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

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.

• The paper introduces a framework where quadratic tensors efficiently unify Clifford, Gaussian, and free-fermion physics via O(n²) parameter scaling.
• It demonstrates efficient classical simulation using tractable tensor network contractions analogous to the Schur complement.
• The formalism systematically generalizes mixed-dimensional quantum codes and stabilizer states, offering new avenues for quantum error correction.

Quadratic Tensors as an Algebraic Unification of Clifford, Gaussian, and Free-Fermion Models

Overview and Motivation

The paper "Quadratic tensors as a unification of Clifford, Gaussian, and free-fermion physics" (2601.15396) establishes a rigorous algebraic framework that unifies several classes of exactly solvable quantum models—most notably qubit/qudit Clifford circuits, bosonic/fermionic free theories (Gaussian and matchgate models), stabilizer codes, rotor and GKP codes—under the formalism of quadratic tensors defined on abelian groups and (super) Hopf algebras. This algebraic unification elucidates deep structural parallels across models traditionally treated as disparate, identifies shared efficiency in classical simulation (via quadratic parameter scaling and tractable tensor contractions), and provides systematic methods for mixed-dimensional compositions and generalizations to higher-order tensor constructions.

Formal Development

1. Quadratic Functions and Tensors

The cornerstone is the generalization of quadratic functions—second-order polynomials on vector spaces—to arbitrary abelian groups (and, further, to super Hopf algebras for fermionic cases). Given a product group $G = \prod_i G_i$ with "elementary" factors representing qubits ($\mathbb{Z}_2$), qudits ($\mathbb{Z}_d$), continuous variables ($\mathbb{R}$), rotors ($U(1)$ or $\mathbb{R}$ modulo $1$), or fermionic modes (super Hopf algebra $\mathcal F$), a quadratic tensor $T: G \to \mathbb{C}$ is defined such that

$b(g, h) := T(g + h) T(g)^{-1} T(h)^{-1}$

is a bilinear form into $\mathbb{C}^\times$. The tensor itself is governed by the exponential of a quadratic function $q(g)$, and the parameterization of $T$ by $O(n^2)$ coefficients is universal across these systems.

2. Efficient Classical Simulation

Quadratic tensors enable efficient classical simulation: states, operators, and evolutions for Clifford, Gaussian, and matchgate circuits can be tracked via $O(n^2)$ coefficient matrices (and displacement vectors, for affine components), with tensor-network contractions generalizing the Schur complement to arbitrary abelian group settings. The paper comprehensively demonstrates that network contraction does not lead to exponential overhead, provided the model remains quadratic; higher-order polynomials, by contrast, scale as $O(n^i)$ and lose this efficiency.

3. Mixed-Dimensional and Generalized Codes

The formalism naturally encompasses systems with mixed degrees of freedom (qudits of different dimension, rotors, continuous variables, fermions), and prescribes rules for interactions and tensor compositions. The generalization of stabilizer codes and Clifford gates to arbitrary abelian groups means that encoding, decoding, logical operations, and measurement procedures (including codes such as GKP) are uniformly described as quadratic tensor manipulations. The authors give explicit constructions for generalized stabilizer states, projectors, Pauli operators, and measurements, and provide the associated tensor data for each class.

4. Fermionic Extension: Super Hopf Algebras

A key result is the treatment of free-fermion models: by "super-linearizing" abelian groups into super Hopf algebras (with $\mathcal F$ as the essential 2-dimensional case), the anti-commutativity and parity structure of fermions is algebraically embedded. Fermionic tensor networks (matchgate circuits and their generalizations) thus become a subset of quadratic tensor networks, with network contraction rules modified for parity and reordering sign.

Strong Numerical and Formal Claims

• Parameter Scaling: All quadratic tensor networks for $n$ modes (regardless of the mixture of local Hilbert space structures) are fully specified by $O(n^2)$ coefficients; this translates directly to tractable (polynomial-in-$n$) simulation complexity on classical hardware.
• Contractibility: Tensor networks of quadratic tensors (including stabilizer circuits, Gaussian models, and all symplectic evolutions) can be contracted analytically via algebraic procedures analogous to the Schur complement, rather than brute-force combinatorics.
• Universality: Free bosonic, free fermionic, and Clifford/stabilizer models are instances of the same structure—the theory of quadratic functions over commutative/cocommutative (super) Hopf algebras—thus unifying their simulation theory.
• Generalization Limits: Extending to $i$-th order tensor networks escalates complexity from $O(n^2)$ to $O(n^i)$ and destroys efficient contractibility, which is consistent with the known barrier between Clifford-magic and universal quantum computation.

Implications

Practical Implications

The formalism supplies a unified platform for the construction and classical simulation of quantum protocols involving mixed systems (qudits, rotors, continuous variables, fermions); e.g., GKP encoding, measurement-based quantum computation, and hybrid error correction strategies. Tensor data manipulation—rather than wavefunction enumeration—becomes foundational, leveraging efficient storage and contraction. This directly impacts the architecture of quantum simulation packages and compilers for stabilizer, Gaussian, and matchgate models.

Theoretical Implications

The generalization of quantum error-correcting codes, Clifford groups, stabilizer states, and logical operations to arbitrary abelian, affine, and super-algebraic contexts provides a blueprint for exploring new codes and circuit classes insensitive to basis choice or Hilbert space dimensionality. The identification of universal algebraic structures for solvable models sharpens our understanding of classical simulatability, quantum entanglement classification, spectral theory, and the Clifford hierarchy (with explicit correspondence between higher-order tensors and hierarchy levels).

Future Developments

Potential future trajectories include:

• Algebraic classification and construction of hybrid codes and circuits involving arbitrary mixtures of physical modes.
• Systematic investigation of the tractability boundary when perturbing quadratic models with higher-order, nonquadratic terms.
• Tensor-theoretic optimizations in simulation algorithms, hardware design, and quantum compiler technology.
• Extension of symmetry-based analysis (e.g., superselection rules, group cohomology) informed by the Hopf algebra perspective.

Conclusion

This work presents a mathematically rigorous, technically explicit framework that reveals and exploits the unifying structure underlying Clifford, Gaussian, and free-fermion models. By abstracting the essence of exact solvability through quadratic tensors over abelian and super Hopf algebras, the authors enable uniform description, efficient classical simulation, and systematic generalization of a broad class of quantum mechanical systems. The implications for quantum information, simulation, and code design are substantial, and the algebraic methodology sets a foundation for further expansion into hybrid and higher-order domains.