Tensor-Network Decoders
Tensor-network decoders approximate the true maximum-likelihood correction by contracting a network that sums over all error configurations.
Maximum likelihood by contraction
The optimal decoder chooses the logical correction whose total probability, summed over all physical errors consistent with the syndrome, is largest. This sum has exponentially many terms. A tensor-network decoder expresses that sum as a two-dimensional network of tensors whose contraction yields the coset probabilities exactly in principle.
Controlled approximation
Exact contraction of a 2D tensor network is itself hard, so the decoder truncates the bond dimension, the size of the indices connecting tensors, to make contraction tractable. Increasing the bond dimension increases accuracy and cost, giving a tunable knob that interpolates between a fast approximate decoder and the true maximum-likelihood decoder.
- Sums over error cosets rather than picking a single most-likely error.
- Bond dimension trades accuracy against runtime.
- Approaches optimal accuracy as the bond dimension grows.
- Naturally incorporates degeneracy, which matching-style decoders handle only crudely.
The main value of tensor-network decoders is as a benchmark: they define the best achievable logical error rate for a given code and noise model, against which faster decoders like matching, union-find, and belief propagation are measured. They also handle correlated errors, such as the correlation between X and Z from a Y error, that simpler decoders split apart.
The cost is speed. Full tensor-network decoding is generally too slow for real-time use on fast hardware, so it is run offline to calibrate and validate. That said, low-bond-dimension variants and clever schedules have been pushed toward practical speeds, and the method remains the reference point for decoder accuracy.