $SO(3)$ and $SU(2)$, we exploit the Lie algebra structure to simplify the matrix, which then allows us to derive an explicit formula for the exact dimension of the group-equivariant and permutation-invariant space. Numerical simulations are provided to show that the proposed method scales linearly instead of exponentially for existing methods in the literature. We also show that for large values of $N$, the number of rotation-equivariant and permutation-invariant basis functions is of a comparable order as the number of permutation-invariant basis functions, while pre-asymptotically, a large gain can be achieved by explicitly enforcing rotation-equivariance on top of permutation-invariance.

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53 pages, 9 figures, 6 tables

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In this article, we propose a practical construction of Lie group-equivariant and permutation-invariant functions of NNN variables from the knowledge of a one-particle basis that is stable with respect to the group action. The construction is generic for any linear Lie group and relies on building a matrix constructed from the Lie algebra whose kernel is spanned by a group-equivariant and permutation-invariant basis. In particular, this construction does not require the knowledge of Clebsch--Gordan coefficients and instead directly builds generalized Clebsch--Gordan coefficients. For specific groups such as SO(3)SO(3)SO(3) and SU(2)SU(2)SU(2), we exploit the Lie algebra structure to simplify the matrix, which then allows us to derive an explicit formula for the exact dimension of the group-equivariant and permutation-invariant space. Numerical simulations are provided to show that the proposed method scales linearly instead of exponentially for existing methods in the literature. We also show that for large values of NNN, the number of rotation-equivariant and permutation-invariant basis functions is of a comparable order as the number of permutation-invariant basis functions, while pre-asymptotically, a large gain can be achieved by explicitly enforcing rotation-equivariance on top of permutation-invariance.
Abstract Symmetries are omnipresent in physics and have been used to reduce the number of degrees of freedom of systems. In this work, we investigate the properties of M SU ( n ) , M -invariant subspaces of the special unitary Lie group SU(n). This group is relevant to quantum computing and quantum systems in general. We demonstrate that for certain choices of M , the subset M SU ( n ) inherits many topological and group properties from SU(n). We then present a combinatorial method for computing the dimension of such subspaces when M is a representation of a permutation group acting on a tensor product of spaces G SU ( n ) , or a Hamiltonian H ( n ) SU ( n ) . The Kronecker product of su ( 2 ) matrices is employed to construct the Lie algebras associated with different permutation-invariant groups GSU(n). Numerical results on the number of dimensions support the developed theory.
Understanding which subclasses of quantum circuits are efficiently classically simulable is fundamental to delineating the boundary between classical and quantum computation. In this context, it is well known that certain tasks based on permutation-equivariant unitaries-i.e., nnn-qubit circuits whose action commutes with the qubit-permuting representation of the symmetric group SnS_nSn​-can be simulated in polynomial time. However, existing approaches scale as O(n7)O(n^7)O(n7), and can rapidly become prohibitively expensive. In this work, we introduce a practical algorithm for simulating SnS_nSn​-equivariant circuits under the assumption that the gate generators are at most kkk-local, with k∈O(1)k\in O(1)k∈O(1). The resulting method runs in O(nω+1)O(n^{ω+1})O(nω+1) time for constant depth, where ωωω is the matrix multiplication exponent, significantly lowering the polynomial degree compared to existing techniques. Finally, we numerically validate this scaling by simulating the dynamical evolution of the Lipkin-Meshkov-Glick model, and show that for n=512n=512n=512 spins, a standard laptop can compute the concurrence of the evolved state in under two minutes.
We describe the implications of permutation symmetry for the state space and dynamics of quantum mechanical systems of matrices of general size NNN. We solve the general 11-parameter permutation invariant quantum matrix harmonic oscillator Hamiltonian and calculate the canonical partition function. The permutation invariant sector of the Hilbert space, for general Hamiltonians, can be described using partition algebra diagrams forming the bases of a tower of partition algebras Pk(N){P}_{k}(N)Pk​(N). The integer kkk is interpreted as the degree of matrix oscillator polynomials in the quantum mechanics. Families of interacting Hamiltonians are described which are diagonalized by a representation theoretic basis for the permutation invariant subspace which we construct for $N\ensuremath{\ge}2k$. These include Hamiltonians for which the low-energy states are permutation invariant and can give rise to large ground state degeneracies related to the dimensions of partition algebras. A symmetry-based mechanism for quantum many body scars discussed in the literature can be realized in these matrix systems with permutation symmetry. A mapping of the matrix index values to lattice sites allows a realization of the mechanism in the context of modified Bose-Hubbard models. Extremal correlators analogous to those studied in AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT are shown to obey selection rules based on Clebsch-Gordan multiplicites (Kronecker coefficients) of symmetric groups.
Efficient classical simulation has matured to a critical component of the quantum computing stack, driving hardware validation, algorithm design, benchmarking, and the study of structured quantum dynamics. Lie-algebraic simulation (g\mathfrak{g}g-sim) offers a compelling approach: it represents Heisenberg-picture dynamics in the adjoint space whose dimension is set by the dynamical Lie algebra (DLA) governing the circuit, enabling efficient simulation of expectation values whenever the DLA grows only polynomially with system size. Despite this promise, existing applications of g\mathfrak{g}g-sim have been confined to free-fermionic settings. It has therefore remained unclear if the method can be applied to other structured circuit families, especially when their generators have large Pauli expansions, and hence whether Lie-algebraic simulability presents a genuinely broader paradigm than free fermions. In this work, we resolve this question by identifying additional non-trivial families of polynomial-dimensional DLAs and introducing symmetry-adapted bases that make the required adjoint-space preprocessing tractable. In particular, we develop an explicit Pauli orbit representation for permutation-equivariant dynamics, enabling efficient processing of cubic-dimensional algebras despite exponential Pauli support, and a modified generalized Gell--Mann representation for bounded Hamming-weight (U(1)U(1)U(1)-equivariant) dynamics, yielding polynomial simulation costs on fixed excitation sectors. Together with streamlined routines for free-fermionic algebras, these constructions significantly broaden the practical scope of g\mathfrak{g}g-sim as a unifying simulation tool for structured quantum circuits. Numerical benchmarks confirm favorable preprocessing scaling and validate large-scale proof-of-concept simulations beyond the reach of state-vector simulation.
Abstract The Symmetric group S n manifests itself in large classes of quantum systems as the invariance of certain characteristics of a quantum state with respect to permuting the qubits. Subgroups of S n arise, among many other contexts, to describe label symmetry of classical images with respect to spatial transformations, such as reflection or rotation. Equipped with the formalism of geometric quantum machine learning, in this study we propose the architectures of equivariant quantum convolutional neural networks (EQCNNs) adherent to S n and its subgroups. We demonstrate that a careful choice of pixel-to-qubit embedding order can facilitate easy construction of EQCNNs for small subgroups of S n . Our novel EQCNN architecture corresponding to the full permutation group S n is built by applying all possible QCNNs with equal probability, which can also be conceptualized as a dropout strategy in quantum neural networks. For subgroups of S n , our numerical results using MNIST datasets show better classification accuracy than non-equivariant QCNNs. The S n -equivariant QCNN architecture shows significantly improved training and test performance than non-equivariant QCNN for classification of connected and non-connected graphs. When trained with sufficiently large number of data, the S n -equivariant QCNN shows better average performance compared to S n -equivariant QNN . These results contribute towards building powerful quantum machine learning architectures in permutation-symmetric systems.
The implementation of physical symmetries into problem descriptions allows for the reduction of parameters and computational complexity. We show the integration of the permutation symmetry as the most restrictive discrete symmetry into quantum circuits. The permutation symmetry is the supergroup of all other discrete groups. We identify the permutation with a SWAP⁡\operatorname{SWAP}SWAP operation on the qubits. Based on the extension of the symmetry into the corresponding Lie algebra, quantum circuit element construction is shown via exponentiation. This allows for ready integration of the permutation group symmetry into quantum circuit ansatzes. The scaling of the number of parameters is found to be O(n3)\mathcal{O}(n^3)O(n3), significantly lower than the general case and an indication that symmetry restricts the applicability of quantum computing. We also show how to adapt existing circuits to be invariant under a permutation symmetry by modification.