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arXiv:2608.22017

Quantifying Nonstabilizerness of Codeword-Stabilized Codes

Yuan Liu, Ke-Mi Xu

quant-phSubmitted Aug 22, 2026• Announced Aug 25, 2026• 0 citations• Revised Oct 19, 2026 (v2)
v1 · Aug 22, 2026v2 · Oct 19, 2026
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Abstract

Universal quantum computation requires non-Clifford gates, which stabilizer codes can provide transversally only up to a finite level of the Clifford hierarchy. Non-stabilizer codes are not subject to this bound and are the natural candidates to fill the gap. However, it is difficult to efficiently evaluate the nonstabilizerness (or magic) of non-stabilizer codes. Here, we resolve this problem for codeword-stabilized (CWS) codes by showing that the nonstabilizerness of a CWS code is determined by the difference structure of its codewords. Whenever a Sidon set of the required size exists, the most magical codes are related to the Sidon sets, whose pairwise differences are all distinct. Structural analysis demonstrates that nonstabilizerness is invariant under coset closure, giving rise to non-stabilizer codes with arbitrarily many logical qubits and constant nonstabilizerness. Nonstabilizerness of whole code families becomes exactly computable, and we obtain closed-form values for the Kerdock codes. Moreover, we reveal a tradeoff between nonstabilizerness and transversal gates. For a code with $k$ logical qubits, a diagonal transversal gate that is non-Clifford on $t$ coordinates forces the nonstabilizerness benchmark to be at most $2(k-t)$. These results reduce the search for codes with high nonstabilizerness and for transversal non-Clifford gates to classical counting problems, to which the standard tools of additive combinatorics apply.

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Universal quantum computation requires non-Clifford gates, which stabilizer codes can provide transversally only up to a finite level of the Clifford hierarchy. Non-stabilizer codes are not subject to this bound and are the natural candidates to fill the gap. However, it is difficult to efficiently evaluate the nonstabilizerness (or magic) of non-stabilizer codes. Here, we resolve this problem for codeword-stabilized (CWS) codes by showing that the nonstabilizerness of a CWS code is determined by the difference structure of its codewords. Whenever a Sidon set of the required size exists, the most magical codes are related to the Sidon sets, whose pairwise differences are all distinct. Structural analysis demonstrates that nonstabilizerness is invariant under coset closure, giving rise to non-stabilizer codes with arbitrarily many logical qubits and constant nonstabilizerness. Nonstabilizerness of whole code families becomes exactly computable, and we obtain closed-form values for the Kerdock codes. Moreover, we reveal a tradeoff between nonstabilizerness and transversal gates. For a code with kkk logical qubits, a diagonal transversal gate that is non-Clifford on ttt coordinates forces the nonstabilizerness benchmark to be at most 2(k−t)2(k-t)2(k−t). These results reduce the search for codes with high nonstabilizerness and for transversal non-Clifford gates to classical counting problems, to which the standard tools of additive combinatorics apply.
Nonstabilizerness (magic) is the quantum resource that, together with stabilizer operations, makes universal quantum computation possible. However, quantifying nonstabilizerness is difficult. The standard measure sums over exponentially many Pauli operators, and for quantum codes no quantitative theory has been available. A structural observation removes this obstacle: 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 41004^{100}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 D4D_4D4​ code; and a cyclic/zero criterion for group multiplication states. For codeword-stabilized (CWS) codes, the companion paper \cite{Liu26arXiv} carries this reduction to its conclusion: the most magical codes are exactly the Sidon sets, and the nonstabilizerness bounds the code's transversal non-Clifford power. 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.
Codeword-stabilized quantum codes give a unified graph-state description of stabilizer and nonadditive quantum error-correcting codes. Although each individual CWS word state is stabilizer, coherent superpositions of different word states can be nonstabilizer. We develop a CWS-adapted magic-witness framework that isolates this codeword coherence and converts it into certified lower bounds on robustness of magic. The main result is an exact reduction of the stabilizer threshold of a natural CWS coherence witness to a finite-geometric problem over the classical CWS word set. For general weighted superpositions, the threshold is computed by enumerating affine intersections and affine-quadratic phases. For equal-weight superpositions, the phase optimization collapses, and the threshold is determined entirely by how many CWS words can lie in affine flats of each dimension. Thus a quantum optimization over stabilizer states becomes a classical affine-incidence problem. This reduction yields a fixed-parameter algorithm, an analytic lower bound for an infinite union-stabilizer family, and exact rational certificates for several standard nonadditive CWS examples. The framework also clarifies why exact enumeration fails for large structured families and identifies the remaining task as an affine-intersection problem. The result provides a geometric mechanism by which nonlinear CWS word sets generate certifiable magic.
A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-König bound for nnn-dimensional topological stabilizer codes. In this work, we extend topological Pauli stabilizer codes to a broad class of nnn-dimensional Clifford hierarchy stabilizer codes. These codes correspond to the (n+1)(n+1)(n+1)D Dijkgraaf-Witten gauge theories with non-Abelian topological order. We construct transversal non-Clifford gates through automorphism symmetries represented by cup products. In 2D, we obtain the first transversal non-Clifford logical gates including T and CS for Clifford stabilizer codes, using the automorphism of the twisted Z23\mathbb{Z}_2^3Z23​ gauge theory (equivalent to D4\mathbb{D}_4D4​ topological order). We also combine it with the just-in-time decoder to fault-tolerantly prepare the logical T magic state in O(d)O(d)O(d) rounds via code switching. In 3D, we construct a transversal logical T\sqrt{\text{T}}T​ gate in a non-Clifford stabilizer code at the third level of the Clifford hierarchy, located on a tetrahedron corresponding to a twisted Z24\mathbb{Z}_2^4Z24​ gauge theory. Our constructions surpass the Bravyi-König bound by achieving the logical gates in the (n+1)(n+1)(n+1)-th level of Clifford hierarchy in nnn spatial dimension.
We study the structure and transversal logical capabilities of stabilizer quantum error correcting codes. Among our results, we identify universal lower bounds on circuit depth to generate a full logical Clifford algebra, and develop novel constructions of logical transversal gates including a new depth-one transversal phase S‾\mathrm{\overline{S}}S gate in the rotated surface code and a depth-one intra-block CZ‾\mathrm{\overline{CZ}}CZ gate in the 2D-toric code that generalizes to all odd distances and all lengths L≥3L\ge3L≥3, respectively. Finally, we construct a high-rate non-LDPC CSS code family with parameters [[n,n,Θ(nβ)]][[n,\sqrt{n},Θ({n^β})]][[n,n​,Θ(nβ)]] where β≈0.2823β\approx 0.2823β≈0.2823 in one demonstrated case, that provably possesses a constant-depth complete 2-local transversal logical Clifford basis instruction set architecture (ISA) composed of all individually targeted S‾\mathrm{\overline{S}}S, SHS‾=X\mathrm{\overline{SHS}} = \sqrt{X}SHS=X​, and CZ‾\mathrm{\overline{CZ}}CZ gates. This ISA is depth-one for certain subfamilies that we design and generally constant-depth under certain conditions. The code family is built from a small code with parameters [[n0,2,d0]][[n_0, 2, d_0]][[n0​,2,d0​]], and is tunable in the standard way: it tiles out to form utility-scale logical qubit counts, and it scales up through concatenation to achieve higher distances and error suppression. We show that this construction preserves the depth-one complete transversal logical Clifford basis ISA when composed with these commuting construction actions, inheriting structure from the core codes so that at scale the complete logical Clifford basis ISA remains depth-one up to depth-two addressable operations between tiled cores. We call these Quantum Logic Codes.
Non-Clifford operations required for universal quantum computation can be implemented by measuring logical Clifford operators. We present a generalized framework to fault-tolerantly measure transversal Clifford logical operators in quantum low-density parity check (QLDPC) codes. Our protocol takes any QLDPC code and deforms the code across additional ancillary qubits such that a product of stabilizers measures the desired logical Clifford operator. We prove that the deformed, Clifford-stabilized code remains LDPC and that its distance is preserved up to a constant factor independent of the number of physical qubits. We further prove that the Clifford measurement has a fault distance against data qubit and measurement errors that grows linearly with the distance of the original code. We apply our methods to implement logical non-Clifford gates in algorithmically relevant subroutines and to prepare magic states in quantum stabilizer codes.
It is a major challenge in quantum fault-tolerance to obtain low-overhead protocols for performing non-Clifford gates. In this vein, we construct quantum codes with low-weight stabilizers that support transversal (i.e. low-depth) implementations of the non-Clifford Cr−1ZC^{r-1}ZCr−1Z gate, for every constant r≥3r\geq 3r≥3. In particular, we obtain length-nnn quantum LDPC codes (with constant-weight stabilizers) of polynomial distance d≥n(1−ε)/rd\geq n^{(1-ε)/r}d≥n(1−ε)/r supporting transversal Cr−1ZC^{r-1}ZCr−1Z gates on a close-to-linear number k≥n1−εk\geq n^{1-ε}k≥n1−ε of disjoint tuples of logical qubits, for arbitrarily small ε>0ε>0ε>0. Our construction is the first with constant-weight stabilizers that obtains dk≫ndk\gg ndk≫n, and as a consequence achieves arbitrarily small magic state overhead exponent γ=log⁡(n/k)/log⁡(d)>0γ=\log(n/k)/\log(d)>0γ=log(n/k)/log(d)>0. Comparable prior constructions instead required at least polylogarithmic stabilizer weight. We also show how to obtain linearly many k=Ω(n)k=Ω(n)k=Ω(n) logical Cr−1ZC^{r-1}ZCr−1Z gates, though with stabilizer weight and physical circuit depth nεn^εnε. We show that our transversal gates also support addressing (i.e. targeting) of specific logical qubits. To obtain our codes, we develop a general transformation based on cup products that maps classical codes satisfying a multiplication property to quantum codes with transversal Cr−1ZC^{r-1}ZCr−1Z. We apply this transformation to a new family of classical Tanner codes that we construct from punctured tensor products of algebraic codes.