# Fixed-Depth Two-Qubit Circuits and the Monodromy Polytope

## TL;DR

This paper uses symplectic geometry to characterize the set of two-qubit quantum programs achievable with fixed-depth circuits, describing it as a convex polytope and identifying cases with particularly large accessible subspaces.

## Contribution

It provides an explicit geometric description of two-qubit program spaces at fixed depth, including a complete set of inequalities and analysis of specific gate families.

## Key findings

- The accessible program space forms a convex polytope describable by linear inequalities.
- Certain XY-family gates allow for larger low-depth program subspaces.
- Explicit inequalities are derived for common example cases.

## Abstract

For a native gate set which includes all single-qubit gates, we apply results from symplectic geometry to analyze the spaces of two-qubit programs accessible within a fixed number of gates. These techniques yield an explicit description of this subspace as a convex polytope, presented by a family of linear inequalities themselves accessible via a finite calculation. We completely describe this family of inequalities in a variety of familiar example cases, and as a consequence we highlight a certain member of the "XY-family" for which this subspace is particularly large, i.e., for which many two-qubit programs admit expression as low-depth circuits.