bibkey: wierichs2025brickwall authors: D. Wierichs; K. Kottmann; N. Killoran year: 2025 title: “Unitary synthesis with optimal brick wall circuits” doi: 10.48550/arXiv.2511.16736 url: https://arxiv.org/abs/2511.16736v1 claim: “Brick wall circuits with the optimal number of parameters parametrize SU(2^n); their Jacobian has full rank almost everywhere for n ≤ 5, and universality would follow from Conjecture 3: the critical values of a parametrized quantum circuit with full rank almost everywhere form an at most (d−2)-dimensional subset of the target group.” strata_touched:

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Wierichs, Kottmann and Killoran, unitary synthesis with optimal brick wall circuits

D. Wierichs, K. Kottmann and N. Killoran, Unitary synthesis with optimal brick wall circuits, arXiv:2511.16736v1 (20 November 2025; quant-ph; the only version).

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DOI: 10.48550/arXiv.2511.16736. Primary version: https://arxiv.org/abs/2511.16736v1. Source: https://arxiv.org/pdf/2511.16736v1 (text read from the PDF): §V, printed page 9 (Conjecture 3), and Appendix F, printed page 15 (Definition 1).

Source statements

§V, page 9: “One path to showing surjectivity of a PQC thus would be to show that the critical values actually form a subspace of G that is even lower-dimensional, namely at most (d−2)-dimensional. … Based on numerical experiments and lower-dimensional examples, we do conjecture this to be true: Conjecture 3. The critical values F(C) of a PQC that has full rank almost everywhere in T^d form an at most (d−2)-dimensional subset of G.”

§V, page 9: “To summarize, if Conj. 3 holds, we know our circuit families to be universal for n≤5.”

Appendix F, Definition 1, page 15: “A parametrized quantum circuit (PQC) is a quantum circuit consisting of Clifford gates and d rotations chosen from the set {R_X, R_Y, R_Z}, with one value of the input vector feeding into one rotation each, without preprocessing. We identify it with the map F: T^d → G ⊂ C^{2^n×2^n}, that results from evaluating the matrix of the circuit in some basis.” The appendix continues: “From here on we will look at PQCs with d = dim(G) ≤ 4^n.”