bibkey: garciafernandez2026openqc
authors: Miguel Garcia Fernandez, Chiara Paletta, Ana L. Retore
year: 2026
title: "Open-boundary integrable quantum circuits with different geometries"
doi: 10.48550/arXiv.2607.02093
url: https://arxiv.org/abs/2607.02093v1
claim: "The paper builds open-boundary Yang-Baxter integrable quantum circuits for every configuration of -kappa inhomogeneities (Theorems 1 and 2) and conjectures that the minimum circuit depth with kappa_- such sites is (N + 3)/2 - kappa_- for odd N (Conjecture 1) and (N + 4)/2 - kappa_- for even N (Conjecture 2)."
strata_touched:
  - D5/S3/Quantum/Dynamics/OpenIntegrableCircuitDepthRefutation
  - D5/S3/Quantum/Dynamics/OpenIntegrableCircuitMinDepth
license: citation-only
triage: anchor

Open-boundary integrable quantum circuits with different geometries

The paper constructs Floquet circuits from the double-row transfer matrix of an open integrable spin chain with inhomogeneities ±κ. For a configuration

$\vec{n}=(n_{{\kappa }_{-}},\cdots ,n_2,n_1)$, with $1\le n_1<n_2<\cdots <n_{{\kappa }_{-}}\le N$

of the sites carrying -κ, Theorem 1 (n_{κ₋} < N) gives the circuit

$\left({\overleftarrow{\prod }}_{1\le r\le {\kappa }_{-}}{U}_{n_r,n_r+1}\right)K_1^R(\kappa )\left({\overrightarrow{\prod }}_{1\le j\le N-1,j\ne n_1,\dots ,n_{{\kappa }_{-}}}{U}_{j,j+1}\right){\tilde{K}}_N^L(\kappa )$

up to a scalar, and Theorem 2 (n_{κ₋} = N) the circuit

${\tilde{K}}_N^L(\kappa )\left({\overleftarrow{\prod }}_{1\le r\le {\kappa }_{-}-1}{U}_{n_r,n_r+1}\right)K_1^R(\kappa )\left({\overrightarrow{\prod }}_{j\ne n_1,\dots ,n_{{\kappa }_{-}}}{U}_{j,j+1}\right)$.

Circuits with the same κ₋ form one equivalence class; the depth is “the numbers of sequential gate layers required to implement the circuit”, and the paper states:

Conjecture 1 For odd $N$ and $0<{\kappa }_{-}\le \frac{N-1}{2}$, the minimum depth is given by $d=\frac{1}{2}(N+3)-{\kappa }_{-}.$

Conjecture 2 For even $N$ and $0<{\kappa }_{-}\le \frac{N}{2}$, the minimum depth is given by $d=\frac{1}{2}(N+4)-{\kappa }_{-}.$

Verified locator

• DOI: 10.48550/arXiv.2607.02093 (resolves through doi.org to https://arxiv.org/abs/2607.02093, checked 2026-09-27).
• URL: https://arxiv.org/abs/2607.02093v1 (the only version listed by the arXiv API on 2026-09-27); source file OpenQC.tex, section “Circuits with minimum depth $d$ and independent geometries”, \paragraph{Conjecture 1} and \paragraph{Conjecture 2}; Theorems 1 and 2 in the section “General construction”.

Integrability conjecture

The TeX source OpenQC.tex, §3.1.2, “From the circuit to $\vec{n}$”, states:

For the open-boundary case, similarly to the periodic setting \cite{Paletta:2025sap}, we conjecture that any circuit in which each gate $U_{i,i+1}$ (constructed from an $\check{R}$-matrix) appears exactly once per period to every nearest-neighbor pair of spins, and where each boundary gate is constructed from a $K$-matrix, is integrable.

Section 5, “Conclusion and Outlook”, states:

First, for two types of inhomogeneities, we conjecture that a circuit is Yang–Baxter integrable if: (1) the bulk gate is a solution of the Yang–Baxter equation; (2) the boundary gates satisfy the boundary Yang–Baxter equations; and (3) each bulk gate is applied exactly once to every pair of consecutive qubits. We have verified this conjecture numerically up to thirty sites. However, an analytic proof, even for the periodic case, remains an open problem.