Tensor-network code[1–5]

Description

Block quantum code constructed using a tensor-network-based graphical framework from atomic tensors a.k.a. quantum Lego blocks [3], which can serve as encoding isometries for smaller quantum codes. The class of codes constructed using the framework depends on the choice of atomic Lego blocks.

The individual Lego blocks and resulting quantum Lego codes can be stabilizer [2,4] or non-stabilizer [3,5]. They need not be isometries, meaning that this class of codes generalizes planar-perfect tensor-network codes. However, both the logical and physical degrees of freedom must have the same local dimension.

For example, any stabilizer code can be built out of atomic blocks like the 2-site repetition code, single-site trivial stabilizer codes, and tensor products of the \(|0\rangle\) state. Specifically, the HaPPY holographic code is a quantum Lego code whose atomic Lego block is the five-qubit perfect qubit code.

Many known codes can be created using this code’s methods in order to further their understanding, including a 6-qubit implementation of the generalized Bacon-Shor code, the toric code, and the \([[7,1,3]]\) Steane code. Finite patches of the toric-code tensor network, equipped with boundary stopper tensors or repetition-code boundary tensors, reproduce surface-code patches and subsystem variants. Local Hadamard modifications of alternating tensors yield XZZX surface-code variants, while re-interpreting alternating rows of surface-code physical legs as logical legs yields 2D Bacon-Shor codes and makes their relation to the quantum compass model explicit [3]. The same framework also yields flat-geometry perfect-tensor codes, holographic Reed-Muller codes with transversal non-Clifford gates, and a 3D subsystem code built from Steane tensors with localized cube-like stabilizers [3]. For example, a simple \( [[4,2,2]] \) stabilizer code can be written as a rank 6 tensor. Attaching two of these via gluing together one logical leg from each can produce a \([[6,4,2]]\) stabilizer code. Code optimization in this framework can be done using reinforcement learning [6,7].

To construct a Lego code, the encoding map \(V\) for each code that is to be used in the construction is converted to a tensor by decomposing it using the formula \begin{align} V = \sum_{i_j} V_{i_1 \ldots i_{n+k}} | i_{k+1} \ldots i_{k+n} \rangle \langle i_1 \ldots i_k |~. \tag*{(1)}\end{align} We then look at the codes graphically, treating each \(i_j\) as an edge dangling out of the tensor vertex \(V_{i_1 \ldots i_{n+k}}\). These edges are either connected to another tensor vertex’s edges or left dangling. If the block codes are stabilizer, then each local tensor has unitary product stabilizers (UPS). The goal is to push each UPS through the tensor network until each dangling edge has only trivial support. Otherwise, a matching value is pushed through the edge and the process is repeated on the next tensor. If a UPS can be pushed through the whole network, then a UPS for the larger network has been found. The dangling legs (edges) and UPS of the whole network can then be converted to physical/logical elements and stabilizers/logical operators for a new quantum code.