Non-Stabilizer Tensors in Quantum Computation

• Non-stabilizer tensors are quantum states that cannot be generated by Clifford circuits, characterized by high stabilizer rank and nontrivial entanglement.
• They are constructed using methods like XS-stabilizer and graph-restricted formalisms, offering concrete pathways to realize computational 'magic' in many-body systems.
• Applications in topological models and holographic tensor networks illustrate how these tensors drive advances in quantum gravity and enhance resource theories in computation.

A non-stabilizer tensor is a multidimensional array (or equivalently, a pure quantum state in a multipartite system) whose entries or structure cannot be realized via the stabilizer formalism, i.e., cannot be generated from the action of Clifford circuits on product states or as linear combinations of stabilizer states. In contrast to stabilizer tensors—key resource states in Clifford-based quantum error correction—non-stabilizer tensors exhibit enriched computational complexity, nontrivial entanglement features, and resourcefulness for universal quantum computation and models of quantum gravity. Non-stabilizer tensors appear in diverse contexts, from XS-stabilizer and graph-restricted tensor formalisms to tensor network models for holography and topologically ordered phases.

1. Definitions and Conceptual Foundations

Non-stabilizer tensors are tensors (or quantum states) that do not admit a representation as stabilizer states, i.e., states of the form $U |0\rangle^{\otimes n}$ for $U$ a Clifford circuit, nor are they finite linear combinations thereof. The stabilizer formalism, based on the action of the Clifford group on tensor products of Pauli eigenstates, leads to states allowing efficient classical description and simulation. Non-stabilizer tensors fall outside this efficiently simulable set and are quantified via stabilizer rank: the minimal number of stabilizer states needed in a linear decomposition (Lovitz et al., 2021).

Key formal concepts include:

• Exact stabilizer rank $\chi(\psi)$: the smallest number of stabilizer states whose linear combination yields $\psi$.
• Approximate (ε)-stabilizer rank $\chi_\epsilon(\psi)$: the minimal stabilizer rank required to approximate $\psi$ within trace distance $\epsilon$.

Non-stabilizer tensors are crucial for universal quantum computation, providing "magic" outside the Clifford set, and underpin many-body phenomena not accessible by stabilizer techniques.

2. Constructions: Beyond the Clifford Paradigm

Multiple frameworks yield explicit families of non-stabilizer tensors:

• XS-Stabilizer Formalism: States stabilized by non-commuting tensor products of elements from the Pauli–S group $G_1 = \langle \alpha I, X, S \rangle$ with $\alpha = e^{i\pi/4}$, $S = \text{diag}(1, i)$, which generalizes Clifford circuits by incorporating cubic and higher phase relations in amplitudes. Notable examples include the 6-qubit cubic-phase state, whose amplitude polynomial structure cannot arise in Pauli stabilizer codes (Ni et al., 2014).
• Product States with Large Amplitude Structure: Leveraging number-theoretic arguments (refined Moulton theorem), one can construct product states whose computational basis amplitudes grow exponentially, resulting in exponential stabilizer rank yet potentially constant approximate rank. For example,

$$\psi_n = \sqrt{3/(4^{2^n} - 1)} \bigotimes_{i=1}^n (e_0 + 2^{2^{i-1}} e_1)$$

exhibits $\chi(\psi_n) \ge 2^n/(4n)$ but $\chi_\epsilon(\psi_n) = O(1)$ for all fixed $\epsilon > 0$ (Lovitz et al., 2021).

• Graph-Restricted Tensors: Tensors whose reduced density matrices are maximally mixed for certain bipartitions defined by the clique structure of an underlying graph, supporting entanglement patterns unattainable in the stabilizer formalism (Bistroń et al., 28 Dec 2025). The analytic construction of pentagonal and hexagonal graph-restricted tensors yields continuous families parameterized by irrational numbers, confirming non-Clifford properties.

3. Topological and Network-Based Realization

Non-stabilizer tensors also emerge as resource states in tensor network models and topological quantum field theories:

• SU(2)$_1$ Chern–Simons Theory: W-states and Dicke states—canonical non-stabilizer states—can be prepared and manipulated through 3-manifold path integrals with Wilson loop insertions. The Wilson loop topology injects "magical" non-Clifford content. This construction generalizes stabilizer path integral protocols from abelian $U(1)_k$ to nonabelian SU(2), realizing non-stabilizer tensor networks corresponding to specified multipartite entanglement structures (Munizzi et al., 16 Oct 2025).
• Holographic Tensor Networks: In networks built from graph-restricted or hexagonal non-stabilizer tensors, the entanglement and correlation properties of the boundary theory (e.g., power-law decay of two-point functions with continuously tunable scaling dimensions) cannot be matched by perfect (stabilizer) tensors, thus supporting a "vast landscape" of solvable holographic toy models with diverse critical properties (Bistroń et al., 28 Dec 2025).

4. Algebraic and Geometric Analysis: Rank, Multiplicativity, and Genericity

Algebraic-geometric and number-theoretic methods provide rigorous understanding and classification of non-stabilizer tensors:

• Lower Bounds via Subset-Sum Complexity: The exact stabilizer rank for states with amplitudes forming exponentially increasing sequences is lower bounded by $p/(4\log_2 p)$, with $p$ the sequence length. Generic product states thus lie at maximal distance from the stabilizer variety (Lovitz et al., 2021).
• Multiplicativity of Rank: Construction of families $\psi_\alpha$ with $\chi(\psi_\alpha) = 2$ and $\chi(\psi_\alpha \otimes \psi_\alpha) = 4$ (strict multiplicativity) disproves the possibility that rank can always be strictly submultiplicative, impacting simulation complexity and resource composability.
• Secant Varieties and Generic Rank: For $n$-fold product states, the stabilizer rank generically obeys $\chi_n = O(2^{n/2})$, and almost every such state achieves this maximum. The associated secant varieties encode the closure properties in projective Hilbert space, and a realification argument bounds the real-generic rank by $\chi_n^R \le 2\chi_n$.

5. Algorithmic and Circuit Synthesis Aspects

For some non-stabilizer tensor families, circuit constructions are explicit and tractable:

• XS-Stabilizer Circuit Preparation: States characterized by cubic- or higher-degree phase polynomials can be generated by Clifford circuits supplemented by diagonal non-Clifford gates, specifically $T$, controlled-$S$, and $CCZ$ gates. Preparation involves a Clifford layer, application of diagonal non-Clifford gates matching phase structure, and a final Clifford correction. Circuit size is $O(n)$ (Ni et al., 2014).
• Efficient Classical Algorithms Under Constraints: For regular XS-stabilizer groups (no $S$ phases in diagonal elements), tasks such as existence checking, dimension computation, finding explicit normal forms, and calculating entanglement entropies reduce to efficient polynomial-time routines (Ni et al., 2014).

6. Physical Examples and Applications

Non-stabilizer tensors underlie key many-body phases and serve as resource states in quantum information and condensed matter:

• Topological Models: The doubled semion model and higher twisted quantum doubles $D^\omega(\mathbb{Z}_2^k)$ are realized via XS-stabilizers, achieving ground spaces supporting non-Abelian anyons unattainable by Pauli stabilizer codes (Ni et al., 2014).
• Entanglement Structures in Holography: Graph-restricted non-stabilizer tensors with tunable scaling dimensions give rise to boundary theories in holographic models with nontrivial, non-stabilizer correlations and entanglement spectra, contrasting the triviality of correlators in perfect tensor (stabilizer) models (Bistroń et al., 28 Dec 2025).
• Topological Preparation and Modular Evolution: Non-stabilizer tensor states (e.g., $W_n$, Dicke) are accessible via topologically robust preparation protocols involving Dehn twists and path integrals in $SU(2)_1$ Chern–Simons theory, enriching the geometric understanding of quantum informational resources (Munizzi et al., 16 Oct 2025).

7. Open Problems and Future Directions

Several conceptual and technical questions remain active:

• Fine-Tuning Rank Bounds: Improving lower and upper bounds for stabilizer rank of physically motivated non-stabilizer tensors, especially beyond linear and square-root regimes (Lovitz et al., 2021).
• Classifying Multiplicative Pairs: Determining necessary and sufficient conditions for strict multiplicativity of stabilizer rank under tensor products, relevant for both simulation and resource theory.
• Systematic Exploration of Graph Constraints: Classifying graph-restricted tensor landscapes, especially for higher-dimensional systems and hypergraph generalizations, to chart the full space of accessible non-stabilizer entanglement (Bistroń et al., 28 Dec 2025).
• Physical Realizability: Investigating experimental platforms where non-stabilizer tensors with resourceful entanglement and computational features are robustly preparable.

The proliferation of non-stabilizer tensor families, enabled by algebraic, topological, and combinatorial constructions, suggests an expansive landscape of many-body quantum states with distinct computational and physical properties. These objects drive progress in understanding quantum computational resource theory, entanglement in many-body physics, and models of quantum gravity.