Fixed-Depth Two-Qubit Circuits and the Monodromy Polytope

Eric C. Peterson, Gavin E. Crooks, and Robert S. Smith

Rigetti Quantum Computing, 2919 Seventh St, Berkeley, CA 94710

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Updated version: The authors have uploaded version v4 of this work to the arXiv which may contain updates or corrections not contained in the published version v3. The authors left the following comment on the arXiv:
Updated after initial publication to fix increasing/decreasing conventions and an incorrect Figure 6


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 ``$\mathrm{XY}$--family'' for which this subspace is particularly large, i.e., for which many two-qubit programs admit expression as low-depth circuits.

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