Optimized Many-Hypercube Codes toward Lower Logical Error Rates and Earlier Realization

TL;DR

This paper investigates smaller many-hypercube quantum error-detecting codes to achieve lower logical error rates and earlier experimental realization, introducing efficient fault-tolerant encoders and demonstrating superior performance in circuit-level noise models.

Contribution

It introduces optimized smaller many-hypercube codes and fault-tolerant encoders, showing they outperform larger codes in logical error rates and resource overhead.

Findings

Smaller codes like D_{6,4,4} achieve lower logical error rates than larger codes.

Efficient fault-tolerant encoders reduce overhead by about 60%.

D_{6,4,4} performs best for logical CNOT gates in circuit-level noise models.

Abstract

Many-hypercube codes, concatenated ${[[n,n-2,2]]}$ quantum error-detecting codes ($n$ is even), have recently been proposed as high-rate quantum codes suitable for fault-tolerant quantum computing. While the original many-hypercube codes with ${n=6}$ can achieve remarkably high encoding rates (about 30% and 20% at concatenation levels 3 and 4, respectively), they have large code block sizes at high levels (216 and 1296 physical qubits per block at levels 3 and 4, respectively), making not only experimental realization difficult but also logical error rates per code block high. Toward earlier experimental realization and lower logical error rates, here we comprehensively investigate smaller many-hypercube codes with $[[6,4,2]]$ and/or $[[4,2,2]]$ codes, where, e.g., $D_{6,4,4}$ denotes the many-hypercube code using $[[6,4,2]]$ at level 1 and $[[4,2,2]]$ at levels 2 and 3. As a result, we found a counterintuitive fact that $D_{6,4,4}$ ($D_{6,6,4,4}$) can achieve lower logical error rates per code block than $D_{4,4,4}$ ($D_{4,4,4,4}$), despite its higher encoding rate and larger code block size. Focusing on level 3, we also developed efficient fault-tolerant encoders realizing about 60% overhead reduction while maintaining or even improving the performance, compared to the original design. Using them, we numerically confirmed that $D_{6,4,4}$ also achieves the best performance for logical controlled-NOT gates in a circuit-level noise model. These results are important for targeting a high-rate code toward early experimental realization of efficient fault-tolerant quantum computing.