Local tensor-network codes

Abstract

Tensor-network codes enable the construction of large stabilizer codes out of tensors describing smaller stabilizer codes. An application of tensor-network codes was an efficient and exact decoder for holographic codes. Here, we show how to write some topological codes, including the surface code and colour code, as simple tensor-network codes. We also show how to calculate distances of stabilizer codes by contracting a tensor network. The algorithm actually gives more information, including a histogram of all logical coset weights. We prove that this method is efficient in the case of stabilizer codes encoded via local log-depth circuits in one dimension and holographic codes. Using our tensor-network distance calculator, we find a modification of the rotated surface code that has the same distance but fewer minimum-weight logical operators by “doping” the tensor network, i.e., we break the homogeneity of the tensor network by locally replacing tensors. For this example, this corresponds to an improvement in successful error correction of almost 2% against depolarizing noise (in the perfect-measurement setting), but comes at the cost of introducing three higher-weight stabilizers. Our general construction lets us pick a network geometry (e.g., a Euclidean lattice in the case of the surface code), and, using only a small set of seed codes (constituent tensors), build extensive codes with the potential for optimisation.

Optimization of tensor network codes with reinforcement learning

“…the joint measurement of X a X b and Z a Z b operators on the two physical qubits, a and b, corresponding to the contracted tensor legs. Tensor fusion creates a new tensor network code, as long as a technical condition described in [12,14] is satisfied. This technical condition essentially requires that the Bell measurement does not measure any logical degree of freedom; that is the X a X b and Z a Z b operators commute with all logical operators of the original code tensors (see appendix A of [14] for more context).…”

“…Tensor fusion creates a new tensor network code, as long as a technical condition described in [12,14] is satisfied. This technical condition essentially requires that the Bell measurement does not measure any logical degree of freedom; that is the X a X b and Z a Z b operators commute with all logical operators of the original code tensors (see appendix A of [14] for more context). This is also equivalent to the existence of an isometry from the two contracted indices plus the logical qubits to the complementary set of indices.…”

“…These are products of the original operators with some stabilizers. Tensor fusion is general enough to handle both CSS and non-CSS codes, and mixtures thereof [14].…”

“…Building on the successful use of tensor networks in other areas of quantum information science [5][6][7][8][9][10][11], tensor-network codes were recently introduced [12] as a method to construct new codes by applying tensor network operations to existing codes, including codes with multiple logical qubits [13]. For efficiently contractible tensor-network codes, this method comes with natural tensor-network decoders and efficient ways to compute the distribution of Pauli-weights of logical operators and stabilizers, allowing one to easily read off the code distance [14].…”

“…In this paper, we treat the search for good codes as a 'game' , in which the starting pieces of the game are a finite initial set of seed code tensors [12,14], which, when contracted together yield various possible stabilizer codes. For our purposes, better codes are those that have high distance for a given number of physical qubits, and so there is a natural performance metric associated with a code.…”