Separating quantum-comb and fixed-wire SDP relaxations

Identify examples in which the quantum-comb relaxation and the fixed-dimension internal-wire semidefinite-programming relaxation produce different upper bounds on the optimal circuit fidelity.

Background

The paper presents two distinct semidefinite-programming relaxations for architecture incompatibility. The main-text relaxation fixes the internal-wire dimension but relaxes the product structure of the gate Choi states using a positive-partial-transpose constraint, whereas the quantum-comb relaxation allows an arbitrarily large memory system and relaxes the gate operations from unitaries to general completely positive trace-preserving maps. The authors report agreement between the two relaxations for the examples studied, but whether there are instances in which they yield different upper bounds remains unresolved.

References

Identifying examples where the two relaxations lead to different upper bounds is an interesting question that we leave for future work.

— Witnessing the architecture of quantum circuits  (2608.13169 - Mothe et al., 13 Aug 2026) in Appendix, Section 2, subsection “Building a witness”