Two-Qubit Circuit Depth and the Monodromy Polytope
Eric C. Peterson, Gavin E. Crooks, Robert S. Smith
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 ``<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">X</mml:mi><mml:mi mathvariant="normal">Y</mml:mi></mml:mrow></mml:math>--family'' for which this subspace is particularly large, i.e., for which many two-qubit programs admit expression as low-depth circuits.