The Lindahl Letter

Nested learning and the illusion of depth

Thank you for tuning in to week 218 of the Lindahl Letter publication. A new edition arrives every Friday. This week the topic under consideration for the Lindahl Letter is, “Nested learning and the illusion of depth.” Just for fun with this nested learning paper we are evaluating today, I downloaded the 52 page PDF and uploaded it to my Google Drive to have Gemini create an audio overview of the paper. That is just a one button request these days. We have reached a point where we can easily listen to a paper recap with very little friction. It’s actually harder to get a complete reading of the PDF as an audio file. I had tried the Adobe Acrobat read aloud feature and I don’t really like the robotic output. Sometimes, I would rather listen to a paper than read it when I am trying to really think deeply about something. The 5 minutes of podcast audio Gemini spit out about the paper are embedded below. It’s interesting to say the least how quickly Gemini turned that paper into a short podcast. It’s entirely possible that my analysis might be less entertaining than the podcast Gemini created on the fly. You will be the judge of that one. This is a paper I actually printed out 2 pages per page using the double sided setting. That is how I used to read papers during graduate school. This paper had a few color elements that is something my graduate school papers never really had. They were all monochromatic. I had to put on my reading glasses and hold the paper a little closer than I used to with the 2 pages per page printing. I’ll have to remember to just print using single page spacing next time around. I really only print out papers I want to keep in my stack of stuff. This one certainly fits that criteria. Trying to make content that is accessible is one of the reasons that I have been recording audio for the Lindahl Letter. Sometimes listening to something is a great unlock. Other times due to complexity and the diagrams included you just have to read academic papers. I try to bring things forward without complex charts in a highly consumable way. My take on research notes is that they need to be generally understandable and communicate a clear take on whatever topic is being covered. The content has to be condensed into something that can be considered in 5-10 minutes. To that end I’m going to do my best to bring this paper on nested learning to life today. This paper matters, it really does, because the research presented undermines one of the core assumptions driving modern AI investment and the endless LLM building and training that has been occurring, namely that stacking more layers reliably produces qualitatively better intelligence [1]. The mantra to just keep scaling maybe will fade away. If many so-called deep models collapse into shallow equivalents during training, then reported gains attributed to architectural depth may instead be artifacts of data scale, regularization, or optimization heuristics rather than true representational progress. This has direct implications for benchmarking, since comparisons that reward parameter count or depth risk overstating advances that do not translate into more robust reasoning or generalization. It also affects hardware and infrastructure strategy, because enormous resources are being allocated to support depth that may not deliver proportional returns. At a deeper level, the result forces a reconsideration of what meaningful learning progress actually looks like, shifting attention from surface complexity toward mechanisms that introduce genuinely new inductive structure and adaptive behavior. Maybe the long term impact of this call out is likely to be gradual rather than abrupt, but it meaningfully shifts the intellectual ground beneath current AI narratives [1]. The paper in question provides a formal vocabulary for a concern many researchers have held intuitively, that architectural depth has become a proxy metric for progress rather than a principled design choice. Over time, this reframing may influence how serious research groups evaluate models, placing more weight on identifiably distinct learning mechanisms, training dynamics, and robustness properties instead of raw scale. It is unlikely to immediately change the minds of investors or vendors whose incentives favor larger systems, but it can shape academic norms, reviewer expectations, and eventually benchmark construction. Historically, results like this matter most not because they halt a paradigm, but because they constrain it, narrowing the space of credible claims and forcing future advances to justify themselves on grounds other than appearance and size. This argument intersects directly with my broader concerns about interpretability and generalization. I am still curious about creating a combiner model, but this might change the mechanics of how that might ultimately work. If performance gains arise primarily from optimization dynamics rather than architectural expressivity, then claims about learned representations should be treated with caution. Apparent abstraction may not correspond to stable semantic structure but to transient equilibria shaped by training order, learning rates, and implicit regularization. This aligns with growing skepticism about whether large models truly learn hierarchical concepts or merely approximate them through iterative adjustment [2]. The implications extend beyond theory. Nested learning reframes debates about model scaling, architectural novelty, and transfer learning. It suggests that progress may come less from ever deeper networks and more from better understanding and controlling learning dynamics. This has practical consequences for reproducibility, safety, and deployment, since nested optimization can introduce path dependence and sensitivity to training regimes that are difficult to observe or audit. In the broader context of the AI marketplace, this work reinforces a recurring theme. Fluency and performance do not necessarily imply understanding. As with recent neuroscience critiques of language models, nested learning highlights how impressive outputs can emerge from mechanisms that lack stable, interpretable internal structure [3]. That gap matters when systems are deployed in high stakes environments where reliability, robustness, and reasoning are essential. We will see how this plays out in 2026 and what new research will ultimately shift the landscape. Footnotes: [1] Behrouz, A., Razaviyayn, M., Zhong, P., & Mirrokni, V. “Nested learning: The illusion of deep learning architectures.” Advances in Neural Information Processing Systems 39 (2025). https://abehrouz.github.io/files/NL.pdf [https://abehrouz.github.io/files/NL.pdf] [2] Raghu, M., Poole, B., Kleinberg, J., Ganguli, S., & Dickstein, J. “On the expressive power of deep neural networks.” Proceedings of the 34th International Conference on Machine Learning (2017). https://arxiv.org/abs/1606.05336 [https://arxiv.org/abs/1606.05336] [3] Riley, B. “Large language mistake: Cutting edge research shows language is not the same as intelligence.” The Verge (2025). https://www.theverge.com/ai-artificial-intelligence/827820/large-language-models-ai-intelligence-neuroscience-problems [https://www.theverge.com/ai-artificial-intelligence/827820/large-language-models-ai-intelligence-neuroscience-problems] What’s next for the Lindahl Letter? New editions arrive every Friday. If you are still listening at this point and enjoyed this content, then please take a moment and share it with a friend. If you are new to the Lindahl Letter, then please consider subscribing. Make sure to stay curious, stay informed, and enjoy the week ahead! This is a public episode. If you would like to discuss this with other subscribers or get access to bonus episodes, visit www.nelsx.com [https://www.nelsx.com?utm_medium=podcast&utm_campaign=CTA_1]

20 de dic de 2025 - 7 min

The 5 biggest unsolved problems in quantum computing

Thank you for tuning in to week 216 of the Lindahl Letter publication. A new edition arrives every Friday. This week the topic under consideration for the Lindahl Letter is, “The biggest unsolved problems in quantum computing.” The field of quantum computing has accelerated rapidly during the last decade, yet its most important breakthroughs remain incomplete. The core research challenges that stand between today’s prototypes and large scale, industrially relevant systems are now visible with unusual clarity. I think we are on the path to seeing this technology realized. These challenges are increasingly framed not as incremental milestones but as structural bottlenecks that shape the entire trajectory of the field. This week’s analysis focuses on the five most critical problems that must be solved for quantum computing to reach fault tolerant, economically meaningful operation. These gaps define where research investment, national strategy, and competitive advantage will be determined in the coming decade. 1. A fully fault tolerant logical qubit with logical error rates below threshold The first and most fundamental problem is the absence of a fully fault tolerant logical qubit. I know, I know, people are getting close, but this technology is not fully realized just yet. Theoretical thresholds for fault tolerance are well studied, and progress has been reported through surface codes, low density parity check codes, and recent advances in magic state distillation. Several groups have demonstrated logical qubits whose performance exceeds their underlying physical qubits, and some trapped-ion experiments now show better than break-even behavior under repeated rounds of error correction. However, no team has yet realized a logical qubit that maintains below-threshold logical error rates in a fully integrated setting that combines encoding, stabilizer measurement, real time decoding, and continuous correction across arbitrarily deep circuits. Experiments such as the University of Osaka’s zero level magic state distillation results and Quantinuum’s recent logical circuit demonstrations illustrate meaningful progress, yet a complete fault tolerant logical qubit build rolling off the assembly line has not been achieved [1]. This missing element prevents reliable execution of deep circuits and stands as the central research challenge of the field. I am also tracking a leaderboard of efforts aimed at increasing the number and stability of logical qubits as new systems emerge [2]. 2. A scalable and manufacturable quantum architecture that supports thousands of high fidelity qubits The second unsolved problem is the absence of a scalable, manufacturable quantum architecture capable of supporting thousands of high fidelity qubits. Superconducting platforms continue to face wiring congestion, cross talk, and fabrication variability across large wafers, which limits reproducibility at scale. Trapped-ion systems achieve some of the highest gate fidelities reported, but their physical footprint, control volume, and relatively slow gate speeds constrain system growth. Neutral atom arrays offer large qubit counts, yet they have not demonstrated uniform, high fidelity two qubit gates across arrays large enough to support fault tolerant codes. Photonic and spin qubits continue to advance but remain earlier in their development for universal, gate based architectures. Across all platforms, the transition from laboratory systems to repeatable, wafer scale manufacturing has not occurred. Most resource estimates indicate that tens of thousands of physical qubits will be required for practically useful, error corrected applications, and no architecture is yet positioned to deliver this scale with consistent fidelity. I am tracking universal gate based physical qubit leaders closely, and I expect to see significant shifts in 2026 as fabrication strategies evolve [3]. 3. Integrated cryogenic classical control systems capable of real time decoding at scale The third unsolved problem concerns the integration of classical control systems capable of operating efficiently at cryogenic temperatures. Quantum processors rely on classical electronics to generate precise control pulses, read measurement outcomes, and perform real time decoding. As devices grow, these classical requirements become a dominant engineering bottleneck. Current systems depend on extensive room temperature hardware and thousands of coaxial lines, an approach that is not viable for scaling beyond a few hundred qubits. Research into cryogenic CMOS, multiplexed readout architectures, and fast low noise routing has shown meaningful progress, and prototype decoders have demonstrated sub microsecond performance. However, the field still lacks a fully integrated classical to quantum control stack that can operate near the device, support large scale decoding throughput, and eliminate the wiring overhead required for million channel systems. Solving this challenge is as essential as improving qubit fidelity, because fault tolerant computation will require tightly coupled classical and quantum subsystems functioning in real time at cryogenic depths. 4. A modular, networked quantum architecture with reliable chip to chip entanglement The fourth major unsolved problem involves modularity and quantum networking. Large scale quantum computers will not be monolithic systems. They will require distributed architectures in which multiple chips or modules exchange entanglement to support error corrected computation across larger systems. Research groups have demonstrated chip to chip photonic links, heralded entanglement generation, and short range coupling between trapped-ion and superconducting devices, but these demonstrations remain small scale and experimental. No team has yet produced a modular architecture capable of sustaining reliable inter module entanglement rates, routing operations, and error corrected logical circuits across networked components. A practical quantum interconnect, whether photonic or microwave based, would redefine system design by enabling large logical qubit counts without relying on a single monolithic wafer. Developing these networked architectures is now seen as one of the highest value targets for national research programs, because modularity is likely the only viable path to systems with millions of physical qubits. 5. A verified quantum advantage tied to a real scientific or industrial workload The fifth unsolved problem is the absence of a widely accepted, independently verified quantum advantage tied to a real scientific or industrial workload. Quantum supremacy experiments have demonstrated that certain random circuit sampling tasks are exceptionally difficult for classical systems to simulate, but these tasks do not translate into chemistry, materials, optimization, or cryptography workloads. Several vendors have recently reported domain specific quantum advantages, including applications in quantum navigation and narrow optimization tasks, but these demonstrations have not yet achieved broad community validation or independent replication under strict verification and resource accounting. A robust demonstration of advantage requires a computation that is infeasible for classical systems within realistic time and energy constraints, produces an output that can be meaningfully verified, and operates using real hardware error rates rather than idealized gates. Achieving this milestone would mark a decisive shift in the strategic landscape of the field and would accelerate commercial investment into fault tolerant platforms. Together, these five problems outline the most important questions I’m tracking that are facing quantum computing today. This is based on my research interests. Please feel free to let me know if something else jumps out when you read this list. Each topic represents an opportunity for technical leadership, research investment, and industrial strategy. That does not mean my list is complete. It’s directionally accurate for late 2025, but things in the quantum computing space are changing rapidly. These elements called out also define the hurdles that stand between early laboratory demonstrations and the large-scale quantum platforms required for transformative scientific progress. What’s next for the Lindahl Letter? New editions arrive every Friday. If you are still listening at this point and enjoyed this content, then please take a moment and share it with a friend. If you are new to the Lindahl Letter, then please consider subscribing. Make sure to stay curious, stay informed, and enjoy the week ahead! Links I’m sharing this week! You may not have watched Linus Torvalds build a computer on your watch list for 2025, but I’m sharing that link anyway. I truly enjoyed watching this video. This video made me chuckle several times and was delightful. Footnotes: [1] Itogawa, T., Takada, Y., Hirano, Y., & Fujii, K. (2024). Even more efficient magic state distillation by zero-level distillation. arXiv preprint arXiv:2403.03991. http://arxiv.org/pdf/2403.03991 [http://arxiv.org/pdf/2403.03991] [2] Top quantum computers by logical qubit [3] Updating my top 10 quantum computer leaderboard This is a public episode. If you would like to discuss this with other subscribers or get access to bonus episodes, visit www.nelsx.com [https://www.nelsx.com?utm_medium=podcast&utm_campaign=CTA_1]

6 de dic de 2025 - 9 min