Fig. 2: Numerical verification of the Dyck construction.
From: Unified framework for open quantum dynamics with memory
Operator norm of \({\tilde{{{{\bf{I}}}}}}_{N}\) (Light) and \({{{{\bf{{{{\mathcal{K}}}}}}}}}_{N}\) (Dark) as a function of NΔt. Lines denote \({\tilde{{{{\bf{I}}}}}}_{N}\) computed from analytic expressions and \({{{{\bf{{{{\mathcal{K}}}}}}}}}_{N}\) from post-processing exact numerical results via the transfer tensor method40. Circles denote \({{{{\bf{{{{\mathcal{K}}}}}}}}}_{N}\) from the Dyck diagrammatic method, and crosses are \({\tilde{{{{\bf{I}}}}}}_{N}\) obtained via the inverse map discussed in Eqs. (13) and (14). Dashed lines denote the operator norm of the crest term of \({{{{\bf{{{{\mathcal{K}}}}}}}}}_{N}\) (the Dyck path diagram with the highest height). Parameters used are: Δ  =  1 (other parameters are expressed relative to Δ), ϵ  =  0, β  =  5, Δt  =  0.1, ωc  = 7.5, and ξ  =  0.1 and s  =  1 (a), ξ  =  0.5 and s  =  1 (b), and ξ  =  0.5 and s  =  0.5 (c).
