Finite-precision implementation of d-ary BPQM

Extend the finite-bit-register approximation analysis for binary belief propagation with quantum messages to d-ary eigen-list registers, and establish that the resulting finite-precision decoder approximation has error that vanishes as the number of bits per register increases.

Background

The efficient implementation is described in a real-register model in which eigen-list entries are stored exactly. Such registers are formally infinite-dimensional, so a physically finite implementation requires approximating each real entry with finitely many bits.

For binary belief propagation with quantum messages, prior work provides an analysis showing that the approximation error vanishes as the register precision increases. The paper uses the analogous d-ary construction but does not prove the corresponding finite-precision guarantee for d-ary eigen lists, leaving this technical question unresolved.

References

For binary BPQM, a representation with finitely many bits per register approximates the decoder with an error that vanishes as the number of bits grows. We expect this analysis to extend to $d$-ary eigen lists and leave it to future work.

— Efficient capacity-achieving entanglement generation with application to pure-loss Bosonic channels  (2610.03674 - Mandal et al., 2 Oct 2026) in Section 6, Section 6.4, “Decoder implementation and complexity”