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Jan 8

Efficient Magic State Cultivation on $\mathbb{RP}^2$

Preparing high-fidelity logical magic states is crucial for fault-tolerant quantum computation. Among prior attempts to reduce the substantial cost of magic state preparation, magic state cultivation (MSC), a recently proposed protocol for preparing T states without magic state distillation, achieves state-of-the-art efficiency. Inspired by this work, we propose a new MSC procedure that would produce a logical T state on a rotated surface code at a further reduced cost. For our MSC protocol, we define a new code family, the RP^2 code, by putting the rotated surface code on RP^2 (a two-dimensional manifold), as well as two self-dual CSS codes named SRP-3 and SRP-5 respectively. Small RP^2 codes are used to hold logical information and checked by syndrome extraction (SE) circuits. We design fast morphing circuits that enable switching between a distance 3 (5) RP^2 code and an SRP-3 (SRP-5) code on which we can efficiently check the correctness of the logical state. To preserve the high accuracy of the cultivated logical T state, we design an efficient and easy-to-decode expansion stage that grows a small RP^2 code to a large rotated surface code in one round. Our MSC protocol utilizes non-local connectivity, available on both neutral atom array and ion trap platforms. According to our Monte Carlo sampling results, our MSC protocol requires about an order of magnitude smaller space-time volume to reach a target logical error rate around 10^{-9} compared to the original MSC protocol.

  • 4 authors
·
Mar 24, 2025

MaGIC: Multi-modality Guided Image Completion

Vanilla image completion approaches exhibit sensitivity to large missing regions, attributed to the limited availability of reference information for plausible generation. To mitigate this, existing methods incorporate the extra cue as a guidance for image completion. Despite improvements, these approaches are often restricted to employing a single modality (e.g., segmentation or sketch maps), which lacks scalability in leveraging multi-modality for more plausible completion. In this paper, we propose a novel, simple yet effective method for Multi-modal Guided Image Completion, dubbed MaGIC, which not only supports a wide range of single modality as the guidance (e.g., text, canny edge, sketch, segmentation, depth, and pose), but also adapts to arbitrarily customized combination of these modalities (i.e., arbitrary multi-modality) for image completion. For building MaGIC, we first introduce a modality-specific conditional U-Net (MCU-Net) that injects single-modal signal into a U-Net denoiser for single-modal guided image completion. Then, we devise a consistent modality blending (CMB) method to leverage modality signals encoded in multiple learned MCU-Nets through gradient guidance in latent space. Our CMB is training-free, thereby avoids the cumbersome joint re-training of different modalities, which is the secret of MaGIC to achieve exceptional flexibility in accommodating new modalities for completion. Experiments show the superiority of MaGIC over state-of-the-art methods and its generalization to various completion tasks. Our project with code and models is available at yeates.github.io/MaGIC-Page/.

  • 5 authors
·
May 19, 2023

What's the Magic Word? A Control Theory of LLM Prompting

Prompt engineering is crucial for deploying LLMs but is poorly understood mathematically. We formalize LLM systems as a class of discrete stochastic dynamical systems to explore prompt engineering through the lens of control theory. We investigate the reachable set of output token sequences R_y(mathbf x_0) for which there exists a control input sequence mathbf u for each mathbf y in R_y(mathbf x_0) that steers the LLM to output mathbf y from initial state sequence mathbf x_0. We offer analytic analysis on the limitations on the controllability of self-attention in terms of reachable set, where we prove an upper bound on the reachable set of outputs R_y(mathbf x_0) as a function of the singular values of the parameter matrices. We present complementary empirical analysis on the controllability of a panel of LLMs, including Falcon-7b, Llama-7b, and Falcon-40b. Our results demonstrate a lower bound on the reachable set of outputs R_y(mathbf x_0) w.r.t. initial state sequences mathbf x_0 sampled from the Wikitext dataset. We find that the correct next Wikitext token following sequence mathbf x_0 is reachable over 97% of the time with prompts of kleq 10 tokens. We also establish that the top 75 most likely next tokens, as estimated by the LLM itself, are reachable at least 85% of the time with prompts of kleq 10 tokens. Intriguingly, short prompt sequences can dramatically alter the likelihood of specific outputs, even making the least likely tokens become the most likely ones. This control-centric analysis of LLMs demonstrates the significant and poorly understood role of input sequences in steering output probabilities, offering a foundational perspective for enhancing language model system capabilities.

  • 4 authors
·
Oct 2, 2023

Physics of Language Models: Part 4.1, Architecture Design and the Magic of Canon Layers

Understanding architectural differences in language models is challenging, especially at academic-scale pretraining (e.g., 1.3B parameters, 100B tokens), where results are often dominated by noise and randomness. To overcome this, we introduce controlled synthetic pretraining tasks that isolate and evaluate core model capabilities. Within this framework, we discover CANON LAYERS: lightweight architectural components -- named after the musical term "canon" -- that promote horizontal information flow across neighboring tokens. Canon layers compute weighted sums of nearby token representations and integrate seamlessly into Transformers, linear attention, state-space models, or any sequence architecture. We present 12 key results. This includes how Canon layers enhance reasoning depth (e.g., by 2times), reasoning breadth, knowledge manipulation, etc. They lift weak architectures like NoPE to match RoPE, and linear attention to rival SOTA linear models like Mamba2/GDN -- validated both through synthetic tasks and real-world academic-scale pretraining. This synthetic playground offers an economical, principled path to isolate core model capabilities often obscured at academic scales. Equipped with infinite high-quality data, it may even PREDICT how future architectures will behave as training pipelines improve -- e.g., through better data curation or RL-based post-training -- unlocking deeper reasoning and hierarchical inference.

facebook AI at Meta
·
Dec 19, 2025 2