Stabilizer Extent in Quantum Computation

• Stabilizer extent is a measure of nonstabilizerness quantifying the minimal squared l1-norm in decomposing a quantum state into stabilizer states.
• It employs advanced methods such as convex optimization, column generation, and tomography to estimate the resource cost for classical simulation of Clifford+T circuits.
• This metric underpins resource theory by assessing the 'magic' within quantum states, thereby guiding simulation strategies and quantum error-correction benchmarks.

Stabilizer extent is a central quantitative measure of nonstabilizerness and magic for pure quantum states, particularly in the resource theory of magic for quantum computation. For an $n$-qubit pure state $\ket{\psi}$, the stabilizer extent $\xi(\ket{\psi})$ is the minimum squared $\ell_1$-norm achievable over all decompositions of $\ket{\psi}$ as a superposition of normalized stabilizer states. This monotone captures the complexity of expressing nonstabilizer states as linear combinations of stabilizer states, governs leading classical simulation costs for Clifford+$T$ circuits, and quantifies the nonstabilizerness resource available in a given state. Stabilizer extent also generalizes to arbitrary model classes and admits algorithmic, geometric, and operational interpretations.

1. Formal Definition and Mathematical Properties

For a pure $n$-qubit state $\ket{\psi}$, the stabilizer extent is defined as

$$\xi(\ket{\psi}) = \min\left\{\left(\sum_{i} |c_i|\right)^2 : \ket{\psi} = \sum_{i} c_i \ket{\phi_i},\, \ket{\phi_i} \text{ stabilizer states}\right\}$$

Alternatively, some recent works use the unsquared $\ell_1$-norm

$\ket{\psi}$0

but this is a convention and the distinction does not affect qualitative results (Arunachalam et al., 7 Oct 2025).

Stabilizer extent is always greater than or equal to the inverse of the stabilizer fidelity and is bounded above by the robustness of magic: $\ket{\psi}$1 where $\ket{\psi}$2 is the maximal squared overlap with a stabilizer state (Haug et al., 2023).

For composite or product states, stabilizer extent is submultiplicative: $\ket{\psi}$3 but, crucially, not multiplicative in general (Heimendahl et al., 2020). For most states, $\ket{\psi}$4, a fact that impacts simulation strategies for large quantum systems.

2. Geometric and Resource-Theoretic Interpretation

Stabilizer extent is a convex-geometric measure of how much a state deviates from the convex hull of stabilizer states (the so-called stabilizer polytope). In resource theory, it is a magic monotone: it is nonincreasing under stabilizer-preserving (Clifford) operations and measures the "amount of magic" available for quantum computational tasks.

It is distinct from stabilizer rank ($\ket{\psi}$5), the minimal number of stabilizer states required in a decomposition: $\ket{\psi}$6 Unlike rank, extent captures "spread" by summing the moduli of coefficients, not just counting terms, making it more sensitive to widely distributed decompositions.

Operationally, stabilizer extent quantifies the cost for classical simulation algorithms based on stabilizer decompositions: the runtime of leading simulators for Clifford+$\ket{\psi}$7 circuits scales with (typically polynomial in) the stabilizer extent of non-Clifford input states (Hamaguchi et al., 2024, Heimendahl et al., 2020).

3. Algorithms for Estimation and Exact Computation

Direct computation of stabilizer extent is intractable for large $\ket{\psi}$8 due to the $\ket{\psi}$9 number of stabilizer states. Several algorithms have been developed to circumvent this scaling:

• Bounded estimation by stabilizer entropy measurements: By measuring the $\xi(\ket{\psi})$0-th Pauli moments $\xi(\ket{\psi})$1 for integer $\xi(\ket{\psi})$2, one obtains efficiently accessible lower bounds: $\xi(\ket{\psi})$3. This approach enables scalable experimental access and benchmarking on noisy quantum hardware (Haug et al., 2023).
• Exact computation via convex optimization (SOCP) and column generation: The stabilizer extent minimization can be formulated as a complex $\xi(\ket{\psi})$4-minimization subject to a linear representability constraint. Recent advances employ "stabilizer pruning" and column generation: only a small, adaptively selected subset of stabilizer states is retained at each iteration, guided by solving primal/dual SOCPs and efficiently computed stabilizer overlaps. This approach has enabled the exact computation of stabilizer extent for Haar random pure states up to $\xi(\ket{\psi})$5 qubits (general) and $\xi(\ket{\psi})$6 (real) on a classical computer, leveraging recursive enumeration and in-place memory optimization (Hamaguchi et al., 2024).
• Learning-theoretic and tomography approaches: The stabilizer extent controls the complexity of quantum state tomography and learning, with sample and time complexity scaling quasi-polynomially or polynomially in the extent (and $\xi(\ket{\psi})$7) under certain algorithmic conjectures (Arunachalam et al., 5 Jun 2026, Arunachalam et al., 7 Oct 2025). Ridge regression and boosting of weak stabilizer agnostic learners are the central algorithmic tools.

Table: Stabilizer Extent—Algorithmic and Operational Aspects

Approach	Scaling	Comments
Bell-measurement bounds	$\xi(\ket{\psi})$8 copies, $\xi(\ket{\psi})$9 classical	Lower bounds via $\ell_1$0 (Haug et al., 2023)
Column-generation SOCP	Superexp. in $\ell_1$1 naive; $\ell_1$2 qubits feasible	Adaptive subset, exact computation (Hamaguchi et al., 2024)
Tomography (learning)	$\ell_1$3	Quasi-poly or poly in $\ell_1$4 (with conjectures) (Arunachalam et al., 5 Jun 2026, Arunachalam et al., 7 Oct 2025)

4. Behavior under Quantum Operations and Amortization

Stabilizer extent can be extended to measure the nonstabilizerness generation capability of unitary channels using an amortized framework. For a unitary $\ell_1$5,

$\ell_1$6

A strict version further restricts to stabilizer inputs. Critically, for stabilizer extent the amortized and strict-amortized versions coincide: $\ell_1$7 for any unitary $\ell_1$8 (Zhu et al., 2024). Amortization (i.e., allowing arbitrary magic input states) cannot enhance the magic-generating capacity measured by stabilizer extent. This sharply contrasts with the amortized $\ell_1$9-stabilizer Rényi entropy, for which magic generation can be enhanced by input nonstabilizerness.

For Clifford+$\ket{\psi}$0 circuits, worst-case upper bounds on stabilizer extent scale as $\ket{\psi}$1 for $\ket{\psi}$2 $\ket{\psi}$3 gates (Arunachalam et al., 7 Oct 2025).

5. Connections to Magic Monotones and Simulation

Stabilizer extent sits in a hierarchy with other monotones: $\ket{\psi}$4 where $\ket{\psi}$5 is robustness of magic, $\ket{\psi}$6 is stabilizer fidelity, and $\ket{\psi}$7 are Pauli moment invariants (Haug et al., 2023). The extent, while operationally similar to robustness for pure states, is strictly weaker in its convexity and sensitivity to decomposition structure, and does not share multiplicativity properties.

For classical simulation of Clifford+$\ket{\psi}$8 circuits, the leading Monte-Carlo–based simulator costs scale with the stabilizer extent of input magic states, making it a key complexity determinant (Heimendahl et al., 2020, Hamaguchi et al., 2024).

6. Non-multiplicativity and Limitations

Stabilizer extent is not multiplicative under tensor products: $\ket{\psi}$9 for generic states, as proved in (Heimendahl et al., 2020). This is established via duality geometry and the effect of the maximally entangled state in the stabilizer dictionary, which breaks the extremality of product dual witnesses. Non-multiplicativity translates directly to several related monotones, including mixed-state extent and generalized robustness. For simulation, this means that naïvely multiplying the extents of subsystems may overestimate the true simulation complexity.

7. Applications and Theoretical Implications

Algorithmic learning and tomography: Stabilizer extent controls the efficiency with which a quantum state can be learned or tomographically reconstructed from data. States with bounded extent $T$0 admit learning and tomography algorithms with sample complexity $T$1 under certain conjectures, and via ridge-regression boosting of weak agnostic learners (Arunachalam et al., 5 Jun 2026, Arunachalam et al., 7 Oct 2025).

Resource-theoretic significance: As a magic monotone, stabilizer extent underpins the quantitative structure of the magic-state resource theory and provides strict guarantees about the feasibility of classical simulation, the formation of magic states, and the transfer properties of magic through quantum channels (Hamaguchi et al., 2024, Zhu et al., 2024).

Relationship to geometric and combinatorial structures: The structure and behavior of stabilizer extent is intimately connected with the geometry of the stabilizer polytope, the enumeration and overlap properties of stabilizer states (García et al., 2017), and the extremal positioning of so-called "maximally magical" states—such as Alltop vectors and MUB-balanced states in prime dimensions—relative to stabilizer bases (Andersson et al., 2014).

A plausible implication is that for quantum error correction, compilation, and benchmarking tasks that depend on the nonstabilizerness of resource states, stabilizer extent provides a nuanced and computationally relevant measure that improves upon crude counts of non-Clifford operations and subsumes both geometric and operational aspects. Theoretical advances in computation and learning of stabilizer extent (Hamaguchi et al., 2024, Arunachalam et al., 5 Jun 2026) are likely to drive new protocols for efficient certification, benchmarking, or simulation in fault-tolerant quantum computation.