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arXiv:2601.15396

Quadratic tensors as a unification of Clifford, Gaussian, and free-fermion physics

Andreas Bauer, Seth Lloyd

quant-phmath-phSubmitted Jan 21, 2026• Announced Jan 23, 2026• 0 citations
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Abstract

Certain families of quantum mechanical models can be described and solved efficiently on a classical computer, including qubit or qudit Clifford circuits and stabilizer codes, free-boson or free-fermion models, and certain rotor and GKP codes. We show that all of these families can be described as instances of the same algebraic structure, namely quadratic functions over abelian groups, or more generally over (super) Hopf algebras. Different kinds of degrees of freedom correspond to different "elementary" abelian groups or Hopf algebras: $\mathbb{Z}_2$ for qubits, $\mathbb{Z}_d$ for qudits, $\mathbb{R}$ for continuous variables, both $\mathbb{Z}$ and $\mathbb{R}/\mathbb{Z}$ for rotors, and a super Hopf algebra $\mathcal F$ for fermionic modes. Objects such as states, operators, superoperators, or projection-operator valued measures, etc, are tensors. For the solvable models above, these tensors are quadratic tensors based on quadratic functions. Quadratic tensors with $n$ degrees of freedom are fully specified by only $O(n^2)$ coefficients. Tensor networks of quadratic tensors can be contracted efficiently on the level of these coefficients, using an operation reminiscent of the Schur complement. Our formalism naturally includes models with mixed degrees of freedom, such as qudits of different dimensions. We also use quadratic functions to define generalized stabilizer codes and Clifford gates for arbitrary abelian groups. Finally, we give a generalization from quadratic (or 2nd order) to $i$th order tensors, which are specified by $O(n^i)$ coefficients but cannot be contracted efficiently in general.

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Certain families of quantum mechanical models can be described and solved efficiently on a classical computer, including qubit or qudit Clifford circuits and stabilizer codes, free-boson or free-fermion models, and certain rotor and GKP codes. We show that all of these families can be described as instances of the same algebraic structure, namely quadratic functions over abelian groups, or more generally over (super) Hopf algebras. Different kinds of degrees of freedom correspond to different "elementary" abelian groups or Hopf algebras: Z2\mathbb{Z}_2Z2​ for qubits, Zd\mathbb{Z}_dZd​ for qudits, R\mathbb{R}R for continuous variables, both Z\mathbb{Z}Z and R/Z\mathbb{R}/\mathbb{Z}R/Z for rotors, and a super Hopf algebra F\mathcal FF for fermionic modes. Objects such as states, operators, superoperators, or projection-operator valued measures, etc, are tensors. For the solvable models above, these tensors are quadratic tensors based on quadratic functions. Quadratic tensors with nnn degrees of freedom are fully specified by only O(n2)O(n^2)O(n2) coefficients. Tensor networks of quadratic tensors can be contracted efficiently on the level of these coefficients, using an operation reminiscent of the Schur complement. Our formalism naturally includes models with mixed degrees of freedom, such as qudits of different dimensions. We also use quadratic functions to define generalized stabilizer codes and Clifford gates for arbitrary abelian groups. Finally, we give a generalization from quadratic (or 2nd order) to iiith order tensors, which are specified by O(ni)O(n^i)O(ni) coefficients but cannot be contracted efficiently in general.
The paper "Quadratic tensors as a unification of Clifford, Gaussian, and free-fermion physics" by Andreas Bauer and Seth Lloyd presents a unified algebraic framework for efficiently simulable quantum models. It demonstrates that Clifford circuits, stabilizer codes, free-boson (Gaussian) models, free-fermion models, and certain rotor/Gottesman-Kitaev-Preskill (GKP) codes are all instances of quadratic functions over abelian groups or (super) commutative Hopf algebras. Quadratic tensors, representing quantum objects like states and operators, are specified by O(n²) coefficients for n degrees of freedom and can be efficiently contracted using a generalized Schur complement. The framework supports mixed degrees of freedom (e.g., qudits, continuous variables, fermions) and extends to generalized stabilizer codes and Clifford operations. A generalization to ith-order tensors is provided, though they lack efficient contractibility. This review summarizes the paper's content, compares it with fractal geometry concepts, and ideates connections to Dr. Syed Muntasir Mamun's quantum articulations. Mamun, SM (2026)……………………………………………1 Unifying Algebraic Structures in Solvable Quantum Models: A Review of Quadratic Tensors with Fractal Interpretations and Emergent Quantum Connections Also available on Academia.edu: https://www.academia.edu/attachments/131608598/download_file?s=portfolio
This paper presents a rigorous mathematical and ontological synthesis of solvable quantum systems and emergent fractal geometries. By leveraging the quadratic tensor framework, we demonstrate that disparate families of quantum models—including Clifford circuits, stabilizer codes, Gaussian states, and free-fermion systems—can be unified as quadratic functions over abelian groups or super-commutative Hopf algebras. This algebraic unification permits classical simulation with O(n2)O(n^2)O(n2) complexity via a generalized Schur complement contraction mechanism, dissolving the boundaries between discrete-variable qudits and continuous-variable bosonic modes. However, physical reality frequently transcends these smooth, integer-dimensional algebraic bases. We integrate a fractal-recursive ontology, identifying that quantum trajectories in path integrals and energy spectra in quantum materials (e.g., Hofstadter's butterfly) exhibit non-integer Hausdorff dimensions and self-similar architectures that challenge traditional linear causality. We further explore the transition from quadratic to iii-th order tensors, identifying the Clifford hierarchy's correspondence to higher-order polynomial maps and the resulting computational boundaries of the Gottesman-Knill theorem. Also available on Academia.edu: https://www.academia.edu/attachments/131609415/download_file?s=portfolio
We introduce group surface codes, which are a natural generalization of the Z2\mathbb{Z}_2Z2​ surface code, and equivalent to quantum double models of finite groups with specific boundary conditions. We show that group surface codes can be leveraged to perform non-Clifford gates in Z2\mathbb{Z}_2Z2​ surface codes, thus enabling universal computation along with established means of performing logical Clifford gates. Moreover, for suitably chosen groups, we demonstrate that arbitrary reversible classical gates can be implemented transversally in the group surface code. We present the logical operations in terms of a set of elementary logical operations, which include transversal logical gates, a means of transferring encoded information into and out of group surface codes, and preparation and readout. By composing these elementary operations, we implement a wide variety of logical gates and provide a unified perspective on recent constructions in the literature for sliding group surface codes and preparing magic states. We furthermore use tensor networks inspired by ZX-calculus to construct spacetime implementations of the elementary operations. This spacetime perspective also allows us to establish explicit correspondences with topological gauge theories. Our work extends recent efforts in performing universal quantum computation in topological orders without the braiding of anyons, and shows how certain group surface codes allow us to bypass the restrictions set by the Bravyi-König theorem, which limits the computational power of topological Pauli stabilizer models.
Abstract Associated to every finite abelian group G , there is a generalization of qubit Clifford group associated to G . We demonstrate that every G -Clifford gate can be decomposed into some distinguished elementary types: Fourier transforms, quadratic phase gates, and automorphism gates. Combined with earlier results on classical simulation of normalizer circuits, this implies a generalized Gottesman-Knill-type result. We additionally provide circuits for a universal quantum computing scheme based on local two-qudit G -Clifford gates and magic states.
We develop a theory of stabilizer codes whose stabilizer groups consist of commuting qudit Paulis, oscillator displacements, and/or planar-rotor displacements. We consider codes whose stabilizer groups form generalized lattices in quantum phase space, a condition that guarantees a finite logical dimension. We construct oscillator-rotor, rotor-qudit, and oscillator-rotor-qubit codes that cannot be decomposed into separate subsystems by generalized Clifford transformations. We also introduce an oscillator-qubit code arising from a partially lifted Golay code, where Construction A is applied to half of the Golay code coordinates. We derive analytical forms of logical operators from the symplectic dual lattice, determine Clifford-Gaussian gates from lattice symmetries, show that the code dimension equals the covolume of the stabilizer lattice, and organize the syndrome subspaces into a vector bundle. Our technique builds on earlier non-commutative geometric results by Rieffel, but it can often be interpreted simply as concatenating a given stabilizer code with a Gottesman-Kitaev-Preskill (GKP) code, applying conventional lattice-theoretic results, and unconcatenating. We formulate distances, optimal decoding, and quantum weight enumerators, recovering the Pauli, qudit, and GKP versions together with known MacWilliams identities. We construct finite-energy codewords whose encoding is an isometry up to an error exponentially small in the inverse damping parameter.
A U(1)U(1)U(1) quantum rotor pairs a periodic angle with an integer-valued conjugate momentum, and occurs in molecular rotation, superconducting phase-charge circuits, and compact gauge fields. Coupling such a rotor coherently to qubits gives a hybrid register whose control structure is not inherited from either the oscillator-qubit or the qudit case. We develop a Clifford theory, together with a universal-control result, for registers of nnn qubits and rrr rotors. We classify all automorphisms of the hybrid phase space F22n×Zr×Tr\mathbb{F}_2^{2n}\times\mathbb{Z}^r\times\mathbb{T}^rF22n​×Zr×Tr that preserve the Weyl commutation relations, and give an explicit finite Clifford circuit for each one. The classification is directional: rotor momentum parity may control qubit Pauli operations within the Clifford group, while every nonzero qubit-controlled rotor momentum shift is non-Clifford. It also yields normal forms for the mixed qubit-rotor couplings and the exact minimum number of elementary mixed gates needed to synthesize them. Adding a rotor cosine potential and one fixed qubit-rotor conditional phase to the local Clifford operations gives universal control on the full Hilbert space in the strong operator topology. We then apply this structure in three settings: an exact controlled-shift realization of gauge-covariant matter hopping, which is necessarily non-Clifford; rotor phase estimation with direct angle readout and probe optimization under momentum-support and energy constraints; and finite Fourier transforms on rotor momentum codes, where the one-rotor transform for d=2sd=2^sd=2s compiles into O(s)O(s)O(s) momentum-selective and controlled-shift instructions and each cross-register Fourier factor is implemented by one quadratic rotor Clifford gate.