Both of the techniques assume bitwise access to the oracle as a classical function.

Finally, we provide mathematical bounds on the interference between superposition channels in MIMOFormer.

We show how to "compile" human-readable programs into standard decoder-only transformer models.

With the advent of deep learning, progressively larger neural networks have been designed to solve complex tasks. We take advantage of these capacity-rich models to lower the cost of inference by exploiting computation in superposition.

This paper presents a text classification algorithm inspired by the notion of superposition of states in quantum physics. By regarding text as a superposition of words, we derive the wave function of a document and we compute the transition probability of the document to a target class according to Born's rule. Two complementary implementations are presented. In the first one, wave functions are calculated explicitly.

This report introduces a novel class of reasoning architectures, termed Quantum Circuit Reasoning Models (QCRM), which extend the concept of Variational Quantum Circuits (VQC) from energy minimization and classification tasks to structured logical inference and reasoning. We posit that fundamental quantum mechanical operations, superposition, entanglement, interference, and measurement, naturally map to essential reasoning primitives such as hypothesis branching, constraint propagation, consistency enforcement, and decision making. The resulting framework combines quantum-inspired computation with differentiable optimization, enabling reasoning to emerge as a process of amplitude evolution and interference-driven selection of self-consistent states. We develop the mathematical foundation of QCRM, define its parameterized circuit architecture, and show how logical rules can be encoded as unitary transformations over proposition-qubit states. We further formalize a training objective grounded in classical gradient descent over circuit parameters and discuss simulation-based implementations on classical hardware. Finally, we propose the Quantum Reasoning Layer (QRL) as a differentiable hybrid component for composable reasoning models applicable to scientific, biomedical, and chemical inference domains.

Classical statistical inference and learning theory often fail to explain the success of modern neural networks. A key reason is that these models are non-identifiable (singular), violating core assumptions behind PAC bounds and asymptotic normality. Singular learning theory (SLT), a physics-inspired framework grounded in algebraic geometry, has gained popularity for its ability to close this theory-practice gap. In this paper, we empirically study SLT in toy settings relevant to interpretability and phase transitions. First, we understand the SLT free energy $\mathcal{F}_n$ by testing an Arrhenius-style rate hypothesis using both a grokking modulo-arithmetic model and Anthropic's Toy Models of Superposition. Second, we understand the local learning coefficient $λ_α$ by measuring how it scales with problem difficulty across several controlled network families (polynomial regressors, low-rank linear networks, and low-rank autoencoders). Our experiments recover known scaling laws while others yield meaningful deviations from theoretical expectations. Overall, our paper illustrates the many merits of SLT for understanding neural network phase transitions, and poses open research questions for the field.

The success of today's large language models (LLMs) depends on the observation that larger models perform better. However, the origin of this neural scaling law, that loss decreases as a power law with model size, remains unclear. We propose that representation superposition, meaning that LLMs represent more features than they have dimensions, can be a key contributor to loss and cause neural scaling. Based on Anthropic's toy model, we use weight decay to control the degree of superposition, allowing us to systematically study how loss scales with model size. When superposition is weak, the loss follows a power law only if data feature frequencies are power-law distributed. In contrast, under strong superposition, the loss generically scales inversely with model dimension across a broad class of frequency distributions, due to geometric overlaps between representation vectors. We confirmed that open-sourced LLMs operate in the strong superposition regime and have loss scaling inversely with model dimension, and that the Chinchilla scaling laws are also consistent with this behavior. Our results identify representation superposition as a central driver of neural scaling laws, providing insights into questions like when neural scaling laws can be improved and when they will break down.

Finding human-understandable circuits in language models is a central goal of the field of mechanistic interpretability. We train models to have more understandable circuits by constraining most of their weights to be zeros, so that each neuron only has a few connections. To recover fine-grained circuits underlying each of several hand-crafted tasks, we prune the models to isolate the part responsible for the task. These circuits often contain neurons and residual channels that correspond to natural concepts, with a small number of straightforwardly interpretable connections between them. We study how these models scale and find that making weights sparser trades off capability for interpretability, and scaling model size improves the capability-interpretability frontier. However, scaling sparse models beyond tens of millions of nonzero parameters while preserving interpretability remains a challenge. In addition to training weight-sparse models de novo, we show preliminary results suggesting our method can also be adapted to explain existing dense models. Our work produces circuits that achieve an unprecedented level of human understandability and validates them with considerable rigor.

Artificial intelligence (AI) is used in numerous fields of science, yet the initial research questions and targets are still almost always provided by human researchers. AI-generated creative ideas in science are rare and often vague, so that it remains a human task to execute them. Automating idea generation and implementation in one coherent system would significantly shift the role of humans in the scientific process. Here we present AI-Mandel, an LLM agent that can generate and implement ideas in quantum physics. AI-Mandel formulates ideas from the literature and uses a domain-specific AI tool to turn them into concrete experiment designs that can readily be implemented in laboratories. The generated ideas by AI-Mandel are often scientifically interesting - for two of them we have already written independent scientific follow-up papers. The ideas include new variations of quantum teleportation, primitives of quantum networks in indefinite causal orders, and new concepts of geometric phases based on closed loops of quantum information transfer. AI-Mandel is a prototypical demonstration of an AI physicist that can generate and implement concrete, actionable ideas. Building such a system is not only useful to accelerate science, but it also reveals concrete open challenges on the path to human-level artificial scientists.