Stabilizer Rank in Quantum Simulation

• Stabilizer Rank is the minimum number of stabilizer states required to represent a quantum state, quantifying its non-stabilizer structure in simulations.
• Its exact and approximate versions directly affect the efficiency of classical simulations for circuits that combine Clifford operations with magic states.
• Recent advances connect stabilizer rank with algebraic structures and higher-order Fourier analysis, establishing bounds for magic-state tensors and simulation complexities.

Stabilizer rank is the minimum number of stabilizer states required to express a target quantum state as a linear combination. In the near-Clifford simulation literature it is a standard measure of how much non-stabilizer structure a state contains relative to Clifford resources, and its approximate variants govern the cost of simulating circuits built from Clifford operations plus magic states or other non-Clifford ingredients (Peleg et al., 2021, Bravyi et al., 2018). Subsequent work has developed a structural theory of stabilizer decompositions, connected stabilizer states with quadratic phase functions and higher-order Fourier analysis, established exact and approximate lower and upper bounds for magic-state tensor powers, and extended the subject toward learning theory, testing, and qudit magic-state orbit classifications (Labib, 2021, Arunachalam et al., 7 Oct 2025, Labib et al., 27 May 2026).

1. Definition and neighboring notions

For a pure state $|\psi\rangle$, the stabilizer rank $\chi(\psi)$ is the smallest integer $r$ such that

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

where each $|\varphi_j\rangle$ is a stabilizer state and $c_j\in\mathbb C$ (Peleg et al., 2021). Equivalent formulations define $\chi(|\phi\rangle)$ as the minimum number of stabilizer states whose span contains $|\phi\rangle$ (Mehraban et al., 2024). For qubits, a stabilizer state is a state of the form $|\varphi\rangle = U|0^n\rangle$ for an $n$-qubit Clifford unitary $\chi(\psi)$0 (Peleg et al., 2021).

The approximate notion replaces exact equality by approximation. One standard definition is

$\chi(\psi)$1

which asks for the minimum exact stabilizer rank among all states within Euclidean distance $\chi(\psi)$2 of the target (Peleg et al., 2021). In qudit weak-simulation work, closely related fidelity-based approximate notions also appear (Huang et al., 2018). This distinction is operationally important because exact and approximate simulation have different complexity behavior.

Two closely related quantities organize much of the literature. Stabilizer fidelity is the best overlap with a single stabilizer state,

$\chi(\psi)$3

or, in alternate notation,

$\chi(\psi)$4

(Arunachalam et al., 7 Oct 2025, Bravyi et al., 2018). Stabilizer extent is an $\chi(\psi)$5-based optimization over stabilizer decompositions. One formulation is

$\chi(\psi)$6

while another writes

$\chi(\psi)$7

equivalently $\chi(\psi)$8 (Arunachalam et al., 7 Oct 2025, Bravyi et al., 2018). The standard inequality

$\chi(\psi)$9

makes extent a smoother proxy for approximate stabilizer rank in simulation algorithms (Bravyi et al., 2018).

2. Algebraic structure of stabilizer states

A central reason stabilizer rank is mathematically tractable is that stabilizer states have rigid algebraic normal forms. For qubits, one representation used in lower-bound work is

$r$0

where $r$1 is an affine subspace, $r$2 is an $r$3-linear function, and $r$4 is a quadratic polynomial over $r$5 (Peleg et al., 2021). In the corresponding function view, a stabilizer function has the form

$r$6

which makes the decomposition problem amenable to tools from analysis of Boolean functions (Peleg et al., 2021).

A refined formulation identifies stabilizer states with quadratic phase functions on affine subspaces. For qubits,

$r$7

where $r$8 is an affine subspace and $r$9 is quadratic in the sense that $|\psi\rangle=\sum_{j=1}^{r} c_j |\varphi_j\rangle,$0 is independent of $|\psi\rangle=\sum_{j=1}^{r} c_j |\varphi_j\rangle,$1 for all $|\psi\rangle=\sum_{j=1}^{r} c_j |\varphi_j\rangle,$2 (Labib, 2021). In characteristic $|\psi\rangle=\sum_{j=1}^{r} c_j |\varphi_j\rangle,$3, these are nonclassical quadratic phase functions rather than merely classical quadratic polynomials.

For odd prime qudit dimension $|\psi\rangle=\sum_{j=1}^{r} c_j |\varphi_j\rangle,$4, the description is simpler: $|\psi\rangle=\sum_{j=1}^{r} c_j |\varphi_j\rangle,$5 where $|\psi\rangle=\sum_{j=1}^{r} c_j |\varphi_j\rangle,$6, $|\psi\rangle=\sum_{j=1}^{r} c_j |\varphi_j\rangle,$7 is an affine subspace, and $|\psi\rangle=\sum_{j=1}^{r} c_j |\varphi_j\rangle,$8 is a quadratic polynomial (Labib, 2021). In this sense, prime-dimensional stabilizer states are literally quadratic polynomials on affine subspaces of $|\psi\rangle=\sum_{j=1}^{r} c_j |\varphi_j\rangle,$9.

This viewpoint places stabilizer rank within higher-order Fourier analysis. Ordinary Fourier analysis studies correlation with characters, whereas higher-order Fourier analysis studies correlation with polynomial phase functions; stabilizer states correspond to degree-$|\varphi_j\rangle$0 nonclassical polynomial phases, graph states give classical quadratic phases, and qubit stabilizer states sit naturally inside the broader nonclassical degree-$|\varphi_j\rangle$1 framework (Labib, 2021). This identification is the basis for importing inverse theorems, derivative arguments, and polynomial-rank techniques into stabilizer-rank lower bounds.

3. Role in classical simulation

Stabilizer rank matters because Clifford circuits acting on stabilizer states are efficiently classically simulable by the stabilizer formalism. If the non-Clifford part of a computation is isolated into a magic-state resource, then a decomposition

$|\varphi_j\rangle$2

reduces simulation to $|\varphi_j\rangle$3 stabilizer computations followed by a linear recombination (Mehraban et al., 2023). In this sense, upper bounds on stabilizer rank imply faster classical simulation, while lower bounds constrain the effectiveness of rank-based simulation strategies.

The modern simulation framework couples exact and approximate decompositions. For Clifford+$|\varphi_j\rangle$4 circuits with $|\varphi_j\rangle$5 $|\varphi_j\rangle$6 gates, the approximate-simulation cost can be bounded by

$|\varphi_j\rangle$7

and more generally for non-Clifford gates $|\varphi_j\rangle$8 the runtime takes the form

$|\varphi_j\rangle$9

(Bravyi et al., 2018). The same work emphasized that direct decompositions of resources such as CCZ can outperform uniform reductions to $c_j\in\mathbb C$0-gates.

Exact upper bounds remain important for strong simulation. An improved asymptotic result showed

$c_j\in\mathbb C$1

improving the previously known asymptotic exponent $c_j\in\mathbb C$2 (Qassim et al., 2021). The construction proceeds through “magic cat” states and a chain-contraction argument yielding $c_j\in\mathbb C$3, rather than through simple blockwise tensoring. Via known techniques, this gives a classical algorithm that approximates output probabilities of an $c_j\in\mathbb C$4-qubit Clifford+$c_j\in\mathbb C$5 circuit with $c_j\in\mathbb C$6 uses of the $c_j\in\mathbb C$7 gate within inverse polynomial relative error in time $c_j\in\mathbb C$8 (Qassim et al., 2021).

The same basic program extends to odd-prime qudits. For $c_j\in\mathbb C$9 copies of a qudit magic state $\chi(|\phi\rangle)$0, one obtains an explicit approximate stabilizer-rank scaling

$\chi(|\phi\rangle)$1

where $\chi(|\phi\rangle)$2 is the single-qudit orbit overlap determined by a cubic Gauss sum (Huang et al., 2018). The qutrit instance gives weak-simulation cost $\chi(|\phi\rangle)$3, and the same work supplied an $\chi(|\phi\rangle)$4 algorithm for the inner product of two $\chi(|\phi\rangle)$5-qudit stabilizer states (Huang et al., 2018).

4. Lower and upper bounds for magic-state tensor powers

The most developed asymptotic results concern tensor powers of magic states. For exact rank, the qubit benchmark is the linear lower bound

$\chi(|\phi\rangle)$6

which improved the earlier $\chi(|\phi\rangle)$7 lower bound (Peleg et al., 2021). The proof uses the explicit quadratic-on-affine-subspace form of stabilizer states, finds a large affine subspace on which all linear and affine pieces are constant, then studies directional derivatives of the remaining quadratics to force equal amplitudes at two inputs of different Hamming weights, contradicting the strictly weight-dependent amplitudes of $\chi(|\phi\rangle)$8 and $\chi(|\phi\rangle)$9 (Peleg et al., 2021).

The qudit analogue was established using higher-order Fourier analysis. If $|\phi\rangle$0 is the single-qudit magic state built from the generalized $|\phi\rangle$1-gate in prime dimension $|\phi\rangle$2, then

$|\phi\rangle$3

for every prime $|\phi\rangle$4 (Labib, 2021). Here

$|\phi\rangle$5

where $|\phi\rangle$6 is a degree-$|\phi\rangle$7 nonclassical polynomial for $|\phi\rangle$8, and a classical cubic polynomial for $|\phi\rangle$9 (Labib, 2021). The argument shows that the magic-state phase is nearly orthogonal to every quadratic phase,

$|\varphi\rangle = U|0^n\rangle$0

then converts that anti-correlation into a stabilizer-rank lower bound via polynomial-rank and affine-restriction arguments (Labib, 2021).

For approximate rank, an early breakthrough proved that for some absolute constant $|\varphi\rangle = U|0^n\rangle$1,

$|\varphi\rangle = U|0^n\rangle$2

the first nontrivial lower bound of this kind (Peleg et al., 2021). The proof converts a low-rank approximation into a low-degree polynomial approximation to a restricted threshold function, then invokes Razborov–Smolensky lower bounds against majority (Peleg et al., 2021).

A later probabilistic argument strengthened the approximate lower-bound landscape dramatically for $|\varphi\rangle = U|0^n\rangle$3. For a wide range of approximation parameters,

$|\varphi\rangle = U|0^n\rangle$4

equivalently $|\varphi\rangle = U|0^n\rangle$5 at fixed $|\varphi\rangle = U|0^n\rangle$6 (Mehraban et al., 2023). The proof combines a strong lower bound for Haar-random states, the fact that any $|\varphi\rangle = U|0^n\rangle$7-qubit state can be approximated using $|\varphi\rangle = U|0^n\rangle$8 many $|\varphi\rangle = U|0^n\rangle$9 gates, and a teleportation argument showing that approximate stabilizer rank cannot increase through the relevant gadget (Mehraban et al., 2023).

Setting	Quantity	Representative bound
Qubit magic states $n$0	Exact rank	$n$1 (Peleg et al., 2021)
Prime-dimensional qudit magic states $n$2	Exact rank	$n$3 (Labib, 2021)
Qubit magic states $n$4	Approximate rank	$n$5 (Peleg et al., 2021)
Qubit $n$6-state $n$7	Exact upper bound	$n$8 (Qassim et al., 2021)
Qubit $n$9-state $\chi(\psi)$00	Approximate lower bound	$\chi(\psi)$01 (Mehraban et al., 2023)

5. Exact versus approximate rank and other structural subtleties

A recurring theme is that exact stabilizer rank, approximate stabilizer rank, stabilizer fidelity, and stabilizer extent are related but not interchangeable. One explicit separation is furnished by product states with exponentially large exact rank but constant approximate rank. A family constructed from a refinement of Moulton’s theorem includes explicit product states $\chi(\psi)$02 with

$\chi(\psi)$03

for every constant $\chi(\psi)$04 (Lovitz et al., 2021). This shows that exact rank can be very large even when fixed-error approximation is easy.

The same work also gave the first nontrivial examples of multiplicative stabilizer rank under tensor product. For

$\chi(\psi)$05

one has $\chi(\psi)$06, and for all but finitely many $\chi(\psi)$07,

$\chi(\psi)$08

so equality can occur in the general submultiplicative bound $\chi(\psi)$09 (Lovitz et al., 2021). The same paper introduced generic stabilizer rank, defining $\chi(\psi)$10, and proved the upper bound $\chi(\psi)$11 (Lovitz et al., 2021).

The relation to stabilizer fidelity is also subtle. A qualitative theorem states that for each $\chi(\psi)$12 there exists $\chi(\psi)$13 such that any state with $\chi(\psi)$14 satisfies $\chi(\psi)$15, and the same conclusion holds for approximate rank $\chi(\psi)$16 whenever $\chi(\psi)$17 (Mehraban et al., 2024). Thus constant exact or approximate rank forces a constant overlap with some stabilizer state. The converse fails: states can have high stabilizer fidelity and still have arbitrarily large stabilizer rank or approximate stabilizer rank (Mehraban et al., 2024). This is one reason recent testing work treats exact rank as a brittle notion and approximate rank as more meaningful for tolerant testing.

A quantitative version of the fidelity–rank connection was obtained using Barnes Wall lattices. If a pure state $\chi(\psi)$18 has stabilizer fidelity $\chi(\psi)$19, and there exists a state $\chi(\psi)$20 with $\chi(\psi)$21 such that

$\chi(\psi)$22

then

$\chi(\psi)$23

so $\chi(\psi)$24 forces $\chi(\psi)$25 (Kalra et al., 6 Mar 2025). Applied to $\chi(\psi)$26, this reproduces a linear-by-log lower bound in a regime where the approximation fidelity may itself be exponentially small (Kalra et al., 6 Mar 2025).

6. Learning, testing, and orbit-dependent refinements

Recent work has reframed stabilizer rank as an algorithmically extractable structure. A structured decomposition theorem shows that for every $\chi(\psi)$27, an arbitrary $\chi(\psi)$28-qubit state can be written as

$\chi(\psi)$29

where $\chi(\psi)$30 has stabilizer rank $\chi(\psi)$31 and the residual satisfies $\chi(\psi)$32 (Arunachalam et al., 7 Oct 2025). The same work algorithmizes an inverse theorem for the Gowers-$\chi(\psi)$33 norm by introducing a self-correction framework for stabilizer states. Given access to the state-preparation unitary $\chi(\psi)$34 and its controlled version, it yields a learning protocol for a structured stabilizer decomposition in runtime $\chi(\psi)$35 under algorithmic PFR, and $\chi(\psi)$36 unconditionally (Arunachalam et al., 7 Oct 2025). For states of stabilizer extent $\chi(\psi)$37, it outputs a constant-close approximation in time $\chi(\psi)$38, improving to polynomial time under APFR; for stabilizer-rank-$\chi(\psi)$39 states it gives an unconditional runtime $\chi(\psi)$40 (Arunachalam et al., 7 Oct 2025).

Testing results point in the same direction. Improved tolerant-testing bounds distinguish whether the maximum fidelity with a stabilizer state is at least $\chi(\psi)$41 or at most $\chi(\psi)$42, provided $\chi(\psi)$43, and the rank-to-fidelity theorem implies that low approximate stabilizer-rank states are not pseudorandom (Mehraban et al., 2024). This suggests that low stabilizer rank is not only a simulation resource notion but also a statistically detectable structural promise.

Orbit dependence has become a further refinement, especially for qutrits. Distinct Clifford orbits of magic states can exhibit different stabilizer ranks at small tensor powers, and explicit decompositions imply orbit-dependent asymptotic exponents: $\chi(\psi)$44 (Labib et al., 27 May 2026). The same work proved the first nontrivial $\chi(\psi)$45 lower bounds for the Hadamard-eigenstate and Norrell qutrit orbits, and found two-copy Clifford conversion protocols with constant success probabilities $\chi(\psi)$46 for $\chi(\psi)$47 and $\chi(\psi)$48 for $\chi(\psi)$49, enabling constant-overhead injection of a non-Clifford diagonal gate in each case (Labib et al., 27 May 2026). All qutrit decomposition identities were formalized in Lean 4 + mathlib4, underscoring the exact algebraic character of these rank statements (Labib et al., 27 May 2026).

Taken together, these developments present stabilizer rank as a meeting point of simulation complexity, additive combinatorics, higher-order Fourier analysis, lattice methods, and quantum learning theory. Exact and approximate versions capture different operational regimes, their interaction with fidelity and extent is highly nontrivial, and recent orbit-sensitive and algorithmic results indicate that stabilizer rank is best viewed not as a single isolated invariant but as part of a larger hierarchy of stabilizer complexity measures.