Probabilistic modeling over permutations using quantum computers

This article presents an argument for why quantum computers could unlock new methods for machine learning. We argue that spectral methods, in particular those that learn, regularise, or otherwise manipulate the Fourier spectrum of a machine learning model, are often natural for quantum computers. For example, if a generative machine learning model is represented by a quantum state, the Quantum Fourier Transform allows us to manipulate the Fourier spectrum of the state using the entire toolbox of quantum routines, an operation that is usually prohibitive for classical models. At the same time, spectral methods are surprisingly fundamental to machine learning: A spectral bias has recently been hypothesised to be the core principle behind the success of deep learning; support vector machines have been known for decades to regularise in Fourier space, and convolutional neural nets build filters in the Fourier space of images. Could, then, quantum computing open fundamentally different, much more direct and resource-efficient ways to design the spectral properties of a model? We discuss this potential in detail here, hoping to stimulate a direction in quantum machine learning research that puts the question of ``why quantum?'' first.

A central design principle in modern machine learning and artificial intelligence is to align a model's inductive bias with the structure of its input data. For matrix-valued inputs, relevant matrix-level relationships can be characterised through spectral values and spectral subspaces; however, common coordinate-wise rotation-gate data-encoding unitaries used in most quantum machine learning models do not explicitly construct such a matrix-level representation. We introduce Quantum Spectral Models (QSMs), in which we construct the generator of the data-encoding unitary directly from each input matrix. We study three QSM variants based on symmetric, global block, and non-overlapping patch-local block Hamiltonians. Their outputs admit truncated Fourier representations in which input-dependent spectral gaps supply candidate phase carriers, while spectral subspaces help determine their coefficients. We evaluate the QSMs and comparison quantum models on two matrix representations of Pendigits and two controlled synthetic tasks defined by spectral statistics. At the largest evaluated circuit depth, QSM variants lead the tested quantum models in mean test accuracy across all four benchmarks. The patch-local QSM leads on Pendigits, whereas the global block-Hamiltonian QSM leads on the controlled spectral tasks. Ablations show a task-dependent reversal: subspace-preserving controls perform better on Pendigits, whereas spectral-value-only controls lead among the tested ablations on the synthetic tasks. Together, these results shed new light on quantum machine-learning model design by showing how input-conditioned spectral representations can provide an analysable inductive bias, while offering a broader perspective on structure-aware model design in machine learning and artificial intelligence.

Structured state-space models (SSMs) achieve linear-time inference and logarithmic-time parallel training via associative scans, yet homogeneous diagonal transitions generate solvable semigroups unable to track non-solvable group words or 3D rigid-body manifolds. We develop a theoretical and empirical framework for Non-Commutative Lie-Algebraic State-Space Models (Lie-SSMs), parameterizing recurrent updates on Lie algebra tangent spaces via the matrix exponential. First, via an o-minimal multi-rate cylindrical flag collar, we establish an unconditional single-layer finite-division criterion for homogeneous contracting $SO(3)$ and $SU(2)$ heads, proving odd-prime Sylow abelianity, odd-$p$-core centrality $[u^2,x]=e$, exponent-30 commutator laws ($[u^{30},v^{30}]=e$ in $SO(3)^m$ and $[u^{30},v^{30}]\in Z(G),[u^{30},v^{30}]^2=e$ in $SU(2)^m,SO(4)^m$, excluding $S_5$), and sharp threshold $d^*(S_5)=d^*(A_6)=5$ in $SO(5)$. Second, Depth-2 semidirect and Schreier $2$-cocycle cascades resolve $S_5$, $2I\cong SL_2(F_5)$, and $\mathrm{Heis}(F_p)$ in $O(\log T)$ depth, whereas unnormalized contracting coupling collapses exponentially without Directional RMSNorm. Third, we characterize the $O(\varepsilon^{c-1})$ Chen--Magnus--Hall--Witt commutator saddle and $1/|[G,G]|$ coset-centroid dilution, proving $\varepsilon=0$ symmetry breaking and local Polyak--Łojasiewicz convergence ($θ_{\max}<π$), solving $S_5$ ($99.90\%/99.40\%$ at $T=128$) and $D_{16}$ ($100.00\%$) via dense prefix loss and factored readouts. Across $Q_8,A_5,S_5,D_{16}$, Dyck-2, $SE_2(3)$ Left-Jacobian odometry, GPU scans ($26.03\times$ speedup), and three 3D IMU cohorts (UCI HAR, HAPT-PT 30-subject CV, and REALDISP), Depth-2 Lie-$SO(3)$ cascades reach $95.69\%\pm 0.14\%$ on HAR, $87.16\%\pm 0.64\%$ ($88.32\%\pm 0.93\%$ 30-subject episodes) on HAPT-PT, and $84.23\%\pm 1.00\%$ under displacement on REALDISP.