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

TL;DR

This work evaluates high-rate many-hypercube codes built from small base codes to enable earlier fault-tolerant quantum computation. It identifies that certain level-3 and level-4 configurations, notably D_{6,4,4} and D_{6,6,4,4}, outperform smaller variants in block and logical-CNOT performance under circuit-level noise. It introduces efficient fault-tolerant encoders for level-3 codes, achieving roughly 60% overhead reduction and marginally better performance. Overall, the results offer a practical path toward experimental FTQC with high encoding-rate codes and guide design choices for high-fidelity logical operations.

Abstract

Many-hypercube codes [H. Goto, Sci. Adv. 10, eadp6388 (2024)], 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 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 notable fact that $D_{6,4,4}$ ($D_{6,6,4,4}$) can achieve lower block error rates than $D_{4,4,4}$ ($D_{4,4,4,4}$), despite its higher encoding rate. 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 will be useful for early experimental realization of fault-tolerant quantum computing with high-rate quantum codes.

Figures (10)

• Figure 1: Definitions of (a) $D_4$ and (b) $D_6$. The circles represent physical qubits. The top two show logical Pauli operators and the bottom two show stabilizers. $Z$ and $X$ operators are shown in blue and red, respectively. Solid and dashed line segments connecting qubits are guides to the eye.
• Figure 2: Visualization of level-2 MHC codes. The circles represent physical qubits. Gray squares correspond to logical Pauli operators. In each figure, the top shows logical Pauli operators for $\mathrm{Q}_{1,1}$ and the bottom shows examples of stabilizers by highlighting them in blue and red for $Z$ and $X$ operators, respectively.
• Figure 3: Visualization of level-3 MHC codes. The circles represent physical qubits. Gray cubes correspond to logical Pauli operators. In each figure, the left shows logical Pauli operators for $\mathrm{Q}_{1,1,1}$ and the right shows examples of stabilizers by highlighting them in blue and red for $Z$ and $X$ operators, respectively.
• Figure 4: All-zero state encoders for (a) $D_4$ and (b) $D_6$.
• Figure 5: Performance comparison of MHC codes. (a--c) Decoding error probabilities at level 2--4, respectively, for bit-flip errors.