The minimum depth of open integrable circuits
Abstract
For the open-boundary integrable quantum circuits of Garcia Fernandez, Paletta and Retore, the least depth over the configurations with kappa of the N sites carrying -kappa is floor(N/(kappa + 1)) + 1 whenever 1 <= kappa and 2 kappa <= N.
Theorem 1.1 (Minimum depth).
Proof. Machine-checked in Lean as D5/S3/Quantum/Dynamics/OpenIntegrableCircuitMinDepth.result (✓ std3). ∎
Source. Repository-derived.
Acknowledgement. Miguel Garcia Fernandez, Chiara Paletta, Ana L. Retore (2026). Open-boundary integrable quantum circuits with different geometries. DOI: 10.48550/arXiv.2607.02093. URL: https://arxiv.org/abs/2607.02093v1.
Commentary.
Read the circuit of a set S of sites as the path of sites 0, 1, …, N, with site 0 for the boundary gate K1, site N for KN and site k in between for U_k: the circuit lists these gates in the order of the sites outside S decreasing, then 0, then the sites of S increasing, and two gates share a site of the chain exactly when their sites are consecutive. So the circuit fits in L layers exactly when some labelling of the sites 0, …, N by numbers below L increases along each step from k to k + 1 with k + 1 in S and decreases along each other step. Along a run of consecutive steps outside S the labels decrease, and each of the |S| steps into S can raise the label by at most L - 1, which forces N < (|S| + 1) L; hence the depth is at least floor(N/(kappa + 1)) + 1. Conversely, cutting the sites 1, …, N into kappa + 1 runs of length at most floor(N/(kappa + 1)) separated by the kappa sites of S gives a labelling below floor(N/(kappa + 1)) + 1, by induction on kappa.