Tensor-Network Codes

We introduce tensor-network stabilizer codes which come with a natural tensor-network decoder. These codes can correspond to any geometry, but, as a special case, we generalize holographic codes beyond those constructed from perfect or block-perfect isometries, and we give an example that corresponds to neither. Using the tensor-network decoder, we find a threshold of 18.8% for this code under depolarizing noise. We show that, for holographic codes, the exact tensor-network decoder (with no bond-dimension truncation) has polynomial complexity in the number of physical qubits, even for locally correlated noise, making this the first efficient decoder for holographic codes against Pauli noise and, also, a rare example of a decoder that is both efficient and exact.

Figure 1

Tensor-network codes: (a) the tensor $T^1(L{)}_{(g_1,\dots ,g_6)}$ describing a six-qubit code, where $L$ denotes logical operators. (b) Applying Theorem 1 to build a new code tensor by contracting the sixth index of $T^1(L{)}_{(g_1,\dots ,g_6)}$ with the first index of $T_{(h_0,\dots ,h_6)}^0$ to get a $⟦11,1,3⟧$ code. (c) A tensor-network decoder calculates a probability for each logical operator $L$ (the correction operator is determined by finding the maximum of these probabilities) by contracting the code tensor with rank-one tensors $p({\sigma }^{e_i}{\sigma }^{r_i})$ for each physical qubit $i$, with index $r_i$ ($e_i$ is fixed by the syndrome and, so, is not summed over). This is an implementation of maximum-likelihood decoding.

Figure 2

(a) Tensor network for a radius-four holographic code. The central tensor $T^1(L)$ (with one logical qubit) has six legs, whereas all other tensors $T^0$ with no logical qubits have seven legs. (b) Contraction order: starting from the outside, contract inwards. After the first contractions (c), we have tensors at radius four, which are enclosed inside the blue boxes. After the next contraction (d), we have tensors at radius three, enclosed in green boxes. After another round of contractions (e), we have the tensor enclosed in the brown box. Finally, these tensors are contracted with the central tensor, colored blue in (b). (f) Generic form of a tensor contraction: $M(k)$ are the results of previous contractions, with bond legs marked by dashed lines.

Figure 3

Failure rates for the (asymptotically zero-rate) six-qubit holographic code versus the single-qubit error probability $p$ for different code radii. Lines are to guide the eye between failure rates within a fixed code size. Inset: expanded view of failure rates close to the threshold of 18.8% [30].