Determine whether low-depth tensor-network simulation remains efficient for larger exchange networks

Determine whether the actual bond-dimension growth of matrix-product-state simulations for low-depth bubble-sort exchange networks is sufficiently mild to make tensor-network simulation useful beyond the system sizes studied.

Background

The paper observes that each exchange pulse is a linear combination of the identity and a SWAP, so a bubble-sort exchange network can be represented as a matrix-product operator whose bond dimension grows at most exponentially with the number of rounds, but independently of the chain length. This raises the possibility of simulating low-depth networks with matrix-product states rather than full state vectors.

The authors identify the practical unresolved issue as the actual, rather than worst-case, growth of the required bond dimension. If that growth is sufficiently mild, tensor-network methods could extend simulations beyond the 24–30-dot systems reached with sector-restricted and GPU-based state-vector methods.

References

Beyond that, the network itself is a matrix-product operator of bond dimension at most $4$ per round (each pulse is $a\mathbb 1+b\,\mathrm{SWAP}$, of operator Schmidt rank at most $4$, and one pulse crosses each cut per round), so the low-depth regime $R\lesssim2N$ that matters for hardware may be open to a matrix-product-state simulation whose bond dimension is bounded by $4{O(R)}$ independently of $N$; whether the actual growth is mild enough to be useful remains to be tested.

— Bubble-Sort Exchange Networks: Symmetry Selection Rules and Language-Model Distillation on Exchange-Only Spin Chains  (2610.22093 - Minato, 15 Oct 2026) in Section 6, paragraph beginning “Scaling”