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2. The Illusion of State in State-Space Models

   Merrill, William; Petty, Jackson; Sabharwal, Ashish (July 2024, International Conference on Machine Learning (ICML) 2024)

   State-space models (SSMs) have emerged as a potential alternative architecture for building large language models (LLMs) compared to the previously ubiquitous transformer architecture. One theoretical weakness of transformers is that they cannot express certain kinds of sequential computation and state tracking (Merrill & Sabharwal, 2023), which SSMs are explicitly designed to address via their close architectural similarity to recurrent neural networks (RNNs). But do SSMs truly have an advantage (over transformers) in expressive power for state tracking? Surprisingly, the answer is no. Our analysis reveals that the expressive power of SSMs is limited very similarly to transformers: SSMs cannot express computation outside the complexity class 𝖳𝖢0. In particular, this means they cannot solve simple state-tracking problems like permutation composition. It follows that SSMs are provably unable to accurately track chess moves with certain notation, evaluate code, or track entities in a long narrative. To supplement our formal analysis, we report experiments showing that Mamba-style SSMs indeed struggle with state tracking. Thus, despite its recurrent formulation, the "state" in an SSM is an illusion: SSMs have similar expressiveness limitations to non-recurrent models like transformers, which may fundamentally limit their ability to solve real-world state-tracking problems. 

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3. Magic-state resource theory for the ground state of the transverse-field Ising model

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4. Learning the closest product state

   Bakshi, A; Bostanci, J; Kretschmer, W; Landau, Z; Li, J; Liu, A; O'Donnell, R; Tang, E (March 2025, https://doi.org/10.48550/arXiv.2411.04283)

   We study the problem of finding a (pure) product state with optimal fidelity to an unknown n-qubit quantum state ρ, given copies of ρ. This is a basic instance of a fundamental question in quantum learning: is it possible to efficiently learn a simple approximation to an arbitrary state? We give an algorithm which finds a product state with fidelity ε-close to optimal, using N=npoly(1/ε) copies of ρ and poly(N) classical overhead. We further show that estimating the optimal fidelity is NP-hard for error ε=1/poly(n), showing that the error dependence cannot be significantly improved. For our algorithm, we build a carefully-defined cover over candidate product states, qubit by qubit, and then demonstrate that extending the cover can be reduced to approximate constrained polynomial optimization. For our proof of hardness, we give a formal reduction from polynomial optimization to finding the closest product state. Together, these results demonstrate a fundamental connection between these two seemingly unrelated questions. Building on our general approach, we also develop more efficient algorithms in three simpler settings: when the optimal fidelity exceeds 5/6; when we restrict ourselves to a discrete class of product states; and when we are allowed to output a matrix product state. 

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5. Formative reasons for state-to-state influences on firearm acquisition in the U.S.

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   Wang, Xu; Macinko, James; Porfiri, Maurizio; Sipahi, Rifat (September 2024, SSM - Population Health)