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arXiv:2408.00436 · Physical Review A

Search for high-threshold qutrit magic-state distillation routines

Shiroman Prakash, Rishabh Singhal

Published Feb 3, 2026• 0 citations
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Abstract

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 $[[n,1]]_3$ qutrit stabilizer codes with a complete set of transversal Clifford gates for $n\leq 23$, and all $[[n,1]]_3$ stabilizer codes with a transversal $H^2$ gate with $n \leq 9$ qudits. For $n=23$, 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.

DOI
10.1103/61q5-f754

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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.
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.
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.
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.
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.
Qudits offer the potential for low-overhead magic state distillation, although previous results for asymptotically good codes have required qudit dimension q≫100q\gg 100q≫100 or code length N≫100\mathcal{N}\gg 100N≫100. These parameters far exceed experimental demonstrations of qudit platforms, and thus motivate the search for better codes. Using a novel lifting procedure, we construct the first family of good triorthogonal codes on the F22m\mathbb{F}_{2^{2m}}F22m​ alphabet with m≥3m \geq 3m≥3 that lies above the Tsfasman-Vladut-Zink bound. These codes yield a family of asymptotically good quantum codes with transversal CCZ gates, enabling constant space overhead magic state distillation with qudit dimension as small as q=64q=64q=64. Further, we identify a promising code with parameters [[42,14,6]]64[[42,14,6]]_{64}[[42,14,6]]64​. Finally, we show that a distilled ∣CCZ⟩22m|{CCZ}\rangle_{2^{2m}}∣CCZ⟩22m​ can be reduced to a ∣CCZ⟩2n|{CCZ}\rangle_{2^n}∣CCZ⟩2n​ state for arbitrary nnn with a constant-depth Clifford circuit of at most 9 computational basis measurements, 12 single-qudit and 9 two-qudit Clifford gates.
Magic state distillation (MSD) is a procedure to purify noisy non-stabilizer states from a quantum code, useful towards fault-tolerant quantum computing. Quantum circuits requiring non-Clifford gates can be realized using special states called magic states that are the eigenstates of both non-Clifford and Clifford gates. The purification of noisy states is realized by projecting these magic states onto the space spanned by the eigenstates of the projection operator of the stabilizer code. Given a noisy non- stabilizer state from a given stabilizer code, not all states can be distilled towards magic states. In this paper, we provide the condition for distillability of magic states specific to the choice of qudit stabilizer codes. We derive the noise threshold for magic state distillation depending on the code parameters. We prove that magic state distillation is a convergent map in the entropic sense. We also give an efficient decoding circuit for the best-known code to distill the∣Hi⟩\left|H_i\right\rangle∣Hi​⟩state, which is an eigenstate of the qutrit Hadamard gate.