Two-Qubit Circuit Depth and the Monodromy Polytope

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.Corollary (Lemma 46). Allowing the parameter of CPHASE to range freely in 0 ≤ α ≤ 2π, the sets P 2 CPHASE and P 3 CPHASE are the same as the corresponding sets for S = {CZ}. Hence, P 2 CPHASE occupies 0% of the volume of all twoqubit programs, and L CPHASE = 3.We find the situation to be quite different for XY:Corollary (Somewhat informal 3 ; Corollary 53, Remark 57). As a function of α, the volume of the set P 2 XYα is maximized at α = 3π/4, where it contains 75% of randomly sampled two-qubit programs. Correspondingly, L XYα is minimized as L XYα = 9/4. Allowing the parameter of XY to range freely, the set P 2 XY contains ≈96% of randomly sampled all two-qubit programs, with corresponding value L XY ≈ 2.04. 4). We include as appendices an introduction to the mathematics underpinning these results as well as a simpler viewpoint that yields similar qualitative results but is quantitatively inexplicit. The geometry of two-qubit programs and the canonical decompositionAs motivation, we include a brief treatment of the Euler decomposition of single-qubit programs 3 In particular, the notion of "volume" is different from the usual Haar volume, and so "randomly sampled" also changes meaning.Accepted in Quantum 2020-03-19, click title to verify. Published under CC-BY 4.0.7 Identifying a useful analogue of Q and of P U (2) ⊗2 is the primary inhibitor of generalizing this to higher qubit counts. See [45, Proposition IV.3] for a list of references concerning the provenance of this operator Q.

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