  Entangling quantum memories, mediated by optical-frequency or microwave
channels, at high rates and fidelities is key for linking qubits across short
and long ranges. All well-known protocols encode up to one qubit per optical
mode, hence entangling one pair of memory qubits per transmitted mode over the
channel, with probability $\eta$, the channel's transmissivity. The rate is
proportional to $\eta$ ideal Bell states (ebits) per mode. The quantum
capacity, $C(\eta) = -\log_2(1-{\eta})$ ebits per mode, which $\approx
1.44\eta$ for high loss, i.e., $\eta \ll 1$, thereby making these schemes near
rate-optimal. However, $C(\eta) \to \infty$ as $\eta \to 1$, making the known
schemes highly rate-suboptimal for shorter ranges. We show that a
cavity-assisted memory-photon interface can be used to entangle matter memories
with Gottesman-Kitaev-Preskill (GKP) photonic qudits, which along with
dual-homodyne entanglement swaps that retain analog information, enables
entangling memories at capacity-approaching rates at low loss. We benefit from
loss resilience of GKP qudits, and their ability to encode multiple qubits in
one mode. Our memory-photon interface further supports the preparation of
needed ancilla GKP qudits. We expect our result to spur research in low-loss
high-cooperativity cavity-coupled qubits with high-efficiency optical coupling,
and demonstrations of high-rate short-range quantum links.
