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arXiv:2603.18560 · Quantum Science and Technology

High-threshold magic state distillation with quantum quadratic residue codes

Michael Zurel, Santanil Jana, Nadish de Silva

quant-phSubmitted Mar 19, 2026• Announced Mar 20, 2026• 0 citations• Revised Sep 30, 2026 (v2)
v1 · Mar 19, 2026v2 · Sep 30, 2026
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Abstract

We present applications of quantum quadratic residue codes in magic state distillation. This includes showing that existing codes which are known to distill magic states, like the $5$-qubit perfect code, the $7$-qubit Steane code, and the $11$-qutrit and $23$-qubit Golay codes, are equivalent to certain quantum quadratic residue codes. We also present new examples of quantum quadratic residue codes that distill qubit $T$ states and qutrit Strange states with high thresholds, and we show that there are infinitely many quantum quadratic residue codes that distill $T$ states with a non-trivial threshold. All of these codes, including the codes with the highest currently known thresholds for $T$ state and Strange state distillation, are unified under the umbrella of quantum quadratic residue codes.

Journal reference
Quantum Sci. Technol. 11 045052 (2026)
DOI
10.1088/2058-9565/aea362
Comments
42 pages, 8 figures

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We present applications of quantum quadratic residue codes in magic state distillation. This includes showing that existing codes which are known to distill magic states, like the 555-qubit perfect code, the 777-qubit Steane code, and the 111111-qutrit and 232323-qubit Golay codes, are equivalent to certain quantum quadratic residue codes. We also present new examples of quantum quadratic residue codes that distill qubit TTT states and qutrit Strange states with high thresholds, and we show that there are infinitely many quantum quadratic residue codes that distill TTT states with a non-trivial threshold. All of these codes, including the codes with the highest currently known thresholds for TTT state and Strange state distillation, are unified under the umbrella of quantum quadratic residue codes.
Determining the best attainable threshold for qudit magic state distillation is directly related to the question of whether or not contextuality is sufficient for universal quantum computation. We show that the performance of a qudit correcting code for magic state distillation is captured by its complete weight enumerator. For the qutrit strange state -- a maximally magic non-stabilizer state -- the performance of a code is captured by its simple weight enumerator. This result allows us to carry out an extensive search for high-threshold magic state distillation routines for the strange state. Our search covers all [[n,1]]3[[n,1]]_3[[n,1]]3​ qutrit stabilizer codes with a complete set of transversal Clifford gates for n≤23n\leq 23n≤23, and all [[n,1]]3[[n,1]]_3[[n,1]]3​ stabilizer codes with a transversal H2H^2H2 gate with n≤9n \leq 9n≤9 qudits. For n=23n=23n=23, we find over 600 CSS codes that can distill the qutrit strange state with cubic noise suppression. While none of these codes surpass the threshold of the 11-qutrit Golay code, their existence suggests that, for large codes, the ability to distill the qutrit strange state is somewhat generic.
Magic states, by allowing non-Clifford gates through gate teleportation, are important building blocks of fault-tolerant quantum computation. Magic state distillation protocols aim to create clean copies of magic states from many noisier copies. However, the prevailing protocols require substantial qubit overhead. We present a distillation protocol based on permutation-invariant gnu codes, as small as two qubits. The two-qubit protocol achieves a 0.5 error threshold and 1/2 distillation rate, surpassing prior schemes for comparable codes. Our protocol furthermore distils magic states with arbitrary magic by varying the position of the ideal input states on the Bloch sphere. We achieve this by departing from the usual magic state distillation formalism, allowing the use of non-Clifford gates in the distillation protocol, and allowing the form of the output state to differ from the input state. Our protocol is compatible for use in tandem with existing magic state distillation protocols to enhance their performance.
The ternary Golay code—one of the first and most beautiful classical error-correcting codes discovered—naturally gives rise to an 11-qutrit quantum error correcting code. We apply this code to magic state distillation, a leading approach to fault-tolerant quantum computing. We find that the 11-qutrit Golay code can distil the ‘most magic’ qutrit state—an eigenstate of the qutrit Fourier transform known as the strange state —with cubic error suppression and a remarkably high threshold. It also distils the ‘second-most magic’ qutrit state, the Norell state, with quadratic error suppression and an equally high threshold to depolarizing noise.
Magic-state distillation is a protocol to purify noisy quantum states using quantum stabilizer codes. In this paper, we provide conditions for identifying magic states along with analytical expressions for computing the noise threshold. Through quantum entropic results, we prove the convergence of the magic-state distillation procedure. We derive analytical results to show how the noise threshold scales as a function of the minimum distance and the number of physical qudits of the underlying quantum stabilizer code. We distill $|T\ensuremath{\rangle}$ and $|S\ensuremath{\rangle}$ states with thresholds 0.498 (theoretical limit is 0.5) and 0.71 (theoretical limit is 0.75) over the depolarizing channel using [[31,1,16]]2{[[31,1,16]]}_{2}[[31,1,16]]2​ quantum Bose-Chaudhuri-Hocquenghem code and [[29,1,15]]3{[[29,1,15]]}_{3}[[29,1,15]]3​ quantum qutrit code, respectively. Finally, encoding algorithms based on quantum Calderbank-Shor-Steane (CSS) and non-CSS qudit stabilizer codes along with circuits are proposed for magic-state distillation.
We show that using qutrits rather than qubits leads to a substantial reduction in the overhead cost associated with an approach to fault-tolerant quantum computing known as magic state distillation. We construct a family of [[9m−k,k,2]]3 triorthogonal qutrit error-correcting codes for any positive integers m and k with k≤3m−2 that are suitable for magic state distillation. In magic state distillation, the number of ancillae required to produce a magic state with target error rate ϵ is O(logγ⁡ϵ−1) , where the yield parameter γ characterizes the overhead cost. For k=3m−2 , our codes have γ=log2⁡(2+63m−2) , which tends to 1 as m→∞ . Moreover, the [[20,7,2]]3 qutrit code that arises from our construction when m=3 already has a yield parameter of 1.51 which outperforms all known qubit triorthogonal codes of size less than a few hundred qubits.
Many proposals for fault-tolerant quantum computation require injection of 'magic states' to achieve a universal set of operations. Some qubit states are above a threshold fidelity, allowing them to be converted into magic states via 'magic state distillation', a process based on stabilizer codes from quantum error correction. We define quantum weight enumerators that take into account the sign of the stabilizer operators. These enumerators completely describe the magic state distillation behavior when distilling T-type magic states. While it is straightforward to calculate them directly by counting exponentially many operator weights, it is also an NP-hard problem to compute them in general. This suggests that finding a family of distillation schemes with desired threshold properties is at least as hard as finding the weight distributions of a family of classical codes. Additionally, we develop search algorithms fast enough to analyze all useful 5 qubit codes and some 7 qubit codes, finding no codes that surpass the best known threshold.