Long-Range Nonstabilizerness (LRN)

• Long-range nonstabilizerness (LRN) is a quantum property where nonstabilizer magic cannot be removed by shallow local circuits, indicating inherent global correlations and topological structure.
• Diagnostic tools such as stabilizer Rényi entropy, information lattice measures, and mutual information analyses are used to quantitatively isolate and assess LRN.
• LRN plays a critical role in distinguishing quantum phase structure and fault-tolerant quantum computing, as it constrains logical operations in topological codes.

Long-range nonstabilizerness (LRN), also called long-range magic (LRM), denotes the component of nonstabilizerness that cannot be removed by shallow local quantum circuits. In the recent literature, this notion appears in several closely related operational forms: impossibility of mapping a family of states to stabilizer states by shallow local circuits, impossibility of preparing a state from product input by shallow circuits augmented with Clifford layers, and correlation-based constructions that isolate the nonstabilizer content stored in nonlocal degrees of freedom (Korbany et al., 26 Feb 2025, Wei et al., 6 Mar 2025, Li, 26 Mar 2026). Across these formulations, the central issue is whether magic is merely local and circuit-removable, or instead encoded in global structure such as distant correlations, topological sectors, or large-scale information (López et al., 2024, Bilinskaya et al., 30 Oct 2025, Korbany et al., 21 May 2026).

1. Definitions and conceptual variants

A widely used definition is asymptotic and circuit-theoretic. A family of states $\{\ket{\psi_N}\}_{N\in\mathbb{N}}$ has short-range nonstabilizerness if for all $\varepsilon_0>0$ and $\alpha>0$ there exists a local quantum circuit $Q_{D_N}$ of depth $D_N = O(\mathrm{polylog}(N))$ and a stabilizer state $\ket{S_N}$ such that, for all sufficiently large $N$,

$$\Delta\bigl(Q_{D_N}\ket{\psi_N},\,\ket{S_N}\bigr)\le \frac{\varepsilon_0}{N^\alpha} = \varepsilon_N.$$

If no such shallow circuit exists, the family has long-range nonstabilizerness (Korbany et al., 26 Feb 2025).

A related formulation, developed for families $\{|\psi_n\rangle\}$ on bounded-degree lattices, says that the family has long-range magic if for any family of constant-depth local circuits $\{U_n\}$, the states $\varepsilon_0>0$0 remain nonstabilizer states for all sufficiently large $\varepsilon_0>0$1. It has short-range magic if there exists a family of constant-depth local circuits $\varepsilon_0>0$2 such that $\varepsilon_0>0$3 are stabilizer states. A stronger notion, strong LRM, requires that for any constant-depth local circuits $\varepsilon_0>0$4 and any stabilizer states $\varepsilon_0>0$5,

$\varepsilon_0>0$6

This framework also makes explicit that LRM implies long-range entanglement (Wei et al., 6 Mar 2025).

A stricter preparation-based notion is two-sided long-range magic. Here a state is said to have two-sided long-range magic if it cannot be prepared from a product state by either $\varepsilon_0>0$7 or $\varepsilon_0>0$8, even approximately. This places the state outside the first level of the magic hierarchy (Li, 26 Mar 2026).

Alongside these circuit definitions, there are correlation-based formulations. For a bipartite mixed state $\varepsilon_0>0$9, the long-range stabilizer Rényi entropy is defined by

$\alpha>0$0

where $\alpha>0$1. This quantity isolates the magic contained in correlations between $\alpha>0$2 and $\alpha>0$3; if $\alpha>0$4, then $\alpha>0$5 (López et al., 2024).

2. Diagnostics and quantitative witnesses

The long-range SRE construction gives a direct correlation measure of LRN. For pure states, the stabilizer Rényi entropy of order $\alpha>0$6 is

$\alpha>0$7

For mixed states, the order-2 form

$\alpha>0$8

removes contributions from ordinary mixedness. The derived quantity $\alpha>0$9 measures the part of magic that resides in correlations and, in the formulation used there, quantifies magic that cannot be removed by short-depth quantum circuits (López et al., 2024).

A different diagnostic is provided by the information lattice. For a contiguous subsystem $Q_{D_N}$0, the total information is

$Q_{D_N}$1

and the local information is the discrete second derivative

$Q_{D_N}$2

For stabilizer states, all local information values $Q_{D_N}$3 are integers. Hence any noninteger $Q_{D_N}$4 is a direct witness of nonstabilizerness. For localized states with an information gap, the large-scale information

$Q_{D_N}$5

is an $Q_{D_N}$6 quantity, and noninteger $Q_{D_N}$7 is a witness of long-range nonstabilizerness. The same framework introduces a folding procedure, with $Q_{D_N}$8, to separate global from edge-to-edge large-scale contributions (Bilinskaya et al., 30 Oct 2025).

For topologically encoded states on a torus, mutual information between disconnected non-contractible regions provides another diagnostic. If $Q_{D_N}$9 is expanded in the basis of minimal-entropy states associated with a loop $D_N = O(\mathrm{polylog}(N))$0, then

$D_N = O(\mathrm{polylog}(N))$1

where $D_N = O(\mathrm{polylog}(N))$2 is the Shannon entropy. Because stabilizer states have integer-valued entropies and mutual informations, a noninteger thermodynamic limit of this quantity is a sufficient condition for LRN. In the toric code this criterion becomes complete: all encoded non-stabilizer states have LRN. In the doubled Fibonacci model it detects LRN for all states except a finite subset with special modular-group transformation properties (Korbany et al., 21 May 2026).

These diagnostics are not identical, but they are structurally aligned. Each isolates a component of magic that survives shallow local basis changes: correlation magic in $D_N = O(\mathrm{polylog}(N))$3, large-scale noninteger information in $D_N = O(\mathrm{polylog}(N))$4, or noninteger mutual information tied to topological flux sectors.

3. One-dimensional gapped systems and phase structure

A general counting argument shows that long-range nonstabilizerness is generic in Hilbert space. The number of stabilizer states scales as $D_N = O(\mathrm{polylog}(N))$5, while the number of states distinguishable at trace-distance scale $D_N = O(\mathrm{polylog}(N))$6 grows as $D_N = O(\mathrm{polylog}(N))$7. Even after including all shallow local circuits of depth $D_N = O(\mathrm{polylog}(N))$8, the fraction of states reachable from stabilizer states vanishes as $D_N = O(\mathrm{polylog}(N))$9 (Korbany et al., 26 Feb 2025).

The physically sharper analysis concerns one-dimensional gapped systems described by translation-invariant matrix product states. For a TI MPS, one can pass to a canonical renormalization-group fixed point

$\ket{S_N}$0

where the $\ket{S_N}$1 are locally orthogonal fixed-point sectors determined by the local MPS tensors. Any TI MPS is related to its RG fixed point by a shallow local circuit, so LR or SR nonstabilizerness of the original state is equivalent to LR or SR nonstabilizerness of the fixed point (Korbany et al., 26 Feb 2025).

The central sufficient condition in this setting is

$\ket{S_N}$2

The reason is that the mutual information between two distant regions in the fixed-point state is exactly the Shannon entropy of the sector weights, while stabilizer states have integer-quantized mutual information. Since shallow circuits do not change this quantity in the thermodynamic limit, a noninteger limit implies long-range nonstabilizerness (Korbany et al., 26 Feb 2025).

The same work gives a necessary condition for exact short-range nonstabilizerness of RG fixed points: $\ket{S_N}$3 This is derived from constraints on the partial transpose spectrum of stabilizer states (Korbany et al., 26 Feb 2025).

Within this MPS framework, LRN is explicitly stronger than long-range entanglement, and LRN cannot occur in short-range entangled MPS (Korbany et al., 26 Feb 2025). This places LRN naturally in the language of phase structure: it refines the usual distinction between short-range and long-range entanglement by asking whether the non-Clifford component of the state is itself shallow-circuit-trivial.

4. Topological codes, topological order, and fault tolerance

One route to LRN runs through fault-tolerant logical structure. For a $\ket{S_N}$4-dimensional topological stabilizer code family with $\ket{S_N}$5 logical qubits, if a logical state $\ket{S_N}$6 satisfies

$\ket{S_N}$7

then the corresponding encoded physical state $\ket{S_N}$8 has long-range magic. The proof uses a robust generalization of the Bravyi–König theorem: if a constant-depth local unitary maps one topological stabilizer code to another, the induced logical operator must lie in the $\ket{S_N}$9-th level of the Clifford hierarchy (Wei et al., 6 Mar 2025).

The toric-code logical $N$0-state $N$1 is the basic explicit example. Its stabilizer group contains the usual local toric-code stabilizers and a global non-Pauli stabilizer $N$2, and this non-Pauli logical structure forces long-range magic (Wei et al., 6 Mar 2025).

The framework extends beyond stabilizer realizable phases. A topological order cannot be realized by any topological stabilizer code if and only if, for any local Hamiltonian realization of this topological order, all ground states exhibit LRM. Under additional assumptions—local terms with bounded norm, $N$3, a thermodynamic spectral gap, and ground-space dimension not a power of 2—every ground-state family exhibits strong LRM. Fibonacci topological order furnishes the canonical example: on suitable manifolds the ground-space dimension is $N$4, so all ground states have strong LRM (Wei et al., 6 Mar 2025).

For toric-code states on a torus, the mutual-information diagnostic has an immediate logical-gate consequence. Since all encoded non-stabilizer states have LRN, a shallow local logical gate cannot map a stabilizer encoded state to a non-stabilizer encoded state. In the toric code, locality-preserving logical gates therefore act at most as logical Clifford gates. In the doubled Fibonacci case, the corresponding protected-gate classification implies that there are essentially no nontrivial protected gates on the torus (Korbany et al., 21 May 2026).

These results establish a tight connection between LRN and fault-tolerant non-Cliffordness. Long-range magic is not merely a property of wavefunctions; it constrains which logical states and logical gates can be realized by locality-preserving circuits.

5. Explicit states, correlations, and stronger obstructions

Several explicit state families exhibit LRN without requiring a full phase classification. One prominent example is the “magical cat” or ZX-cat state

$N$5

This state cannot be prepared from a product state by either $N$6 or $N$7, even allowing constant approximation error, and therefore has two-sided long-range magic. The proofs combine stabilizer-superposition structure, mutual-information arguments, and approximate quantum error-correcting code techniques (Li, 26 Mar 2026).

The same work proves two-sided long-range magic for ground states of certain nonabelian topological orders. For quantum double and string-net models on closed surfaces of genus $N$8 that admit no topologically transversal gate, no ground state can be prepared by $N$9 within error $$\Delta\bigl(Q_{D_N}\ket{\psi_N},\,\ket{S_N}\bigr)\le \frac{\varepsilon_0}{N^\alpha} = \varepsilon_N.$$0 for fixed $$\Delta\bigl(Q_{D_N}\ket{\psi_N},\,\ket{S_N}\bigr)\le \frac{\varepsilon_0}{N^\alpha} = \varepsilon_N.$$1. Combined with earlier obstructions to $$\Delta\bigl(Q_{D_N}\ket{\psi_N},\,\ket{S_N}\bigr)\le \frac{\varepsilon_0}{N^\alpha} = \varepsilon_N.$$2, this places those states outside the entire first level of the magic hierarchy (Li, 26 Mar 2026).

A complementary route to LRN uses distant two-point correlations. For short-range magic states $$\Delta\bigl(Q_{D_N}\ket{\psi_N},\,\ket{S_N}\bigr)\le \frac{\varepsilon_0}{N^\alpha} = \varepsilon_N.$$3, where $$\Delta\bigl(Q_{D_N}\ket{\psi_N},\,\ket{S_N}\bigr)\le \frac{\varepsilon_0}{N^\alpha} = \varepsilon_N.$$4 is constant-depth local and $$\Delta\bigl(Q_{D_N}\ket{\psi_N},\,\ket{S_N}\bigr)\le \frac{\varepsilon_0}{N^\alpha} = \varepsilon_N.$$5 is stabilizer, the reduced state on distant regions can be generated from $$\Delta\bigl(Q_{D_N}\ket{\psi_N},\,\ket{S_N}\bigr)\le \frac{\varepsilon_0}{N^\alpha} = \varepsilon_N.$$6 EPR pairs by local CPTP maps. This restricts the feasible two-point correlator region. Using this, the hypergraph-state family

$$\Delta\bigl(Q_{D_N}\ket{\psi_N},\,\ket{S_N}\bigr)\le \frac{\varepsilon_0}{N^\alpha} = \varepsilon_N.$$7

is shown to have LRM, because for any distinct qubits $$\Delta\bigl(Q_{D_N}\ket{\psi_N},\,\ket{S_N}\bigr)\le \frac{\varepsilon_0}{N^\alpha} = \varepsilon_N.$$8,

$$\Delta\bigl(Q_{D_N}\ket{\psi_N},\,\ket{S_N}\bigr)\le \frac{\varepsilon_0}{N^\alpha} = \varepsilon_N.$$9

which eventually lies outside the finite-EPR feasible region (Wei et al., 6 Mar 2025).

The same correlation criterion applies to GHZ-type nonstabilizer states

$\{|\psi_n\rangle\}$0

For these,

$\{|\psi_n\rangle\}$1

and if $\{|\psi_n\rangle\}$2 is not a finite-length binary fraction, the family has LRM (Wei et al., 6 Mar 2025).

These constructions show that LRN can arise from at least three distinct mechanisms: encoded logical non-Cliffordness, intrinsic non-stabilizer topological order, and nonlocal correlation patterns that cannot be generated from bounded entanglement plus shallow local processing.

6. Dynamical generation and related spreading phenomena

Long-range SRE provides an exactly solvable dynamical proxy for LRN in a dual-unitary XXZ circuit. If two regions $\{|\psi_n\rangle\}$3 and $\{|\psi_n\rangle\}$4 are separated by distance $\{|\psi_n\rangle\}$5, then the long-range SRE vanishes whenever their backward light cones do not intersect, $\{|\psi_n\rangle\}$6, and becomes nonzero once $\{|\psi_n\rangle\}$7. For the special bipartition

$\{|\psi_n\rangle\}$8

the reduced states $\{|\psi_n\rangle\}$9 and $\{U_n\}$0 are maximally mixed and the long-range SRE reduces to $\{U_n\}$1. In the symmetric case $\{U_n\}$2, $\{U_n\}$3 saturates to $\{U_n\}$4 for any non-Clifford $\{U_n\}$5, while $\{U_n\}$6 for the Clifford points $\{U_n\}$7 and $\{U_n\}$8 (López et al., 2024). This is an explicit light-cone-based creation and saturation of long-range nonstabilizerness.

Related work on nonstabilizerness dynamics, though not always phrased directly as LRN, gives a hydrodynamic backdrop. In the XXZ chain initialized in a domain-wall state, participation entropy and stabilizer Rényi entropy grow as

$\{U_n\}$9

with $\varepsilon_0>0$00 in the ballistic regime, $\varepsilon_0>0$01 in the KPZ-type superdiffusive regime, and $\varepsilon_0>0$02 in the diffusive regime (Tirrito et al., 13 Jun 2025). This suggests that the spatial support of nonstabilizerness is transported by the same hydrodynamic modes that govern conserved charges.

A second hydrodynamic result concerns late-time relaxation in symmetry-constrained random dynamics. In $\varepsilon_0>0$03-symmetric one-dimensional random circuits, the stabilizer Rényi entropy approaches its random-state value with a gap closing as $\varepsilon_0>0$04, and the same scaling is verified in an energy-conserving nonintegrable Ising chain (Xiao et al., 11 Jun 2026). A plausible implication is that the build-up of long-scale nonstabilizer structure can inherit diffusive bottlenecks even when entanglement growth itself is not the limiting factor.

Taken together, these dynamical studies indicate that LRN is not only a static obstruction class. It can be generated, transported, and saturated under local dynamics, with onset set by causal overlap and long-time behavior shaped by hydrodynamics, conservation laws, and circuit architecture.