Structured Stabilizer Decomposition

• Structured stabilizer decomposition is a method that reorganizes quantum stabilizer objects into components based on locality, tensor-product structure, and low-rank simulation.
• It employs techniques such as commutation matrix formalisms and generating tableau methods to factorize multipartite entanglement and isolate non-Clifford operations.
• The approach enables efficient simulation, state recovery, and circuit synthesis by decomposing complex stabilizer systems into manageable algebraic components.

Structured stabilizer decomposition refers to a family of constructions that reorganize stabilizer-theoretic objects into components adapted to locality, tensor-product structure, translation invariance, logical-versus-syndrome separation, or low-rank simulation ansätze. Across the cited works, the decomposed object may be a multipartite stabilizer state, an infinite stabilizer process, a bosonic GKP code space, a Clifford or non-Clifford circuit component, or an unknown quantum state whose stabilizer structure is to be recovered from data (Englbrecht et al., 2022, Comfort et al., 4 Jul 2026, Shaw et al., 2022, Bravyi et al., 2018, Arunachalam et al., 7 Oct 2025). The resulting representations are correspondingly diverse: tuples of alternating bilinear forms up to simultaneous congruence, generating tableaux over $F_p(\delta)$, logical–stabilizer subsystem factorizations, and exact or approximate expansions into stabilizer states or stabilizer projectors.

1. Formal notions and problem classes

The cited literature does not use a single standardized definition of structured stabilizer decomposition. Instead, several formal notions recur. One class of problems asks for tensor-product decompositions of multipartite stabilizer states under a specified local equivalence relation. In that setting, decomposition means deciding when a state is PLC- or LU-equivalent to $\ket{\phi}\otimes\ket{\phi'}$, identifying indecomposable building blocks, and organizing those blocks into an entanglement generating set (EGS) (Englbrecht et al., 2022, Wong et al., 12 Jul 2025).

A second class of problems treats decomposition as a low-rank linear expansion. For pure states, the exact stabilizer rank is

$$\chi(\psi)=\min\left\{k:\ |\psi\rangle=\sum_{\alpha=1}^{k} c_\alpha |\phi_\alpha\rangle,\ |\phi_\alpha\rangle\in \mathrm{STAB}_n\right\},$$

while approximate decomposition is governed by the approximate stabilizer rank $\chi_\delta(\psi)$ and by the stabilizer extent $\xi(\psi)$, defined as the minimum squared $\ell_1$-norm of decomposition coefficients (Bravyi et al., 2018). Related operator-level constructions write a non-Clifford unitary as

$U=\sum_i c_i \ket{\phi_i}\bra{\phi_i},$

with $\ket{\phi_i}$ stabilizer states, so that branching occurs at the operator level rather than only in a state-vector expansion (Huang et al., 2020).

A third class of problems encodes infinite or bosonic stabilizer structure through finite algebraic data. In translation-invariant settings, delayed ZX-diagrams are mapped to generating tableaux $H=[X\mid Z]$ over $F_p(\delta)$ satisfying $\ket{\phi}\otimes\ket{\phi'}$0, and then to canonical normal forms (Comfort et al., 4 Jul 2026). For GKP codes, the oscillator Hilbert space is factorized as a logical subsystem tensored with a stabilizer subsystem,

$\ket{\phi}\otimes\ket{\phi'}$1

with the defining property that tracing out $\ket{\phi}\otimes\ket{\phi'}$2 reproduces ideal decoding (Shaw et al., 2022).

A fourth class is algorithmic. Here the target is not merely existence of a decomposition, but efficient recovery of one. For arbitrary $\ket{\phi}\otimes\ket{\phi'}$3-qubit states, one formulation seeks

$\ket{\phi}\otimes\ket{\phi'}$4

where the extracted part has stabilizer rank $\ket{\phi}\otimes\ket{\phi'}$5 and the residual has stabilizer fidelity $\ket{\phi}\otimes\ket{\phi'}$6 (Arunachalam et al., 7 Oct 2025). By contrast, some structurally adjacent results identify nearby stabilizer structure without outputting a linear decomposition; tolerant testing based on Bell-difference or $\ket{\phi}\otimes\ket{\phi'}$7-type statistics is of that kind (Arunachalam et al., 2024).

2. Multipartite state decomposition and entanglement structure

For multipartite qubit stabilizer states, the most systematic decomposition framework in the cited literature is the commutation matrix formalism. Fix a partition of the qubits into parties. Choosing a basis of the maximally isotropic stabilizer subspace $\ket{\phi}\otimes\ket{\phi'}$8 yields, for each party $\ket{\phi}\otimes\ket{\phi'}$9, an alternating commutation matrix $$\chi(\psi)=\min\left\{k:\ |\psi\rangle=\sum_{\alpha=1}^{k} c_\alpha |\phi_\alpha\rangle,\ |\phi_\alpha\rangle\in \mathrm{STAB}_n\right\},$$0. PLC equivalence is then reduced to simultaneous congruence of the tuple $$\chi(\psi)=\min\left\{k:\ |\psi\rangle=\sum_{\alpha=1}^{k} c_\alpha |\phi_\alpha\rangle,\ |\phi_\alpha\rangle\in \mathrm{STAB}_n\right\},$$1, and tensor-product decomposition is reduced to simultaneous block diagonalization (Englbrecht et al., 2022). The central criterion is

$$\chi(\psi)=\min\left\{k:\ |\psi\rangle=\sum_{\alpha=1}^{k} c_\alpha |\phi_\alpha\rangle,\ |\phi_\alpha\rangle\in \mathrm{STAB}_n\right\},$$2

The same paper also identifies the tuples that actually arise from stabilizer states by the rank-saturation condition

$$\chi(\psi)=\min\left\{k:\ |\psi\rangle=\sum_{\alpha=1}^{k} c_\alpha |\phi_\alpha\rangle,\ |\phi_\alpha\rangle\in \mathrm{STAB}_n\right\},$$3

This reorganization yields a precise notion of indecomposability: indecomposable stabilizer states correspond exactly to indecomposable tuples of alternating forms under simultaneous congruence. The associated endomorphism ring is defined by

$$\chi(\psi)=\min\left\{k:\ |\psi\rangle=\sum_{\alpha=1}^{k} c_\alpha |\phi_\alpha\rangle,\ |\phi_\alpha\rangle\in \mathrm{STAB}_n\right\},$$4

and Fitting’s lemma implies that decomposability is equivalent to the existence of a nontrivial idempotent self-adjoint endomorphism. The same framework gives a sharp structural threshold: the EGS is finite for $$\chi(\psi)=\min\left\{k:\ |\psi\rangle=\sum_{\alpha=1}^{k} c_\alpha |\phi_\alpha\rangle,\ |\phi_\alpha\rangle\in \mathrm{STAB}_n\right\},$$5, but infinite for every $$\chi(\psi)=\min\left\{k:\ |\psi\rangle=\sum_{\alpha=1}^{k} c_\alpha |\phi_\alpha\rangle,\ |\phi_\alpha\rangle\in \mathrm{STAB}_n\right\},$$6; for four parties, the paper exhibits an explicit infinite family of indecomposable spiral graph states (Englbrecht et al., 2022). It also establishes a qudit contrast: for prime dimensions $$\chi(\psi)=\min\left\{k:\ |\psi\rangle=\sum_{\alpha=1}^{k} c_\alpha |\phi_\alpha\rangle,\ |\phi_\alpha\rangle\in \mathrm{STAB}_n\right\},$$7, every state decomposes uniquely into indecomposable EGS elements, whereas for qubits the uniqueness of decomposition remains open in general (Englbrecht et al., 2022).

Graph states admit a parallel graph-theoretic description. PLC equivalence among graph states is generated by local complementations together with within-party edge additions and removals, denoted LCE. This gives a direct move set on graph representatives, while the commutation matrices supply the finite-field algebraic normal form (Englbrecht et al., 2022).

For tripartite pure qudit stabilizer states, a different decomposition theory appears. Any tripartite pure stabilizer state of arbitrary local dimension $$\chi(\psi)=\min\left\{k:\ |\psi\rangle=\sum_{\alpha=1}^{k} c_\alpha |\phi_\alpha\rangle,\ |\phi_\alpha\rangle\in \mathrm{STAB}_n\right\},$$8 can be reduced by local unitaries to a tensor product of $$\chi(\psi)=\min\left\{k:\ |\psi\rangle=\sum_{\alpha=1}^{k} c_\alpha |\phi_\alpha\rangle,\ |\phi_\alpha\rangle\in \mathrm{STAB}_n\right\},$$9-level GHZ states, pairwise $\chi_\delta(\psi)$0-level EPR pairs, and unentangled $\chi_\delta(\psi)$1-level qudits (Wong et al., 12 Jul 2025). The key invariant is a family of subsystem phase matrices $\chi_\delta(\psi)$2, one for each party, with

$\chi_\delta(\psi)$3

These matrices encode subsystem commutation phases of stabilizer generators. Dimension reduction occurs when all projected phase matrices vanish modulo $\chi_\delta(\psi)$4, while GHZ or EPR extraction is triggered by span and intersection conditions on the $\chi_\delta(\psi)$5 (Wong et al., 12 Jul 2025).

An important subtlety is that local Clifford unitaries are not sufficient in arbitrary prime-power dimensions. The prime-power case $\chi_\delta(\psi)$6 requires local non-Clifford unitaries to access embedded $\chi_\delta(\psi)$7-level entanglement structure that the $\chi_\delta(\psi)$8-level Clifford group cannot rearrange. The cited examples at $\chi_\delta(\psi)$9 and $\xi(\psi)$0 are explicitly used to demonstrate that LC equivalence is strictly weaker than LU equivalence for this decomposition problem (Wong et al., 12 Jul 2025). A common misconception is therefore that tripartite stabilizer entanglement normal forms in arbitrary qudit dimension should follow from local Clifford theory alone; the cited results show that this is false in non-squarefree prime-power dimensions.

3. Infinite, translation-invariant, and bosonic decompositions

For infinite translation-invariant stabilizer systems, the delayed stabilizer ZX-calculus provides a finite graphical language built by adding a single delay generator to the odd-prime stabilizer ZX-calculus (Comfort et al., 4 Jul 2026). The delay has two semantics. In the behavioural semantics, a finite delayed diagram is interpreted through all finite truncations, and two such unrollings are identified when they have the same information content. In the algebraic semantics, the delay is interpreted as multiplication by a formal variable $\xi(\psi)$1, so that translated Pauli families become rational generating functions in $\xi(\psi)$2 (Comfort et al., 4 Jul 2026).

The main algebraic object is the generating tableau. A shifted affine Lagrangian subspace is represented by

$\xi(\psi)$3

with

$\xi(\psi)$4

Commutation against all shifts is captured by the shifted symplectic form

$\xi(\psi)$5

The paper then gives a canonical tableau form

$\xi(\psi)$6

with unique data $\xi(\psi)$7 (Comfort et al., 4 Jul 2026). This is a canonical structured decomposition of a translation-invariant stabilizer relation into finite algebraic pieces.

The rewrite theory is equally central. Every delayed diagram reduces to graph-like form, then to AP-form, and finally to a unique reduced AP-form by generalized local complementation, pivoting, Schur complementation, and Gaussian elimination over $\xi(\psi)$8. The paper proves soundness, universality, and completeness for the generating tableau semantics. A necessary caveat is that completeness is established for the generating tableau or shifted-affine-Lagrangian semantics, not for the full channel-sequence semantics; the relation to the full infinite stabilizer group is only

$\xi(\psi)$9

(Comfort et al., 4 Jul 2026). It would therefore be inaccurate to interpret the completeness theorem as a full behavioural completeness statement.

For bosonic GKP codes, the stabilizer subsystem decomposition gives a different kind of structure. For a code specified by $\ell_1$0, one writes

$\ell_1$1

where $\ell_1$2 labels stabilizer eigenvalues in a primitive cell of the dual lattice (Shaw et al., 2022). The basis states $\ell_1$3 are displaced ideal codewords and simultaneous stabilizer eigenstates. In this representation, the decisive identity is

$\ell_1$4

so partial trace over the nonlogical subsystem is equivalent to ideal primitive-cell decoding (Shaw et al., 2022).

This subsystem factorization is decoder-dependent, since the primitive cell $\ell_1$5 fixes how displacements are reduced modulo the dual lattice. The paper also gives explicit cell, Gaussian, and dimension transformations between decompositions, and shows that full unfolding recovers Zak-state decompositions. In single-mode square GKP codes, this framework supports efficient logical-noise simulation for loss, Gaussian displacement, and dephasing, avoiding the large Fock cutoffs required by direct Fock-basis numerics (Shaw et al., 2022).

4. State, gate, and circuit decompositions for simulation and synthesis

In classical simulation, structured stabilizer decomposition most commonly means replacing the hard non-Clifford part of a problem by a low-rank stabilizer ansatz. For pure states, the foundational quantitative statement is the sparsification bound

$\ell_1$6

where $\ell_1$7 is the stabilizer extent (Bravyi et al., 2018). This converts a dense but low-$\ell_1$8-norm stabilizer expansion into a low-rank approximate decomposition. The same paper proves multiplicativity of $\ell_1$9 for tensor products of factors acting on at most three qubits and develops exact or optimal decompositions for $U=\sum_i c_i \ket{\phi_i}\bra{\phi_i},$0, $U=\sum_i c_i \ket{\phi_i}\bra{\phi_i},$1-type states, and CCZ states and gates. A particularly important structural step is the lifting lemma: if a diagonal resource state $U=\sum_i c_i \ket{\phi_i}\bra{\phi_i},$2 has a stabilizer decomposition into equatorial stabilizer states, then the diagonal unitary $U=\sum_i c_i \ket{\phi_i}\bra{\phi_i},$3 itself lifts to a sum-over-Cliffords decomposition (Bravyi et al., 2018). This reorients decomposition from state space to operator space.

A more explicit operator-level version is the stabilizer projector decomposition of unitaries,

$U=\sum_i c_i \ket{\phi_i}\bra{\phi_i},$4

used to build the SPIR and SPC simulation algorithms (Huang et al., 2020). Here only non-Clifford layers are decomposed, while Clifford subcircuits are handled exactly by Gottesman–Knill propagation. The stabilizer projector rank

$U=\sum_i c_i \ket{\phi_i}\bra{\phi_i},$5

controls cost. SPIR is a polynomial-space recursive algorithm with runtime $U=\sum_i c_i \ket{\phi_i}\bra{\phi_i},$6 when the largest non-Clifford layer has projector rank $U=\sum_i c_i \ket{\phi_i}\bra{\phi_i},$7, while SPC is an exponential-space contraction algorithm with runtime $U=\sum_i c_i \ket{\phi_i}\bra{\phi_i},$8 (Huang et al., 2020). The cited work emphasizes that these decompositions are exact, gate-specific, and advantageous only in certain dense-non-Clifford regimes.

Structured decomposition also appears at the circuit-normal-form level. Using Bruhat decomposition of the binary symplectic group, arbitrary stabilizer circuits over $U=\sum_i c_i \ket{\phi_i}\bra{\phi_i},$9 admit the exact 7-stage form

$\ket{\phi_i}$0

as well as the 9-stage form

$\ket{\phi_i}$1

(Maslov et al., 2017). The same work proves a depth-$\ket{\phi_i}$2 implementation on the linear nearest-neighbor architecture and interprets the resulting circuit as a normal form that is optimal in Hadamard count and asymptotically optimal in parameter count (Maslov et al., 2017).

A subgroup-theoretic alternative avoids explicit symplectic-group factorization and instead derives two normal forms directly inside the Clifford group: $\ket{\phi_i}$3 and

$\ket{\phi_i}$4

In the second form, both $\ket{\phi_i}$5 layers are depth $\ket{\phi_i}$6 and together contain at most $\ket{\phi_i}$7 $\ket{\phi_i}$8 gates (Bataille, 2020). The corresponding decomposition of the non-Hadamard subgroup is unique: $\ket{\phi_i}$9 with $H=[X\mid Z]$0 and $H=[X\mid Z]$1 a symmetric zero-diagonal binary matrix (Bataille, 2020). This is a circuit-level analogue of structured stabilizer decomposition: the entangling, linear-reversible, phase, and Pauli content are isolated into algebraically distinct layers.

5. Learning, testing, and recovering stabilizer structure

Algorithmic recovery of structured stabilizer decompositions for unknown states is developed most explicitly in the learning framework of (Arunachalam et al., 7 Oct 2025). The target is, for arbitrary $H=[X\mid Z]$2 and $H=[X\mid Z]$3, to output

$H=[X\mid Z]$4

where each $H=[X\mid Z]$5 is a stabilizer state, $H=[X\mid Z]$6, and the residual has stabilizer fidelity $H=[X\mid Z]$7. The central technical primitive is self-correction with respect to the class of stabilizer states: from copies of a state having fidelity at least $H=[X\mid Z]$8 with some stabilizer state, the algorithm outputs a stabilizer state with fidelity at least $H=[X\mid Z]$9, for a universal constant $F_p(\delta)$0 (Arunachalam et al., 7 Oct 2025).

The decomposition algorithm iterates this self-corrector on successively prepared residual states, using access to a preparation unitary $F_p(\delta)$1 and its controlled version $F_p(\delta)$2. Under the algorithmic polynomial Freiman–Ruzsa conjecture (APFR), the paper gives a polynomial-time protocol for learning a structured decomposition; without APFR, it gives a quasipolynomial-time protocol (Arunachalam et al., 7 Oct 2025). For states of stabilizer extent $F_p(\delta)$3, the resulting learner runs in time $F_p(\delta)$4 unconditionally and polynomial time under APFR; for stabilizer-rank-$F_p(\delta)$5 states, the abstract states an unconditional runtime $F_p(\delta)$6 (Arunachalam et al., 7 Oct 2025). In this setting, structured stabilizer decomposition is explicitly an algorithmic analogue of structure-versus-pseudorandomness.

A related but distinct line of work studies certification rather than output decomposition. Polynomial-time tolerant testing of stabilizer states is based on a quantum Gowers-$F_p(\delta)$7 norm and on Bell-difference statistics (Arunachalam et al., 2024). The key structural implication is that large $F_p(\delta)$8-type signal yields a structured subset $F_p(\delta)$9 of Pauli labels with small doubling,

$\ket{\phi}\otimes\ket{\phi'}$00

from which one derives subgroup structure and a stabilizer covering of the relevant Pauli set. This eventually implies existence of a nearby stabilizer state (Arunachalam et al., 2024). The paper is explicit, however, that it is not a paper on stabilizer rank or on explicit linear-combination decompositions of arbitrary states into stabilizer states. Its output is a tester and a Pauli-side covering theory, not a low-rank stabilizer expansion (Arunachalam et al., 2024).

An adjacent structural result concerns the class of stabilizer states themselves rather than decompositions of general states. For $\ket{\phi}\otimes\ket{\phi'}$01-qubit stabilizer states, the optimal sample complexity of cloning is $\ket{\phi}\otimes\ket{\phi'}$02, matching the known $\ket{\phi}\otimes\ket{\phi'}$03 sample complexity of learning (Bansal et al., 16 Apr 2026). The proof uses hidden linear subspaces, Bell-basis support, and a structured random purification channel, but it does not study stabilizer rank or sparse expansions of non-stabilizer states (Bansal et al., 16 Apr 2026). This clarifies a boundary of the subject: not every efficient structural representation of stabilizer states is a stabilizer decomposition in the rank-theoretic sense.

6. Orbit-sensitive finite-copy decompositions and open questions

The most explicit recent finite-copy decomposition formulas are orbit-sensitive stabilizer-rank identities for magic-state orbits (Labib et al., 27 May 2026). For qutrits, the cited work treats the Strange, Norrell, Hadamard-eigenstate, and $\ket{\phi}\otimes\ket{\phi'}$04 orbits separately and proves that distinct Clifford orbits can exhibit different stabilizer ranks at small tensor powers. The resulting upper bounds on asymptotic stabilizer-rank exponents are

$\ket{\phi}\otimes\ket{\phi'}$05

all strictly below the prior $\ket{\phi}\otimes\ket{\phi'}$06 baseline (Labib et al., 27 May 2026).

These bounds arise from exact, short decomposition identities. The Strange orbit satisfies $\ket{\phi}\otimes\ket{\phi'}$07, implemented by a difference of two quadratic-phase stabilizer states whose support-selective interference reproduces the target sparse support (Labib et al., 27 May 2026). The Hadamard-eigenstate orbit satisfies $\ket{\phi}\otimes\ket{\phi'}$08, using the factorization

$\ket{\phi}\otimes\ket{\phi'}$09

to reduce amplitude matching to a handful of zero-versus-nonzero support classes (Labib et al., 27 May 2026). The Norrell orbit satisfies $\ket{\phi}\otimes\ket{\phi'}$10, and its 4-copy decomposition uses seven stabilizer states organized by indicator variables $\ket{\phi}\otimes\ket{\phi'}$11 and the count $\ket{\phi}\otimes\ket{\phi'}$12 of coordinates equal to $\ket{\phi}\otimes\ket{\phi'}$13 (Labib et al., 27 May 2026). In the qubit case, the paper gives a direct 4-copy algebraic identity with $\ket{\phi}\otimes\ket{\phi'}$14, matching the known $\ket{\phi}\otimes\ket{\phi'}$15 exponent without using the earlier contracted cat-state construction (Labib et al., 27 May 2026).

The same paper also proves the first nontrivial $\ket{\phi}\otimes\ket{\phi'}$16 asymptotic lower bounds for the Hadamard-eigenstate and Norrell orbits, showing that the finite-copy upper bounds do not yet characterize the true exponent (Labib et al., 27 May 2026). Operationally, two copies of $\ket{\phi}\otimes\ket{\phi'}$17 or $\ket{\phi}\otimes\ket{\phi'}$18 can be converted by explicit two-qutrit Cliffords into an injectable phase state with constant success probability, whereas no analogous two-copy conversion to a non-Clifford diagonal gate exists for the Strange orbit (Labib et al., 27 May 2026). This demonstrates that good decomposability and good injectability need not coincide.

Several broad open issues remain visible across the cited literature. For qubit multipartite stabilizer states, uniqueness of decomposition into EGS elements is unresolved in general, even though the prime-qudit case $\ket{\phi}\otimes\ket{\phi'}$19 is unique (Englbrecht et al., 2022). For delayed ZX, completeness holds for the generating tableau semantics rather than the full channel-sequence semantics (Comfort et al., 4 Jul 2026). For learning arbitrary states, the strongest polynomial-time guarantees still depend on APFR (Arunachalam et al., 7 Oct 2025). For arbitrary qudit dimensions, local Clifford theory is insufficient to capture tripartite LU decomposition at prime powers (Wong et al., 12 Jul 2025). These are not incidental technicalities; they delimit where structured stabilizer decomposition is currently a canonical algebraic theory, where it is a constructive but nonunique synthesis method, and where it remains only partially understood.