arXiv subjects

Zimu Li

Publications and source records attributed to Zimu Li.

At least 19 recordsLinked to original sources

arXiv2026

Non-Abelian sheaf quantum LDPC codes: good and magical

Non-Abelian quantum codes connect quantum error correction, phases of matter, and computational resources. In this work, we develop a general framework for constructing non-Abelian quantum low-density parity-check (qLDPC) codes by gauging sheaf codes via cup products and use it to obtain families with constant encoding rate and linear distance. We resolve the coupled logical constraints using explicit representatives to characterize the full gauged code space. We provide a fundamental treatment of code distance based on the general Knill--Laflamme condition and combine expansion with cleaning to establish protection against arbitrary low-weight errors. We further construct an almost-good family whose entire code space exhibits long-range magic. Gauging and ungauging also enable logical Clifford measurements that prepare encoded magic states. These results extend good qLDPC codes beyond the Pauli stabilizer setting and provide a concrete foundation for exploring non-Abelian phases beyond geometric locality and pursuing the no low-energy trivial magic conjecture.

quant-ph↗

arXiv2026

Transversal non-Clifford gates on good quantum locally testable codes

We achieve nontrivial transversal logical multi-controlled-$Z$ gates with asymptotically optimal parameters simultaneously on quantum low-density parity check codes and quantum locally testable codes, by applying the gate framework of [arXiv:2604.01874] to the recent good qLTC construction of [arXiv:2609.20780]. To this end, we use the covering space method to construct a nonzero cup product pairing on a finite arithmetic cubical complex. This differs from the previous construction of almost-good codes, whose base space is a hypergraph product. We express the pairing as a coefficient in a product of Moore determinants and prove polynomial nonvanishing by a bipartite multigraph specialization. We then construct covering spaces to obtain an asymptotic family of good qLTCs on which the pulled-back pairing induces the desired nontrivial transversal logical action. We further establish polynomial lower bounds on the logical tensor subrank, yielding polynomially many independent logical non-Clifford gates and enabling sublogarithmic-overhead magic state distillation with good qLTCs.

quant-ph↗

arXiv2026

Theory of low-weight quantum codes

Low check weight is a crucial code property for fault-tolerant quantum computing, which underlies the strong interest in quantum low-density parity-check (qLDPC) codes. Here, we initiate the theory of weight-constrained stabilizer codes from various foundational perspectives including the complexity of computing code weight and the explicit boundary of feasible low-weight codes in both theoretical and practical settings. We first prove that computing the optimal generator weight of a stabilizer code is $\mathsf{NP}$-hard, motivating efficiently computable bounds. We derive analytical lower bounds on check weight in terms of code rate and distance, identifying the minimum weights needed for single-qubit error detection and correction, as well as the sharp distance and rate limits of weight-three error-detecting codes. Matching constructions show that these bounds are tight in several regimes. To establish refined finite-size constraints, we develop a linear programming framework based on quantum weight enumerators subject to generator-weight constraints, yielding exact optimal weights for all parameter combinations with $n\le9$. Finally, we show that the same framework can incorporate architecture-dependent constraints, using the 127-qubit IBM Eagle chip as a concrete example. Our study brings the weight as a crucial parameter into coding theory and provides guidance for code design and utility in practical scenarios.

quant-ph↗

arXiv2026

Transversal non-Clifford gates on almost-good quantum LDPC and quantum locally testable codes

We exhibit nontrivial transversal logical multi-controlled-$Z$ gates on $[\![N,Θ(N),\tildeΘ(N)]\!]$ quantum low-density parity-check (qLDPC) codes with soundness $\tildeΘ(1)$, combining nearly optimal code parameters with fault-tolerant non-Clifford gates on qLDPC and quantum locally testable codes for the first time. Remarkably, our proofs proceed through highly general algebraic arguments. Building on insights from [Li et al.,~arXiv:2603.25831], we develop a general covering space framework for constructing and computing a rich family of cohomological invariant forms on sheaf codes that induce transversal logical multi-controlled-$Z$. To certify their nontriviality, we further demonstrate the existence of two-way product-expanding punctured Reed--Solomon codes, which is striking in light of the many negative examples for the product expansion behavior of ordinary Reed--Solomon codes. This approach directly overcomes the previous obstruction to realizing nontrivial logical operations while simultaneously preserving the code parameters. The claimed almost-good code results follow immediately as examples.

quant-ph↗

arXiv2026

LithoDreamer: A Physics-Informed World Model for Multi-Stage Computational Lithography

As semiconductor technology nodes scale, computational lithography is essential for ensuring yield and performance. However, lithography is a continuous physical process involving mask optimization, optical imaging, resist exposure, and development, which existing models fail to capture. To overcome this limitation, we present LithoDreamer, the first physics-informed World Model (WM) framework for computational lithography, which formulates the ``Layout-Mask-Resist Image-After Development Image (ADI)'' pipeline as a decision-driven multi-step evolution system. LithoDreamer captures feature changes between adjacent states to model stage-specific physics-informed latent spaces, in which it controls process intervention exploration and drives subsequent state transitions. To achieve interpretable intervention optimization without continuous supervision, we propose a contrastive variational optimization paradigm that contrasts the latent differences between intervention paths with variational evolution constraints, guiding the model to generate evolutions consistent with real lithography physics. Experiments show LithoDreamer achieves state-of-the-art performance in forward evolution and inverse planning. Our lithography dataset is publicly available at GitHub (https://github.com/7jiangyq/lithodreamer.git).

cs.AI↗

arXiv2026

S-Cheetah: A Novel Quadrupedal Robot with a 3-DOF Active Spine Learning Agile Locomotion

The biological spine of quadrupeds enables sagittal flexion/extension, lateral bending, and axial rotation, playing a crucial role in highly agile and dexterous locomotion. While numerous studies have integrated active spinal joints into quadrupedal robots to enhance agility, most designs simplify control complexity by reducing spinal degrees of freedom (DOF), failing to achieve the spatial tri-axial rotation characteristic of biological spines. Consequently, replicating a multi-DOF biomimetic spine and effectively leveraging it to empower the agile locomotion of quadrupedal robots remains a significant research challenge. In this study, we present S-Cheetah, a quadrupedal robot featuring a 3-DOF bio-inspired serial active spine capable of biomimetic spatial tri-axial rotation. To empower the robot to fully utilize this active spine, we developed a specialized reinforcement learning framework to actively promote the engagement of the introduced spine and maximize the robot's locomotive capabilities by integrating an acceleration curriculum learning strategy with tailored reward functions, such as a gallop gait reward, a spine undulation reward, and a spine steering reward. Experimental results demonstrate that S-Cheetah can achieve a peak speed of 6.9 m/s using the rotary G2 gallop gait and an in-place turning rate of 7.2 rad/s. Besides, the system exhibits an emergent, feline-inspired aerial self-righting capability, allowing it to land stably on four feet from arbitrary orientations during free fall. Finally, through extensive evaluations across diverse locomotion tasks, we prove that the introduction of the proposed 3-DOF spine comprehensively enhances the locomotive agility of quadrupedal robots. Project website: himmy-robotics.github.io/scheetah

cs.RO↗

arXiv2026

TriSplat: Simulation-Ready Feed-Forward 3D Scene Reconstruction

Sparse-view 3D reconstruction is increasingly addressed with feed-forward splatting networks that predict explicit primitives directly from images. Yet most existing methods remain centered on Gaussian primitives and expose surfaces only indirectly: extracting a usable mesh for downstream simulation, physics reasoning, or embodied interaction still requires expensive post-hoc steps that break the feed-forward promise. This limitation is especially pronounced in pose-free settings, where scene structure and camera parameters must be estimated jointly from sparse observations. We present TriSplat, a feed-forward reconstruction network that represents scenes with oriented triangle primitives and directly exports simulation-ready mesh scenes from a single forward pass. Given input images, the network predicts local 3D point maps, triangle attributes, camera poses, and optional intrinsics. Rather than regressing triangle orientation as an unconstrained latent variable, our approach constructs geometry normals from the predicted point maps, refines them with an image-conditioned normal head, and converts them into stable local frames for triangle parameterization. A mono-normal bootstrap schedule further stabilizes early training, while opacity and blur scheduling progressively sharpens the learned surface representation for direct mesh extraction. Experiments on RealEstate10K and DL3DV show that this representation produces more geometry-faithful reconstructions than Gaussian feed-forward baselines while maintaining competitive novel-view rendering quality. Because the rendering primitives are themselves surface triangles, the output can be directly ingested by physics engines, collision detectors, and standard rendering pipelines without any conversion, making it a practical simulation-ready solution for feed-forward 3D scene reconstruction.

cs.CV↗

arXiv2026

Theory of (Co)homological Invariants on Quantum LDPC Codes

With recent breakthroughs in the construction of good qLDPC codes and nearly good qLTCs, the study of (co)homological invariants of quantum code complexes, which fundamentally underlie their logical operations, has become evidently important. In this work, we establish a systematic framework for mathematically analyzing these invariants across a broad spectrum of constructions, from HGP codes to sheaf codes, by synthesizing advanced math tools. We generalize the notion of canonical logical representatives from HGP codes to the sheaf code setting, resolving a long-standing challenge in explicitly characterizing sheaf codewords. Building on this foundation, we present the first comprehensive computation of cup products within the intricate framework of sheaf codes. Given Artin's primitive root conjecture which holds under the generalized Riemann hypothesis, we prove that $\tildeΘ(N)$ independent cup products can be supported on almost good qLDPC codes and qLTCs of length N, opening the possibility of achieving linearly many parallel, nontrivial, constant-depth multi-controlled-Z gates. Moreover, by interpreting sheaf codes as covering spaces of HGP codes via graph lifts, we propose a scheme that inductively generates families of both HGP and sheaf codes in an interlaced fashion from a constant-size HGP code. Notably, the induction preserves all (co)homological invariants of the initial code. This provides a general framework for lifting invariants or logical gates from small codes to infinite code families, and enables efficient verification of such features by checking on small instances. Our theory provides a substantive methodology for studying invariants in HGP codes and extends it to sheaf codes. In doing so, we reveal deep and unexpected connections between qLDPC codes and math, thereby laying the groundwork for future advances in quantum coding, fault tolerance, and physics.

quant-ph↗

arXiv2026

Noise-immune and AI-enhanced DNA storage via adaptive partition mapping of digital data

Encoding digital information into DNA sequences offers an attractive potential solution for storing rapidly growing data under the information age and the rise of artificial intelligence. However, practical implementations of DNA storage are constrained by errors introduced during synthesis, preservation, and sequencing processes, and traditional error-correcting codes remain vulnerable to noise levels that exceed predefined thresholds. Here, we developed a Partitioning-mapping with Jump-rotating (PJ) encoding scheme, which exhibits exceptional noise resilience. PJ removes cross-strand information dependencies so that strand loss manifests as localized gaps rather than catastrophic file failure. It prioritizes file decodability under arbitrary noise conditions and leverages AI-based inference to enable controllable recovery of digital information. For the intra-strand encoding, we develop a jump-rotating strategy that relaxes sequence constraints relative to conventional rotating codes and provides tunable information density via an adjustable jump length. Based on this encoding architecture, the original file information can always be decoded and recovered under any strand loss ratio, with fidelity degrading smoothly as damage increases. We demonstrate that original files can be effectively recovered even with 10% strand loss, and machine learning datasets stored under these conditions retain their classification performance. Experiments further confirmed that PJ successfully decodes image files after extreme environmental disturbance using accelerated aging and high-intensity X-ray irradiation. By eliminating reliance on prior error probabilities, PJ establishes a general framework for robust, archival DNA storage capable of withstanding the rigorous conditions of real-world preservation.

cs.IT↗

arXiv2025

Poincaré Duality and Multiplicative Structures on Quantum Codes

Quantum LDPC codes have attracted intense interest due to their advantageous properties for realizing efficient fault-tolerant quantum computing. In particular, sheaf codes represent a novel framework that encompasses all well-known good qLDPC codes with profound underlying mathematics. In this work, we generalize Poincaré duality from manifolds to both classical and quantum codes defined via sheaf theory on $t$-dimensional cell complexes. Viewing important code properties including the encoding rate, code distance, local testability soundness, and efficient decoders as parameters of the underlying (co)chain complexes, we rigorously prove a duality relationship between the $i$-th chain and the $(t-i)$-th cochain of sheaf codes. We further build multiplicative structures such as cup and cap products on sheaved chain complexes, inspired by the standard notions of multiplicative structures and Poincaré duality on manifolds. This immediately leads to an explicit isomorphism between (co)homology groups of sheaf codes via a cap product. As an application, we obtain transversal disjoint logical $\mathrm{C}Z$ gates with $k_{\mathrm{C}Z}=Θ(n)$ on families of good qLDPC and almost-good quantum locally testable codes. Moreover, we provide multiple new methods to construct transversal circuits composed of $\mathrm{C}\mathrm{C}Z$ gates as well as for higher order controlled-$Z$ that are provably logical operations on the code space. We conjecture that they generate nontrivial logical actions, pointing towards fault-tolerant non-Clifford gates on nearly optimal qLDPC sheaf codes. Mathematically, our results are built on establishing the equivalence between sheaf cohomology in the derived-functor sense, Čech cohomology, and the cohomology of sheaf codes, thereby introducing new mathematical tools into quantum coding theory.

quant-ph↗

arXiv2025

Toward Super-polynomial Quantum Speedup of Equivariant Quantum Algorithms with SU($d$) Symmetry

We introduce a framework of the equivariant convolutional quantum algorithms which is tailored for a number of machine-learning tasks on physical systems with arbitrary SU$(d)$ symmetries. It allows us to enhance a natural model of quantum computation -- permutational quantum computing (PQC) -- and define a more powerful model: PQC+. While PQC was shown to be efficiently classically simulatable, we exhibit a problem which can be efficiently solved on PQC+ machine, whereas no classical polynomial time algorithm is known; thus providing evidence against PQC+ being classically simulatable. We further discuss practical quantum machine learning algorithms which can be carried out in the paradigm of PQC+.

quant-ph↗

arXiv2025

No-go theorems for logical gates on product quantum codes

Quantum error-correcting codes are essential to the implementation of fault-tolerant quantum computation. Homological products of classical codes offer a versatile framework for constructing quantum error-correcting codes with desirable properties, especially quantum low-density parity check (qLDPC) codes. Based on extensions of the Bravyi--König theorem that encompass codes without geometric locality, we establish a series of general no-go theorems for fault-tolerant logical gates supported by hypergraph product codes. Specifically, we show that non-Clifford logical gates cannot be implemented transversally on hypergraph product codes of all product dimensions, and that the dimensions impose various limitations on the accessible level of the Clifford hierarchy gates by constant-depth local circuits. We also discuss examples both with and without geometric locality which attain the Clifford hierarchy bounds. Our results reveal fundamental restrictions on logical gates originating from highly general algebraic structures, extending beyond existing knowledge only in geometrically local, finite logical qubits, transversal, or 2-dimensional product cases, and may guide the vital study of fault-tolerant quantum computation with qLDPC codes.

quant-ph↗

arXiv2025

SU(d)-Symmetric Random Unitaries: Quantum Scrambling, Error Correction, and Machine Learning

Quantum information processing in the presence of continuous symmetry is of wide importance and exhibits many novel physical and mathematical phenomena. SU(d) is a continuous group of particular interest since it represents a fundamental type of non-Abelian symmetry and also plays a vital role in quantum computation. Here, we explicate three particularly interesting applications of symmetric random unitaries in diverse contexts ranging from physics to quantum computing: information scrambling with non-Abelian conserved quantities, covariant quantum error correcting random codes, and geometric quantum machine learning. First, we show that, in the presence of SU(d) symmetry, the local conserved quantities would exhibit residual values even at $t \rightarrow \infty$ which decays as $Ω(1/n^{3/2})$ under local Pauli basis for qubits and $Ω(1/n^{(d+2)^2/2})$ under symmetric basis for general qudits with respect to the system size, in contrast to O(1/n) decay for U(1) case and the exponential decay for no-symmetry case in the sense of out-of-time ordered correlator. Second, we show that SU(d)-symmetric unitaries can be used to construct asymptotically optimal (in the sense of saturating the fundamental limits on the code error, or the approximate Eastin--Knill theorems) SU(d)-covariant codes that encode any constant number of logical qudits, extending [Kong & Liu; PRXQ 3, 020314 (2022)]. Finally, we derive an overpartameterization threshold via the quantum neural tangent kernel required for exponential convergence guarantee of generic ansatz for geometric quantum machine learning, which reveals that the number of parameters required scales only with the dimension of desired subspaces rather than the entire Hilbert space. Our work invites further research on quantum information with continuous symmetries, where the mathematical tools developed in this work are expected to be useful.

quant-ph↗

arXiv2025

Depth3DLane: Monocular 3D Lane Detection via Depth Prior Distillation

Monocular 3D lane detection is challenging due to the difficulty in capturing depth information from single-camera images. A common strategy involves transforming front-view (FV) images into bird's-eye-view (BEV) space through inverse perspective mapping (IPM), facilitating lane detection using BEV features. However, IPM's flat-ground assumption and loss of contextual information lead to inaccuracies in reconstructing 3D information, especially height. In this paper, we introduce a BEV-based framework to address these limitations and improve 3D lane detection accuracy. Our approach incorporates a Hierarchical Depth-Aware Head that provides multi-scale depth features, mitigating the flat-ground assumption by enhancing spatial awareness across varying depths. Additionally, we leverage Depth Prior Distillation to transfer semantic depth knowledge from a teacher model, capturing richer structural and contextual information for complex lane structures. To further refine lane continuity and ensure smooth lane reconstruction, we introduce a Conditional Random Field module that enforces spatial coherence in lane predictions. Extensive experiments validate that our method achieves state-of-the-art performance in terms of z-axis error and outperforms other methods in the field in overall performance. The code is released at: https://anonymous.4open.science/r/Depth3DLane-DCDD.

cs.CV↗

arXiv2024

Designs from Local Random Quantum Circuits with SU(d) Symmetry

The generation of $k$-designs (pseudorandom distributions that emulate the Haar measure up to $k$ moments) with local quantum circuit ensembles is a problem of fundamental importance in quantum information and physics. Despite the extensive understanding of this problem for ordinary random circuits, the crucial situations where symmetries or conservation laws are in play are known to pose fundamental challenges and remain little understood. We construct, for the first time, explicit local unitary ensembles that can achieve high-order unitary $k$-designs under transversal continuous symmetry, in the particularly important SU$(d)$ case. Specifically, we define the Convolutional Quantum Alternating group (CQA) generated by 4-local SU$(d)$-symmetric Hamiltonians as well as associated 4-local SU$(d)$-symmetric random unitary circuit ensembles, and prove that they form and converge to SU$(d)$-symmetric $k$-designs, respectively, for all $k < n(n-3)/2$ with $n$ being the number of qudits. A key technique that we employ to obtain the results is the Okounkov--Vershik approach to $S_n$ representation theory. To study the convergence time of the CQA ensemble, we develop a numerical method using the Young orthogonal form and $S_n$ branching rule. We provide strong evidence for a subconstant spectral gap and certain convergence time scales of various important circuit architectures, which contrast with the symmetry-free case. We also provide comprehensive explanations of the difficulties and limitations in rigorously analyzing the convergence time using methods that have been effective for cases without symmetries, including Knabe's local gap threshold and Nachtergaele's martingale methods. This suggests that a novel approach is likely necessary for understanding the convergence time of SU$(d)$-symmetric local random circuits.

quant-ph↗

arXiv2024

Efficient quantum pseudorandomness under conservation laws

The efficiency of locally generating unitary designs, which capture statistical notions of quantum pseudorandomness, lies at the heart of wide-ranging areas in physics and quantum information technologies. While there are extensive potent methods and results for this problem, the evidently important setting where continuous symmetries or conservation laws (most notably U(1) and SU(d)) are involved is known to present fundamental difficulties. In particular, even the basic question of whether any local symmetric circuit can generate 2-designs efficiently (in time that grows at most polynomially in the system size) remains open with no circuit constructions provably known to do so, despite intensive efforts. In this work, we resolve this long-standing open problem for both U(1) and SU(d) symmetries by explicitly constructing local symmetric quantum circuits which we prove to converge to symmetric unitary 2-designs in polynomial time using a combination of representation theory, graph theory, and Markov chain methods. As a direct application, our constructions can be used to efficiently generate near-optimal covariant quantum error-correcting codes, confirming a conjecture in [PRX Quantum 3, 020314 (2022)].

quant-ph↗

arXiv2024

Convergence efficiency of quantum gates and circuits

We consider quantum circuit models where the gates are drawn from arbitrary gate ensembles given by probabilistic distributions over certain gate sets and circuit architectures, which we call stochastic quantum circuits. Of main interest in this work is the speed of convergence of stochastic circuits with different gate ensembles and circuit architectures to unitary t-designs. A key motivation for this theory is the varying preference for different gates and circuit architectures in different practical scenarios. In particular, it provides a versatile framework for devising efficient circuits for implementing $t$-designs and relevant applications including random circuit and scrambling experiments, as well as benchmarking the performance of gates and circuit architectures. We examine various important settings in depth. A key aspect of our study is an "ironed gadget" model, which allows us to systematically evaluate and compare the convergence efficiency of entangling gates and circuit architectures. Particularly notable results include i) gadgets of two-qubit gates with KAK coefficients $\left(\fracπ{4}-\frac{1}{8}\arccos(\frac{1}{5}),\fracπ{8},\frac{1}{8}\arccos(\frac{1}{5})\right)$ (which we call $χ$ gates) directly form exact 2- and 3-designs; ii) the iSWAP gate family achieves the best efficiency for convergence to 2-designs under mild conjectures with numerical evidence, even outperforming the Haar-random gate, for generic many-body circuits; iii) iSWAP + complete graph achieve the best efficiency for convergence to 2-designs among all graph circuits. A variety of numerical results are provided to complement our analysis. We also derive robustness guarantees for our analysis against gate perturbations. Additionally, we provide cursory analysis on gates with higher locality and found that the Margolus gate outperforms various other well-known gates.

quant-ph↗

arXiv2024

Unifying O(3) Equivariant Neural Networks Design with Tensor-Network Formalism

Many learning tasks, including learning potential energy surfaces from ab initio calculations, involve global spatial symmetries and permutational symmetry between atoms or general particles. Equivariant graph neural networks are a standard approach to such problems, with one of the most successful methods employing tensor products between various tensors that transform under the spatial group. However, as the number of different tensors and the complexity of relationships between them increase, maintaining parsimony and equivariance becomes increasingly challenging. In this paper, we propose using fusion diagrams, a technique widely employed in simulating SU($2$)-symmetric quantum many-body problems, to design new equivariant components for equivariant neural networks. This results in a diagrammatic approach to constructing novel neural network architectures. When applied to particles within a given local neighborhood, the resulting components, which we term "fusion blocks," serve as universal approximators of any continuous equivariant function defined in the neighborhood. We incorporate a fusion block into pre-existing equivariant architectures (Cormorant and MACE), leading to improved performance with fewer parameters on a range of challenging chemical problems. Furthermore, we apply group-equivariant neural networks to study non-adiabatic molecular dynamics of stilbene cis-trans isomerization. Our approach, which combines tensor networks with equivariant neural networks, suggests a potentially fruitful direction for designing more expressive equivariant neural networks.

cs.LG↗