# The First Law of Complexity

## TL;DR

This paper introduces the first law of complexity, showing that the variation in holographic complexity depends only on the endpoint of the optimal trajectory, with implications for understanding quantum circuit boundaries in holography.

## Contribution

It establishes a new principle, the first law of complexity, linking the variation of holographic complexity to endpoint dependence using Nielsen's geometric approach.

## Key findings

- Variation depends only on the end point of the optimal trajectory.
- In the complexity=action conjecture, gravitational contributions cancel out.
- The boundary term from scalar field action determines the complexity variation.

## Abstract

We investigate the variation of holographic complexity for two nearby target states. Based on Nielsen's geometric approach, we find the variation only depends on the end point of the optimal trajectory, a result which we designate the first law of complexity. As an example, we examine the complexity=action conjecture when the AdS vacuum is perturbed by a scalar field excitation, which corresponds to a coherent state. Remarkably, the gravitational contributions completely cancel and the final variation reduces to a boundary term coming entirely from the scalar field action. Hence the null boundary of Wheeler-DeWitt patch appears to act like the "end of the quantum circuit".