[Paper Review] Numerical study of hypergraph product codes

Antoine Grospellier, Anirudh Krishna|arXiv (Cornell University)|Oct 8, 2018

Quantum Computing Algorithms and Architecture19 references18 citations

TL;DR

This paper numerically evaluates the performance of hypergraph product codes under independent bit and phase flip noise using the small-set-flip decoding algorithm. It reports a threshold of approximately 4.6% for a low-rate family (rate ~1.6%) and 2% for a higher-rate family (rate 20%), demonstrating that hypergraph product codes outperform the toric code in logical error rate for large code sizes, with gains of several orders of magnitude at 3600 logical qubits.

ABSTRACT

Hypergraph product codes introduced by Tillich and Zémor are a class of quantum LDPC codes with constant rate and distance scaling with the square-root of the block size. Quantum expander codes, a subclass of these codes, can be decoded using the linear time small-set-flip algorithm of Leverrier, Tillich and Zémor. In this paper, we numerically estimate the performance for the hypergraph product codes under independent bit and phase flip noise. We focus on two families of hypergraph product codes. The first family has rate $1/61 \sim 1.6\%$, has qubits of weight $10$ or $12$ and stabilizers of weight $11$. We report a threshold near $4.6\%$ for the small-set-flip decoder. We also show that for similar rate, the performance of the hypergraph product is better than the performance of the toric code as soon as we deal with more than $500$ logical qubits and that for $3600$ logical qubits, the logical error rate for the hypergraph product code is several orders of magnitude smaller. The second family has rate $0.2$, qubits of weight $10$ and $20$ and stabilizers of weight $15$. We report a threshold near $2\%$ for the small-set-flip decoder.

Motivation & Objective

• To estimate the numerical threshold of hypergraph product codes under independent bit and phase flip noise using the small-set-flip decoder.
• To compare the logical error rate performance of hypergraph product codes with the toric code for increasing code sizes.
• To evaluate the practical advantages of hypergraph product codes in terms of fault-tolerant overhead for large-scale quantum computation.
• To assess the effectiveness of the small-set-flip decoder in correcting stochastic errors in quantum LDPC codes.

Proposed method

• The authors construct two families of hypergraph product codes by taking the product of randomly generated biregular bipartite graphs with degrees (5,6) and (5,10).
• They simulate independent bit and phase flip noise at varying physical error rates across different code block sizes.
• The small-set-flip decoding algorithm is applied to correct errors, with logical failure rates computed as a function of physical error rate and code size.
• Logical error rates are benchmarked against the toric code with side length L=8 for the (5,6) family and L=3 for the (5,10) family, using equivalent logical qubit counts.
• The threshold is estimated as the physical error rate at which the logical error rate begins to decay exponentially with increasing code size.
• Simulations are performed on high-performance computing clusters using the Calcul Québec and Compute Canada infrastructure.

Experimental results

Research questions

• RQ1 What is the numerical threshold of hypergraph product codes under independent bit and phase flip noise when decoded with the small-set-flip algorithm?
• RQ2 How does the logical error rate of hypergraph product codes compare to that of the toric code for increasing numbers of logical qubits?
• RQ3 At what code size does the hypergraph product code become superior to the toric code in terms of logical error rate?
• RQ4 How do the performance characteristics of hypergraph product codes vary with code rate and stabilizer weight?

Key findings

• The hypergraph product code family with rate ~1.6% and qubit weights 10 or 12 achieves a numerical threshold of approximately 4.6% under independent bit and phase flip noise.
• The higher-rate hypergraph product code family with rate 20% and qubit weights 10 or 20 achieves a threshold of approximately 2% under the same noise model.
• For code sizes exceeding 500 logical qubits, the hypergraph product code outperforms the toric code, with logical error rates several orders of magnitude lower at 3600 logical qubits.
• The performance of the small-set-flip decoder improves significantly with increasing code size, showing a clear decay in logical error rate beyond the threshold.
• The results indicate that hypergraph product codes can achieve substantial overhead savings in fault-tolerant quantum computation for large-scale implementations.
• The numerical thresholds are consistent with the analytical lower bound of 2.7×10⁻¹⁶ but significantly higher than previous estimates, suggesting practical viability.