Subsystem many-hypercube codes: High-rate concatenated codes with low-weight syndrome measurements

TL;DR

The paper tackles the encoding-rate vs. fault-tolerance trade-off in quantum error-correcting codes by introducing subsystem many-hypercube (MHC) codes built from the smallest stabilizer code, the D4, to keep syndrome measurements weight at a constant value $4$ across concatenation levels. It provides a detailed construction of the subsystem $D_{4^r}$ codes, analyzes their gauge-qubit structure, and develops two decoders—an optimal block MAP decoder (up to $r=3$) and a scalable neural-network decoder (up to $r=4$)—to contend with the large gauge-qubit space. Empirical results show the NN decoder outperforms the bounded-distance and BP-OSD decoders across bitflip error rates, with an estimated threshold near $2\%$, while the block MAP decoder remains optimal up to level $3$. The work demonstrates a practical, high-rate, low-weight-syndrome approach to large-scale QECCs and suggests extensions to other MHC codes, such as those based on the $[ obreaklu ext{...}]$ family, while noting the continued challenge posed by gauge-qubit-induced decoding complexity.

Abstract

Quantum error-correcting codes (QECCs) require high encoding rate in addition to high threshold unless a sufficiently large number of physical qubits are available. The many-hypercube (MHC) codes defined as the concatenation of the [[6,4,2]] quantum error-detecting code have been proposed as high-performance and high-encoding-rate QECCs. However, the concatenated codes have a disadvantage that the syndrome weight grows exponentially with respect to the concatenation level. To address this issue, here we propose subsystem quantum codes based on the MHC codes. In particular, we study the smallest subsystem MHC codes, namely, subsystem codes derived from the concatenated [[4,2,2]] error-detecting codes. The resulting codes have a constant syndrome-measurement weight of 4, while keeping high encoding rates. We develop the block-MAP and neural-network decoders and show that they demonstrate superior performance to the bounded-distance decoder.

Figures (4)

• Figure 1: A circuit for an encoder of $|\psi_1\psi_2\rangle_L$ state and the syndrome measurement of the $D_4$ code.
• Figure 2: The operators of (a) the $D_{4,4}$ code and (b) the subsystem $D_{4,4}$ code. The logical-qubit operators are the same for the two codes. (c) Operators of the subsystem $D_{4,4,4}$ code. Note that only some of the stabilizer generators are depicted.
• Figure 3: The performance of (a) the subsystem $D_{4,4,4}$ code and (b) the subsystem $D_{4,4,4,4}$ code, and (c) the subsystem $D_{4,4}$ to $D_{4,4,4,4}$ codes with the neural network (NN) decoder. In (a) and (b), the dashed lines are the theoretical performance of the bounded-distance (BD) decoder (\ref{['eq:bdd_prob']}) and solid lines are numerically evaluated by BP-OSD (black), the NN decoder (red), and the block MAP decoder (green), respectively. The vertical axis of (a) and (b) is the relative logical error rate compared with that of the BD decoder (\ref{['eq:bdd_prob']}), while the insets show the bare logical error rate in log scale. In (c), we plot the logical error rate estimated with the NN decoder from level 2 to 4. The inset shows the schematic picture of the NN at level 2.
• Figure 4: The performance of the $D_{4^r}$ codes by (a) the BP-OSD and (b) the level-by-level minimum distance decoder doi:10.1126/sciadv.adp6388. Numerically evaluated logical error rates are plotted from level 2 to 4.