Quantum state transfer through a lossy bus: fundamental error limits for every control pulse, with application to IBM's meter-scale interconnects

How small can pulse shaping make the error of quantum state transfer between two qubits, such as superconducting qubits in different modules, through a lossy bus (a resonator, a waveguide or a cable mode)? We give bounds that hold for every control pulse, every duration and any number of bus modes, knowing only the maximal coupling strength G, the qubit loss rate γ and the bus loss rate κ ≥ γ. The population delivered in time T obeys P(T) ≤ e−γT/[1 + (κ−γ)/(G²T)]: going slowly costs qubit loss, going fast costs bus loss, and no pulse escapes both. The end qubits cannot be bypassed either: in the frame where the qubits are lossless, every transfer keeps the excitation in them for an integrated time bounded below by a quantity that tends to π/(2G). As γ/(κ−γ) → 0 and √(γ(κ−γ))/G → 0, the error cannot fall below 2√(2γ(κ−γ))/G (1 − o(1)); this constant was known as the optimum of perturbative and quasi-adiabatic pulse optimisation; here it bounds every control, and an explicit pulse attains it with explicit error terms. Markovian dephasing enters through the same equations with rates 2/T₂; with a different coherence time for each element the bounds give the floor of a real device, and a reciprocity relation makes the direction of transfer irrelevant. For the four links of IBM's meter-scale relay experiment (arXiv:2610.29986), no pulse can bring the error below 6.03–10.05 × 10⁻⁴ (certified with interval arithmetic) if dephasing is Markovian at the measured T₂*, and not below 1.5–1.8 × 10⁻⁴ for any dephasing noise; the measured errors are 11–16 times larger. Doubling the coupling halves the floor; the dephasing of the worse qubit is the next lever; a 26-ns trapezoidal pulse at 18 MHz brings all four links below 10⁻³ within the model. The floor, the occupation lemma, the frontier lemma and the window comparisons are machine-checked in Isabelle/HOL, and every number comes with the script that produces it. Part II of a study whose Part I is «Certified Speed-Limit Diagnostics for Deployed Superconducting Quantum Processors». Companion site (blueprint, with the laboratory «Check your link»): https://karlesmarin.github.io/blueprints/relay-floors/. English and Spanish versions; sources and supplementary package (scripts, data, Isabelle theories, certificates).