Bin Cheng, Naixu Guo, and Feng Pan have certified a circuit distance of 16 for IBM’s two-gross memory circuits using a standard schedule, a finding that corrects the previously assumed distance of 18. Their new method, which evaluates quantum code distances as tropical partition functions, also revealed discrepancies in 10 of 26 recent papers, finding overestimated distances with a maximum difference of at least eight.

The work certifies distances for codes with up to 31,200 qubits and memory circuits with up to 800,000 fault mechanisms, demonstrating a significant improvement in verifying quantum error correction performance. At 99.98% junction yield, keeping gross-code memory blocks with dead checks that are certified at distance 10 or more raises the estimated usable share from 53% for perfect blocks alone to 94% with an improved schedule.

Tropical Partition Functions Certify Quantum Code Distances

For circuits, the weight of the lightest logical error was verified through simulation, confirming the accuracy of the calculated distances and providing a direct measure of the code’s resilience. This finding highlights the importance of rigorous verification methods like the one presented, as previously published results may not accurately reflect the true capabilities of quantum error correction codes.

Rank-Based Dynamic Programming for Distance Evaluation

The method detailed in this work surpasses existing exact solvers, certifying more instances on two benchmarks than all others combined. This advancement relies on expressing quantum code and circuit distances as tropical partition functions, then evaluating them with rank-based dynamic programming alongside a weight-bounded search. Benchmarking revealed that the time required to certify a distance remained below 10 seconds up to a distance of 25 on a single processing thread, significantly outperforming other general-purpose exact solvers which stalled beyond distance 15.

The researchers employed three algorithms, dense rank-based dynamic programming, a truncated program, and a search, each with costs dependent on specific problem features, and coordinated them using a cost model predicting the work of the dynamic programs. The work also certifies distances for IBM’s two-gross circuit, confirming a distance of 16.

Certifying Distances for 31,200-Qubit Google Codes

Circuit reductions and relaxations proved instrumental in certifying exact values for quantum code distances, exceeding the performance of all other tested exact solvers on two benchmarks. For CSS codes, the minimization process is streamlined by splitting into two smaller problems with unit costs, using binary matrices to define the X-type and Z-type generators.

“We confirm all code distances reported as exact, together with hundreds of additional values,” the researchers state in their published work. Where multiple thresholds were tested, the lower bound consistently held, and upper bounds were verified through explicit error checks against the matrices.

Auditing Overestimated Distances in Published Papers

The team’s analysis, focused on instances they could reconstruct from published data or circuit files, identified instances where reported distances did not align with values certified through their new methodology. These discrepancies, while not invalidating the upper bounds, highlight a potential for misinterpretation when those bounds are presented as exact values. The team’s work extends to complex codes like modular hyperbolic codes, resolving previously unresolved distances within those systems.

Beyond simply verifying existing claims, the audit also demonstrated the practical implications of accurate distance certification for circuit design and hardware use. While a redesigned schedule can achieve a distance of 18, it requires one additional timestep per cycle, a trade-off between performance and reliability.

More significantly, the analysis revealed that 94% of gross-code memory blocks remain usable when certified at a distance of 10 or more with the improved schedule, a substantial increase from the 53% usability rate when relying on only perfect blocks at 99.98% junction yield.

Circuit Distance of 16 for IBM’s Two-Gross Memory Circuits

The analysis, detailed in a new report, extends beyond identifying inaccuracies to demonstrate the practical impact of accurate distance certification on resource allocation and chip usability. This finding is significant because it challenges previously held assumptions about the performance of this widely studied system, and highlights the importance of rigorous verification in quantum computing research. The researchers certify circuit distance 16 for IBM’s two-gross memory circuits with the standard schedule, not the assumed 18, whereas redesigned schedules reach 18 at one noisy cycle, at the cost of one more timestep per cycle.

These findings not only refine their understanding of error correction codes but also offer valuable insights for optimizing circuit design and improving the reliability of future quantum computers. Their approach, expressed as tropical partition functions and evaluated with rank-based dynamic programming, offers a robust method for verifying and refining quantum error correction strategies.

Circuit Distance vs. Code Distance in qLDPC Architectures

Circuit distance, the minimum number of faults needed to flip a logical observable in a quantum circuit, can differ significantly from code distance, a measure of error suppression, particularly within low-density parity-check (qLDPC) codes. This discrepancy arises because a single fault can propagate across multiple qubits in the circuit, reducing the circuit distance below the code distance. This trade-off highlights the detailed relationship between circuit design and computational cost, demonstrating that maximizing distance does not always equate to optimal performance.

A dynamic programming prediction, designed to complete within 10 seconds after a 1-second head start, determines the distance, with a traceback or search at a single threshold confirming the result. This scale of certification exceeds the capabilities of all other tested exact solvers combined.

👉 More information
🗞 Certifying quantum code and circuit distances at scale
✍️ Bin Cheng, Naixu Guo and Feng Pan
🧠 ArXiv: https://arxiv.org/abs/2611.08247v1

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