Quantifying Nonstabilizerness of Quantum Codes by Removing the Inert Background

The gist

Quantifying nonstabilizerness by removing an inert background resolves difficulties in quantifying quantum magic by reducing it to finite classical counting problems for broad classes of codes.

Quantifying Nonstabilizerness

Nonstabilizerness, or magic, quantifies the departure of a quantum state from the stabilizer polytope, which is a resource enabling universal quantum computation alongside stabilizer operations [1–4]. A convenient measure is the second-order stabilizer R´enyi entropy (SRE), defined by M2(ψ⟩) = − log2 2−n X P ∈Pn ⟨ψPψ⟩ 4, where Pn is the n-qubit Pauli group [5, 6]. Direct evaluation of this measure is impractical for generic states because the sum runs over all 4n Pauli operators [7, 8]. Prior work has provided exact values only in special cases and qualitative facts for code families [9–13].

The Peeling Principle

The central observation is that a Clifford unitary conjugates the Pauli group into itself and leaves M2 invariant, meaning any part of a state generated by Clifford operations is invisible to M2 and removable [21]. For a broad range of states, this inert part is precisely the quadratic (order-two) sector, which can be removed because a stabilizer state has a quadratic phase (−1)q(x) in the computational basis [21]. Removing this background isolates an order-three (cubic) object that carries all the magic [22]. This reduction yields closed forms for several code families, such as a formula for all Dicke states and bounds for cubic-phase codes [24–25].

Two Fourth-Moment Dictionaries

The reduction is realized through two fourth-moment dictionaries derived from Parseval’s identity [6]:

1. On the indicator side, the magic of a subset state S⟩ is given by M2(S⟩) = 4 log2 S − log2 E(1)(S), where E(1)(S) = X x∈F n 2 E(S), and E(S) is the additive energy of S [8, 9]. This dictionary applies to any code whose state is Clifford-equivalent to a subset state [23].

2. On the phase side, for a phase state Cf⟩, M2(Cf⟩) = n − log2 2−3n X u,v∈F n 2 X x (−1)∂ 2 f(x;u,v) / 4, where f is the phase function [10]. For cubic functions f, the second derivative is affine in x, leading to M2(Cf⟩) = 2n − log2 N(B), where N(B) counts singular directions of the Hessian [11].

Results for Code Families

The dictionaries yield specific results for four code families:

)&CWS codes the reduction reproduces the companion dictionary [23] and makes nonstabilizerness of Kerdock codes exactly computable [24, 25]. The magic is governed by the additive combinatorial nonlinearity of the classical label code [15].

)&Permutation-invariant Codes and Dicke States A permutation-invariant code has a code space closed under arbitrary qubit permutations [34], and its canonical basis states are the Dicke states [35]. The magic of a 100-qubit Dicke state reduces from a sum over 4100 terms to a short binomial sum [26–28].

)&Cubic-phase Codes and non-Abelian Topological Order For cubic-phase codes, the decoupled form of twisted quantum doubles and non-Abelian topological order yields the exact magic together with a lower bound saturated only by the determinant on F 3 2 (the D4 code) [17]. The minimal instance is equivalent to the D4 topological order [17].

)&Group Algebra Group States For group multiplication states, M2(ψG⟩) = 0 if and only if the group law is cyclic, and the minimal non-cyclic case carries exactly the magic of the ControlledControlled-Z (CCZ) state [24]. The minimal non-cyclic group is the Klein four-group, which exposes the bridge to the cubic picture [37].

Conclusion

In every case, an order-three (cubic) object carries magic after removing an order-two background via a Clifford transformation [15]. The unifying identity is a Parseval fourth moment, which on the indicator side becomes E(1) and on the phase side the Hessian-rank count N(B) [22]. This analysis reduces nonstabilizerness to finite combinatorial problems for these families.

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Quantifying Nonstabilizerness of Quantum Codes by Removing the Inert Background Yuan Liu and Ke-Mi Xu∗ MIIT Key Laboratory of Complex-field Intelligent Exploration, School of Optics and Photonics, Beijing Institute of Technology, Beijing 100081, China Nonstabilizerness (magic) is the quantum resource that, together with stabilizer operations, makes universal quantum computation possible. It appears in fault-tolerant codes and topological order. However, quantifying nonstabilizerness is difficult. The standard measure sums over exponentially many Pauli operators, and for quantum codes no quantitative theory has been available. We resolve this by a structural observation: the Clifford sector carries no magic and can be removed by a Clifford transformation, leaving an order-three object whose magic is an exactly evaluable fourth moment. Removing this inert background yields closed forms for several code families: a formula for all Dicke states, reducing a 100-qubit case from 4100 terms to a short binomial sum; a bound for cubic-phase codes (twisted quantum doubles and non-Abelian topological order), saturated only by the D4 code; and a cyclic/zero criterion for group multiplication states. Together these results reduce the computation of nonstabilizerness for a broad class of codes to finite classical counting problems, and identify the maximally magical codes among them.

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Nonstabilizerness, or magic, quantifies the departure of a quantum state from the stabilizer polytope. Together with stabilizer operations, it is the resource that enables universal quantum computation [1–4]. A convenient measure of nonstabilizerness is the second-order stabilizer R´enyi entropy (SRE) [5, 6] M2(ψ⟩) = − log2 2−n X P ∈Pn ⟨ψPψ⟩ 4, (1) where Pn is the n-qubit Pauli group. The global phases of P are suppressed, since they do not enter the fourth moment. For a generic state the sum runs over all 4n Pauli operators, so a direct evaluation is impractical already for a few tens of qubits [7, 8]. Prior work gives exact values only in special cases and qualitative facts only for code families. Closed forms are known for the W state [9, 10] and for fixedexcitation Dicke [11] and hypergraph states [12], and the permutation-invariant machinery of [13] evaluates symmetric states efficiently in a Dicke basis, but these do not extend to full families or to large codes. Twisted quantum doubles furnish non-Pauli topological stabilizer codes [14, 15], whose SRE encodes fusion rules [16] and whose non-Abelian orders carry extensive, long-ranged magic [17–20], but no exact values have been reported. The exact magic of these code families has thus remained out of reach.

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The central observation of this work is that for a broad and physically relevant class of states the sum of Eq. (1) collapses. A Clifford unitary conjugates the Pauli group into itself and leaves M2 invariant, so any part of a state generated by Clifford operations is invisible to M2 and removable. For a wide range of states, this inert part is precisely the quadratic (order-two) sector: a stabilizer state has a quadratic phase (−1)q(x) in the computational basis [21], and quadratic phases are Clifford-implementable. Removing this background isolates an order-three (cubic) object that carries all the magic. In the language of the Pauli and Clifford hierarchy [22], the Pauli group is a central extension of F 2n 2 classified by the symplectic form, the Clifford layer is the quadratic sector, and magic is the order-three obstruction to quadraticity. Removing the inert background thus reduces the nonstabilizerness of each family to a finite, exactly evaluable quantity. We realize this by two fourth-moment dictionaries: on the indicator side a code state peels to a subset state whose magic is the additive energy of its support, and on the phase side to a phase state whose magic is the rank data of a cubic Hessian. Applied to four code families, the dictionaries yield several results. For Codeword-stabilized (CWS) codes the reduction reproduces the companion dictionary [23] and makes nonstabilizerness of Kerdock codes exactly computable [24, 25]. For permutation-invariant codes we obtain a closed form for all Dicke states, valid for every excitation number k and block length n, so that the magic of a 100-qubit Dicke state reduces from a sum over 4100 terms to a short binomial sum. For cubic-phase codes, the decoupled form of twisted quantum doubles and non-Abelian topological order, we obtain the exact magic together with a lower bound saturated only by the determinant on F 3 2 (the D4 code).

1. Better quantification of quantum resource utilization by using a structural observation: the Clifford sector carries no magic and can be removed by a Clifford transformation, leaving an order-three object whose magic is an exactly evaluable fourth moment. This allows for the reduction of complex sums to finite classical counting problems, enabling exact computation of nonstabilizerness for broad code families.

2. Exact identification of maximally magical quantum codes: The paper provides tools to identify the maximally magical codes among them by yielding closed forms and bounds for specific code families, such as finding that the bound is attainable only if every Mu with u != 0 is nonsingular and identifying the CCZ state as the maximally magic single cubic.

3. Efficient analysis of permutation-invariant states: The derived formula for Dicke states reduces the computation of nonstabilizerness from a sum over 4100 terms to a short binomial sum, which significantly accelerates characterization for large systems like 100-qubit codes.

4. Precise classification of topological order magic: For cubic-phase codes, the magic is quantified by the cubic departure from stabilizer quadraticity, allowing systems with non-Abelian topological order to be characterized by the rank data of a Hessian, specifically identifying that for a benchmark family, M2 = 6 - log2 22.

5. Group theory diagnostics for non-Abelian codes: For group multiplication states, the magic is determined by whether the group law is cyclic, providing an immediate diagnostic criterion to determine if the state is non-stabilizer by checking if the diagonal expectation is not confined to 0, 1.