Low-Rank Stabilizer Decomposition

• Low-rank stabilizer decomposition is the representation of an n-qubit quantum state as a sum of a few stabilizer states, quantified by the exact and approximate stabilizer ranks.
• It is applied in classical simulations by decomposing non-Clifford resources into sums of Clifford circuits, thereby reducing the simulation complexity of quantum circuits.
• Recent research establishes lower bounds and explores orbit-dependent and approximate decompositions, linking these techniques to simulation cost, magic state fidelity, and quantum property testing.

Low-rank stabilizer decomposition denotes the representation of a quantum state as a superposition of a small number of stabilizer states. For an $n$-qubit state $|\psi\rangle$, the associated minimal number of terms is the stabilizer rank $\chi(\psi)$, and the corresponding approximate notion is the $\delta$-approximate stabilizer rank $\chi_\delta(\psi)$. In the literature on classical simulation of Clifford-dominated quantum circuits with non-Clifford resources, these quantities are central because the running time of several simulation methods is determined by the stabilizer rank of tensor powers of magic states; consequently, the subject has developed along two tightly linked directions: constructive low-rank decompositions and lower bounds showing when such decompositions must fail (Bravyi et al., 2018, Peleg et al., 2021).

1. Formal framework

The exact stabilizer rank of a pure state is defined as the smallest integer $r$ such that

$|\psi\rangle=\sum_{j=1}^r c_j|\varphi_j\rangle,$

where each $c_j\in\mathbb{C}$ and each $|\varphi_j\rangle$ is a stabilizer state. A stabilizer state is a state of the form $U|0^n\rangle$ for a Clifford unitary $|\psi\rangle$0 (Peleg et al., 2021). The approximate version, $|\psi\rangle$1, is the minimal stabilizer rank among all states $|\psi\rangle$2 such that $|\psi\rangle$3 (Peleg et al., 2021).

A closely related quantity is the stabilizer extent. For a normalized state $|\psi\rangle$4,

$|\psi\rangle$5

where the minimum is taken over all stabilizer decompositions $|\psi\rangle$6 (Bravyi et al., 2018). The stabilizer fidelity

$|\psi\rangle$7

measures maximal overlap with a stabilizer state (Bravyi et al., 2018).

These notions separate several different questions. Exact stabilizer rank concerns exact synthesis in the stabilizer basis. Approximate stabilizer rank concerns simulation and approximation under norm error. Stabilizer extent is tailored to sparsification and simulation cost bounds. Stabilizer fidelity connects decomposition complexity to proximity to a single stabilizer state. A recurrent theme in the literature is that these measures are related but not interchangeable.

2. Role in classical simulation of quantum circuits

The algorithmic importance of low-rank stabilizer decomposition was systematized in the simulation framework for circuits containing Clifford gates and non-Clifford diagonal gates (Bravyi et al., 2018). One route is a sum-over-Cliffords method: if a circuit is written as

$|\psi\rangle$8

with $|\psi\rangle$9 Clifford and $\chi(\psi)$0 diagonal, then each $\chi(\psi)$1 can be decomposed as

$\chi(\psi)$2

over Clifford operators. The full circuit is then expressed as a superposition of Clifford circuits, and the output state becomes a superposition of stabilizer states (Bravyi et al., 2018).

A second route is gadget-based simulation. In Clifford+$\chi(\psi)$3 and Clifford+CCZ settings, non-Clifford gates are replaced by state-injection gadgets using magic states, and the magic-resource state is itself decomposed into stabilizer states. Two variants were described: a fixed-sample approach that requires exact decompositions, and a random-sample approach that can use approximate decompositions (Bravyi et al., 2018).

The resulting complexity bounds are expressed in terms of stabilizer extent. For Clifford+diagonal circuits, the strong-simulation cost scales as

$\chi(\psi)$4

and the weak-simulation cost as

$\chi(\psi)$5

(Bravyi et al., 2018). This framework extended earlier Clifford+$\chi(\psi)$6 methods to arbitrary diagonal gates and yielded simulations of 40–50 qubits with over 60 non-Clifford gates without high-performance computers. The same work reported a QAOA simulation processing superpositions of $\chi(\psi)$7 stabilizer states and sampling from the full $\chi(\psi)$8-bit output distribution, and Hidden Shift simulations with circuits including up to 64 $\chi(\psi)$9 gates or 16 CCZ gates (Bravyi et al., 2018).

The significance of these results is operational rather than merely representational: low-rank stabilizer decompositions are not just compact descriptions of states, but algorithmic surrogates for the non-Clifford content of a circuit.

3. Approximate decompositions, sparsification, and extent

Approximate low-rank stabilizer decomposition is governed by a sparsification principle. If

$\delta$0

with normalized stabilizer states $\delta$1, then for integer $\delta$2 there exists a state $\delta$3 that is a sum of $\delta$4 stabilizer states sampled according to $\delta$5 such that

$\delta$6

This yields the bound

$\delta$7

and hence

$\delta$8

(Bravyi et al., 2018).

For Clifford magic states, the extent is especially tractable: $\delta$9 (Bravyi et al., 2018). For the single-qubit $\chi_\delta(\psi)$0 state, since $\chi_\delta(\psi)$1, one obtains

$\chi_\delta(\psi)$2

(Bravyi et al., 2018). The same work also established multiplicativity of stabilizer extent for tensor products of single- or few-qubit states, up to three qubits per factor: $\chi_\delta(\psi)$3 (Bravyi et al., 2018).

These results clarify why approximate decompositions became central to simulation practice. Exact rank can be too rigid for algorithm design, whereas extent and approximate rank admit probabilistic sparsification with explicit error control. At the same time, the literature later showed that approximate rank can behave very differently from exact rank, so any interpretation of “low rank” must specify which notion is meant.

4. Lower bounds and limitations

The lower-bound theory developed around tensor powers of single-qubit magic states. For $\chi_\delta(\psi)$4 equal to $\chi_\delta(\psi)$5 or $\chi_\delta(\psi)$6, a 2021 result proved

$\chi_\delta(\psi)$7

improving the previous $\chi_\delta(\psi)$8 lower bound, and also established that for a sufficiently small constant $\chi_\delta(\psi)$9,

$r$0

(Peleg et al., 2021). The same work represented stabilizer states as quadratic functions over affine subspaces of $r$1: $r$2 and recast the amplitudes of $r$3 through the function

$r$4

The proofs combined directional derivatives of quadratic polynomials, combinatorial arguments using Kleitman’s theorem, and complexity-theoretic tools including Razborov–Smolensky low-degree approximations and correlation bounds against majority (Peleg et al., 2021).

A complementary 2021 paper introduced number-theoretic and algebraic-geometric techniques. It showed that if the computational-basis coordinates of a state contain an exponentially increasing subsequence of length $r$5, then

$r$6

and in particular

$r$7

More generally, for any non-stabilizer qubit state $r$8,

$r$9

and for some constant $|\psi\rangle=\sum_{j=1}^r c_j|\varphi_j\rangle,$0,

$|\psi\rangle=\sum_{j=1}^r c_j|\varphi_j\rangle,$1

(Lovitz et al., 2021).

A later probabilistic approach strengthened approximate lower bounds further. It stated that for $|\psi\rangle=\sum_{j=1}^r c_j|\varphi_j\rangle,$2 and a wide range of approximation parameters, the approximate stabilizer rank satisfies a $|\psi\rangle=\sum_{j=1}^r c_j|\varphi_j\rangle,$3 lower bound, using a combination of Haar-random-state arguments, a magic-state teleportation analysis, and a result about trading Clifford operations with $|\psi\rangle=\sum_{j=1}^r c_j|\varphi_j\rangle,$4 gates (Mehraban et al., 2023).

Family or setting	Lower bound	Source
$|\psi\rangle=\sum_{j=1}^r c_j|\varphi_j\rangle,$5 or $|\psi\rangle=\sum_{j=1}^r c_j|\varphi_j\rangle,$6	$|\psi\rangle=\sum_{j=1}^r c_j|\varphi_j\rangle,$7	(Peleg et al., 2021)
$|\psi\rangle=\sum_{j=1}^r c_j|\varphi_j\rangle,$8 or $|\psi\rangle=\sum_{j=1}^r c_j|\varphi_j\rangle,$9, constant-error approximation	$c_j\in\mathbb{C}$0	(Peleg et al., 2021)
Any non-stabilizer qubit $c_j\in\mathbb{C}$1	$c_j\in\mathbb{C}$2	(Lovitz et al., 2021)
Any non-stabilizer qubit $c_j\in\mathbb{C}$3, some constant $c_j\in\mathbb{C}$4	$c_j\in\mathbb{C}$5	(Lovitz et al., 2021)
$c_j\in\mathbb{C}$6, wide range of approximation parameters	$c_j\in\mathbb{C}$7 for approximate rank	(Mehraban et al., 2023)

The conceptual consequence is that low-rank stabilizer decomposition is a powerful simulation primitive only up to a point. The lower-bound papers explicitly connect these limitations to classical simulation hardness and to representation problems for sums of quadratic phases over $c_j\in\mathbb{C}$8 (Peleg et al., 2021, Mehraban et al., 2023).

5. Structural phenomena: separations, multiplicativity, and genericity

Beyond asymptotic lower bounds, the theory uncovered several structural effects. One paper refined a theorem of Moulton to exhibit an explicit sequence of product states with exponential stabilizer rank but constant approximate stabilizer rank, and then gave a stronger field-theoretic construction in which an $c_j\in\mathbb{C}$9-qubit product state has exact stabilizer rank $|\varphi_j\rangle$0 while $|\varphi_j\rangle$1 for sufficiently large $|\varphi_j\rangle$2 (Lovitz et al., 2021). This established an extreme separation between exact and approximate notions.

The same work gave the first non-trivial examples of multiplicative stabilizer rank under tensor product. For the explicitly constructed two-qubit family

$|\varphi_j\rangle$3

one has, for all but finitely many $|\varphi_j\rangle$4,

$|\varphi_j\rangle$5

and in particular this holds for transcendental $|\varphi_j\rangle$6 (Lovitz et al., 2021).

That paper also introduced the generic stabilizer rank

$|\varphi_j\rangle$7

together with its real analogue. Using algebraic geometry, it showed that for each $|\varphi_j\rangle$8, all but finitely many qubit states modulo phase saturate $|\varphi_j\rangle$9, because the set of $U|0^n\rangle$0th tensor powers forms a 1-dimensional irreducible projective variety whose intersection with low-degree secant varieties is finite unless it is the whole variety (Lovitz et al., 2021). The same framework improved an upper bound on $U|0^n\rangle$1 from $U|0^n\rangle$2 to $U|0^n\rangle$3.

These results changed the interpretation of low-rank stabilizer decomposition. They imply that rank behavior is not exhausted by standard magic states; product states can already display maximal exact rank, constant approximate rank, tensor-product multiplicativity, and generic maximality.

6. Orbit-dependent decompositions and higher-dimensional extensions

Recent work generalized the subject from qubit magic states to Clifford orbits of magic states in qubit and qutrit systems. For qubits, the asymptotic per-copy exponent is

$U|0^n\rangle$4

with $U|0^n\rangle$5 for qubits and $U|0^n\rangle$6 for qutrits (Labib et al., 27 May 2026). Lower $U|0^n\rangle$7 corresponds to lower asymptotic stabilizer rank growth.

In the qubit case, an explicit decomposition of the qubit $U|0^n\rangle$8-type orbit at four copies gives

$U|0^n\rangle$9

matching the existing best-known exponent but through a direct algebraic identity rather than an entangled cat-state construction (Labib et al., 27 May 2026). The same paper emphasized orbit dependence at small tensor powers: at $|\psi\rangle$00, $|\psi\rangle$01 whereas $|\psi\rangle$02.

For qutrits, four inequivalent single-qutrit Clifford orbits were analyzed: Strange, Norrell, Hadamard-eigenstate, and qutrit $|\psi\rangle$03-state. Explicit decompositions yielded

$|\psi\rangle$04

all strictly below the prior $|\psi\rangle$05 baseline (Labib et al., 27 May 2026). The same work also proved the first nontrivial $|\psi\rangle$06 asymptotic lower bounds for the Hadamard-eigenstate and Norrell orbits, and described two-qutrit Clifford circuits that convert two copies of these states into an injectable phase state with constant success probability. By contrast, for the Strange orbit, exhaustive search found no two-copy or three-copy Clifford protocol generating a non-Clifford diagonal gate (Labib et al., 27 May 2026).

This orbit-by-orbit analysis shows that low-rank stabilizer decomposition is sensitive not only to the amount of magic resource but also to its Clifford orbit. The work formalized this through explicit decompositions, asymptotic exponents, and machine-checked verification in the open-source library stabrank with Lean 4 proof formalizations (Labib et al., 27 May 2026).

7. Testing low stabilizer complexity and relation to fidelity

The most recent direction connects low-rank stabilizer decomposition to quantum property testing and pseudorandomness. A 2024 work improved tolerant testing of stabilizer states by showing that one can efficiently distinguish whether the maximum fidelity of a quantum state with a stabilizer state is at least $|\psi\rangle$07 or at most $|\psi\rangle$08, provided $|\psi\rangle$09 (Mehraban et al., 2024). The proof combines a random Clifford map with tools from higher-order Fourier analysis.

The same paper studied low stabilizer rank states directly. It showed that if, for an infinite family of quantum states, the stabilizer rank is lower than a constant independent of system size, then stabilizer fidelity is lower bounded by an absolute constant (Mehraban et al., 2024). Using a result of [GIKL22], the paper drew the implication that low approximate stabilizer rank states are not pseudo-random. In a more quantitative formulation, for every constant $|\psi\rangle$10 there exists $|\psi\rangle$11, independent of system size, such that $|\psi\rangle$12 implies $|\psi\rangle$13 (Mehraban et al., 2024).

This line of work places low-rank stabilizer decomposition within a broader complexity landscape. Low stabilizer rank is not only a simulation resource measure; it also constrains how “random-looking” a state can be, links to Gowers-norm methods, and supports efficient testing primitives. A plausible implication is that future progress on low-rank stabilizer decomposition will continue to move simultaneously in three directions: sharper constructive decompositions, sharper lower bounds, and sharper certification procedures for identifying states of low stabilizer complexity.