Tensor Networks and Quantum Error Correction

We establish several relations between quantum error correction (QEC) and tensor network (TN) methods of quantum many-body physics. We exhibit correspondences between well-known families of QEC codes and TNs, and demonstrate a formal equivalence between decoding a QEC code and contracting a TN. We build on this equivalence to propose a new family of quantum codes and decoding algorithms that generalize and improve upon quantum polar codes and successive cancellation decoding in a natural way.

Figure 1

(a) General decoding problem. This TN encodes the probability distribution $Q_4(E|s)$ of qubit 4 given that the 5th and 6th qubits revealed the syndrome $s=(0,1)$. (b)–(e) Action of the cnot gate on the bimodal indicator functions. The truth table for (b) $(\sigma ,{\sigma }^{\prime })\to (\kappa ,{\kappa }^{\prime })$ is given by $(I,{\sigma }_x)\to (I,{\sigma }_x)$, $({\sigma }_x,I)\to ({\sigma }_x,{\sigma }_x)$, $(I,{\sigma }_z)\to ({\sigma }_z,{\sigma }_z)$, and $({\sigma }_z,I)\to ({\sigma }_z,I)$. Identities (c)–(e) follow from the application of (b) to bimodal indicator functions.

Figure 2

Graphical definitions of various unitary TNs and encoding circuits of QEC codes: wires represent one or a few qubits, rectangles represent unitary transformation with time running from top to bottom. The size of each circuit can be varied in an obvious way. For coding applications, location of the data qubits are indicated by triangles, other qubits are initialized to $|0⟩$ (or $|+⟩$). (a) MPS TN and convolutional codes. (b) Tree TN and concatenated quantum block codes. (c) MERA TN and topological code. (d) Branching tree TN and polar codes. (e) Branching MERA TN and codes introduced in this Letter.

Figure 3

Tensor network representation of the branching MERA code. (a) This TN encodes the $x$ error probability of the seventh qubit (from right) knowing that among the first six qubits, only the first and third had $x$ errors, among the last four qubits, only qubit 14 had a $z$ error, and ignoring everything about qubits 8 to 12. The decoder can choose to ignore the information about $x$ errors, in which case the four tensors $b_x$ on the left are replaced by $\mathbf{e}$. Using the circuit identities of Fig. 1, these two TNs for $z$ decoder and the symmetric $x\text{−}z$ decoder become equivalent to (b) and (c).

Figure 4

Performance of the polar and branching-MERA codes under the depolarizing channel. The block-error rate (BER) is the probability of incorrectly decoding a block on $n$ qubits. (a) Comparison of the various codes and decoding algorithms for codes of size $2^{12}$ qubits with encoding rate $1/2$. (b) Upper bound on the BER for codes with depolarizing rate 9.92% and encoding-rate $1/8$ as a function of system size. The probability of error decreases strongly with code size. Both the symmetric decoder and the branching-MERA improve beyond the standard quantum polar code.