A Sweep-Plane Algorithm for the Computation of the Volume of a Union of Polytopes

Optimal synthesis into fixed XX interactions

“…The naive application of inclusion-exclusion is described by The terms on the right-hand side are all volumes of convex bodies, hence are individually approachable, but there are 2 |𝐽 | such summands. These summands can be culled in two ways: 18 See [1] for a notable exception.…”

“…Lemma A.1. Let 𝑎 = (𝑎 1 , 𝑎 2 , 𝑎 3 ) be a positive canonical triple, and let 𝑃 ⊆ 𝔄 𝐶 2 be a polyhedron satisfying the reflection-closure property the bounds 0 ≤ 𝑎 2 , 𝑏 2 , 𝑎 3 , 𝑏 3 ≤ 𝜋 4 entail that the first summand is bounded from below by 1 4 cos 2 (𝑎 1 −𝑏 1 ) and the second summand is bounded from above by 1 4 sin 2 (𝑎 1 −𝑏 1 ). Notice that replacing 𝑏 with its reflection ( 𝜋 2 − 𝑏 1 , 𝑏 2 , 𝑏 3 ) trades the positions in the expression of cos 2 (𝑎 1 − 𝑏 1 ) and sin 2 (𝑎 1 − 𝑏 1 ).…”

“…Notice that replacing 𝑏 with its reflection ( 𝜋 2 − 𝑏 1 , 𝑏 2 , 𝑏 3 ) trades the positions in the expression of cos 2 (𝑎 1 − 𝑏 1 ) and sin 2 (𝑎 1 − 𝑏 1 ). Additionally, notice that the expression is maximized when cos 2 (𝑎 1 − 𝑏 1 ) takes on the larger of the two values, i.e., when cos 2 (𝑎 1 − 𝑏 1 ) ≥ 1 2 . This then holds exactly when the conclusion of the Lemma does.…”

“…Though not required for universal computation, the availability of these "overcomplete" basis gates has the potential to yield more efficient synthesized circuits, particularly if the error magnitude of CX 𝛼 correlates with 𝛼: while the universally programmable CX circuit invokes CX three times, the universally programmable fractional CX circuit invokes CX 𝛼 , CX 𝛽 , and CX 𝛾 with ⟨𝛼 + 𝛽 + 𝛾⟩ = 3/2. 1 In practice, however, these parametric families are difficult to operate. The relationship between the degree of truncation and the value 𝛼 is often nonlinear and prone to imperfect measurement, and constraints in the steering electronics (e.g., waveform sample rate) can make truncations unavailable below some threshold.…”

“…, 𝛼 𝑛 maximizes (average-case) device performance? How does one efficiently find expressions for generic two-qubit unitaries in terms of these 1 See [4] for numerical justification.…”