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1. Quantum Error Correction beyond SU(2): Spin, Bosonic, and Permutation-Invariant Codes from Convex Geometry

   https://doi.org/10.1103/kx3b-4nrp  

   Aydin, Arda; Albert, Victor V; Barg, Alexander (February 2026, PRX Quantum)

   We develop a framework for constructing quantum error-correcting codes and logical gates for three types of spaces—composite permutation-invariant spaces of many qubits or qudits, composite constant-excitation Fock state spaces of many bosonic modes, and monolithic nuclear state spaces of atoms, ions, and molecules. By identifying all three spaces with discrete simplices and representations of the Lie group $SU\left(q\right)$ , we prove that many codes and their gates in $SU\left(q\right)$ can be interconverted between the three state spaces. We construct code instances for all three spaces using classical ${\ell }_1$ codes and Tverberg’s theorem, a classic result from convex geometry. We obtain families of quantum codes with distance that scales almost linearly with the code length $N$ by constructing ${\ell }_1$ codes based on combinatorial patterns called Sidon sets and utilizing their Tverberg partitions. This compares favorably with the existing designs for all the state spaces. We present explicit constructions of codes with shorter length or lower total spin/excitation than known codes with similar parameters, bosonic codes with exotic Gaussian gates, as well as examples of short codes with distance larger than the known constructions. 

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2. Designs from Magic-Augmented Clifford Circuits

   https://doi.org/10.1103/myrb-nyhf  

   Zhang, Yuzhen; Vijay, Sagar; Gu, Yingfei; Bao, Yimu (March 2026, PRX Quantum)

   We introduce magic-augmented Clifford circuits—architectures in which Clifford circuits are preceded and/or followed by constant-depth circuits of non-Clifford (“magic”) gates—as a resource-efficient way to realize approximate $k$ -designs, with reduced circuit depth and usage of magic. We prove that shallow-Clifford circuits, when augmented with constant-depth circuits of magic gates, can generate approximate unitary and state $k$ -designs with $\epsilon$ relative error. The total circuit depth for these constructions on $N$ qubits is $O(\log (N/\epsilon \left)\right)+2^{O(k\log k)}$ in one dimension and $O(\log \log (N/\epsilon \left)\right)+2^{O(k\log k)}$ in all-to-all circuits using ancillas, which improves upon previous results for small $k\ge 4$ . Furthermore, our construction of relative-error state $k$ -designs only involves states with strictly local magic. The required number of magic gates is parametrically reduced when considering $k$ -designs with bounded additive error. As an example, we show that shallow-Clifford circuits followed by $O\left(k^2\right)$ single-qubit magic gates, independent of system size, can generate an additive-error state $k$ -design. We develop a classical statistical mechanics description of our random circuit architectures, which provides a quantitative understanding of the required depth and number of magic gates for additive-error state $k$ -designs. We also prove no-go theorems for various architectures to generate designs with bounded relative error. 

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3. Error-correcting codes for fermionic quantum simulation

   https://doi.org/10.21468/SciPostPhys.16.1.033  

   Chen, Yu-An; Gorshkov, Alexey V.; Xu, Yijia (January 2024, SciPost Physics)

   Utilizing the framework of\mathbb{Z}_2 ${\mathbb{Z}}_2$ lattice gauge theories in the context of Pauli stabilizer codes, we present methodologies for simulating fermions via qubit systems on a two-dimensional square lattice. We investigate the symplectic automorphisms of the Pauli module over the Laurent polynomial ring. This enables us to systematically increase the code distances of stabilizer codes while fixing the rate between encoded logical fermions and physical qubits. We identify a family of stabilizer codes suitable for fermion simulation, achieving code distances of d=2,3,4,5,6,7, allowing correction of any\lfloor \frac{d-1}{2} \rfloor $\lfloor \frac{d-1}{2}\rfloor$ -qubit error. In contrast to the traditional code concatenation approach, our method can increase the code distances without decreasing the (fermionic) code rate. In particular, we explicitly show all stabilizers and logical operators for codes with code distances of d=3,4,5. We provide syndromes for all Pauli errors and invent a syndrome-matching algorithm to compute code distances numerically. 

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4. Magic Tricycles: Efficient Magic-State Generation with Finite Block-Length Quantum LDPC Codes

   https://doi.org/10.1103/ghhp-cytl  

   Menon, Varun; Bonilla_Ataides, J Pablo; Mehta, Rohan; Gu, Andi; Tan, Daniel Bochen; Lukin, Mikhail D (April 2026, Physical Review X)

   The preparation of high-fidelity non-Clifford (magic) states is an essential subroutine for universal quantum computation but imposes substantial space-time overhead. Magic-state factories based on high-rate and -distance quantum low-density parity-check (LDPC) codes equipped with transversal non-Clifford gates can potentially reduce these overheads significantly, by circumventing the need for multiple rounds of distillation and by producing a large number of magic states in a single code block. As a step toward realizing efficient, fault-tolerant magic-state production, we introduce a class of finite block-length quantum LDPC codes which we name tricycle codes, generalizing the well-known bicycle codes to three homological dimensions. These codes can support constant-depth physical circuits that implement logical $CCZ$ gates between three code blocks. To construct these constant-depth $CCZ$ circuits, we develop analytical and numerical techniques that apply to a broad class of three-dimensional homological and balanced-product codes. We further show that tricycle codes enable single-shot state preparation and error correction, leading to a highly efficient magic-state generation protocol. Numerical simulations of specific codes confirm robust performance under circuit-level noise, demonstrating a high circuit-noise threshold of $>0.5\%$ . With modest postselection, certain tricycle codes of block lengths of only approximately 50–100 qubits are shown to produce $|\overline{CCZ}⟩$ magic states with logical error rates ranging from $2\times {10}^{-8}$ to below $3\times {10}^{-11}$ . Finally, we construct optimal depth syndrome extraction circuits for tricycle codes and present a protocol for implementing them efficiently on a reconfigurable neutral-atom platform. 

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5. Fast and Parallelizable Logical Computation with Homological Product Codes

   https://doi.org/10.1103/PhysRevX.15.021065  

   Xu, Qian; Zhou, Hengyun; Zheng, Guo; Bluvstein, Dolev; Ataides, J_Pablo Bonilla; Lukin, Mikhail D; Jiang, Liang (May 2025, Physical Review X)

   Quantum error correction is necessary to perform large-scale quantum computation but requires extremely large overheads in both space and time. High-rate quantum low-density-parity-check (qLDPC) codes promise a route to reduce qubit numbers, but performing computation while maintaining low space cost has required serialization of operations and extra time costs. In this work, we design fast and parallelizable logical gates for qLDPC codes and demonstrate their utility for key algorithmic subroutines such as the quantum adder. Our gate gadgets utilize transversal logical s between a data qLDPC code and a suitably constructed ancilla code to perform parallel Pauli product measurements (PPMs) on the data logical qubits. For hypergraph product codes, we show that the ancilla can be constructed by simply modifying the base classical codes of the data code, achieving parallel PPMs on a subgrid of the logical qubits with a lower space-time cost than existing schemes for an important class of circuits. Generalizations to 3D and 4D homological product codes further feature fast PPMs in constant depth. While prior work on qLDPC codes has focused on individual logical gates, we initiate the study of fault-tolerant compilation with our expanded set of native qLDPC code operations, constructing algorithmic primitives for preparing $k$ -qubit Greenberger-Horne-Zeilinger states and distilling or teleporting $k$ magic states with $O\left(1\right)$ space overhead in $O\left(1\right)$ and $O\left(\sqrt{k}\log k\right)$ logical cycles, respectively. We further generalize this to key algorithmic subroutines, demonstrating the efficient implementation of quantum adders using parallel operations. Our constructions are naturally compatible with reconfigurable architectures such as neutral atom arrays, paving the way to large-scale quantum computation with low space and time overheads. Published by the American Physical Society2025 

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