Quantum Routing and Entanglement Dynamics Through Bottlenecks

To implement arbitrary quantum circuits in architectures with restricted interactions, one may effectively simulate all-to-all connectivity by routing quantum information. We consider the entanglement dynamics and routing between two regions only connected through an intermediate “bottleneck” region with few qubits. In such systems, where the entanglement rate is restricted by a vertex boundary rather than an edge boundary of the underlying interaction graph, existing results such as the small incremental entangling theorem give only a trivial constant lower bound on the routing time (the minimum time to perform an arbitrary permutation). We significantly improve the lower bound on the routing time in systems with a vertex bottleneck. Specifically, for any system with two regions $L, R$ with $N_L, N_R$ qubits, respectively, coupled only through an intermediate region $C$ with $N_C$ qubits, for any $δ> 0$ we show a lower bound of $Ω(N_R^{1-δ}/\sqrt{N_L}N_C)$ on the Hamiltonian quantum routing time when using piecewise time-independent Hamiltonians, or time-dependent Hamiltonians subject to a smoothness condition. We also prove an upper bound on the average amount of bipartite entanglement between $L$ and $C,R$ that can be generated in time $t$ by such architecture-respecting Hamiltonians in systems constrained by vertex bottlenecks, improving the scaling in the system size from $O(N_L t)$ to $O(\sqrt{N_L} t)$. As a special case, when applied to the star graph (i.e., one vertex connected to $N$ leaves), we obtain an $Ω(\sqrt{N^{1-δ}})$ lower bound on the routing time and on the time to prepare $N/2$ Bell pairs between the vertices. We also show that, in systems of free particles, we can route optimally on the star graph in time $Θ(\sqrt{N})$ using Hamiltonian quantum routing, obtaining a speed-up over gate-based routing, which takes time $Θ(N)$.

💡 Research Summary

The paper investigates the fundamental limits of quantum routing and entanglement distribution in architectures where two large regions, L and R, are connected only through a small intermediate region C—a “vertex bottleneck.” While previous work on edge‑bottleneck systems (where the number of edges crossing a bipartition limits the entanglement rate via the Small Incremental Entangling theorem) yields non‑trivial lower bounds on routing time, vertex‑bottleneck systems have only been known to admit a constant‑order lower bound, which is far from tight.

The authors consider both the gate‑based routing model (where only local 1‑ and 2‑qubit gates respecting the connectivity graph are allowed) and the continuous‑time Hamiltonian routing model (where the system evolves under a Hamiltonian whose terms respect the same graph). They focus on piecewise time‑independent Hamiltonians with a minimum segment width Δ, as well as smooth time‑dependent Hamiltonians.

Main technical contributions

1. Improved lower bound on Hamiltonian routing time.
   For a tripartite system with |L| = N_L, |R| = N_R, |C| = N_C (assumed N_L ≥ N_R ≥ N_C) and any δ > 0, the worst‑case Hamiltonian routing time satisfies
   \