  Large Language Models (LLMs) exhibit various emergent abilities. Among these
abilities, some might reveal the internal working mechanisms of models. In this
paper, we uncover a novel emergent capability in models: the intrinsic ability
to perform extended sequences of calculations without relying on
chain-of-thought step-by-step solutions. Remarkably, the most advanced models
can directly output the results of two-digit number additions with lengths
extending up to 15 addends. We hypothesize that the model emerges Implicit
Discrete State Representations (IDSRs) within its hidden states and performs
symbolic calculations internally. To test this hypothesis, we design a sequence
of experiments that look into the hidden states. Specifically, we first confirm
that IDSRs exist. Then, we provide interesting observations about the formation
of IDSRs from layer, digit, and sequence perspectives. Finally, we confirm that
models indeed use IDSRs to produce the final answers. However, we also discover
that these state representations are far from lossless in current open-sourced
models, leading to inaccuracies in their final performance. Our work presents a
novel exploration of LLMs' symbolic calculation abilities and the underlying
mechanisms.
