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Scientists have developed an exact approach to a key quantum error correction problem once believed to be unsolvable, and have shown that what appeared to be hardware-related errors may in fact be due to suboptimal decoding.

The new algorithm, called PLANAR, achieved a 25% reduction in logical error rates when applied to Google Quantum AI's experimental data. This discovery revealed that a quarter of what the tech giant attributed to an "error floor" was actually caused by their decoding method, rather than genuine hardware limitations.

Quantum computers are extraordinarily sensitive to errors, making quantum error correction essential for practical applications.

Minimum-weight perfect matching (MWPM) algorithms are commonly used to tackle the decoding problem in quantum error correction. Although they are computationally efficient, their suboptimal nature introduces extra algorithmic errors that can obscure true hardware performance.

The theoretical alternative—maximum-likelihood decoding—has long been known to be optimal but was considered computationally impossible for practical systems, belonging to the notoriously difficult #P complexity class.

However, researchers have now discovered that repetition codes under circuit-level noise have a special mathematical structure that enables exact solutions.

ÌÇÐÄÊÓƵ spoke with the co-authors of the study, Prof. Feng Pan from the Singapore University of Technology and Design, Prof. Pan Zhang from the Chinese Academy of Sciences, and Dr. Huikai Xu, a Research Scientist at the Beijing Academy of Quantum Information Sciences (BAQIS).

"Our group has studied Ising spin-glass models with different topologies for many years, focusing on two core problems: finding ground states and calculating partition functions," the researchers explained.

"When we entered quantum error correction, we discovered a profound connection: Most likely error decoding mirrors ground-state search in spin glasses, while most likely decoding mirrors partition function calculation."

The research, in ÌÇÐÄÊÓƵical Review Letters, represents the first exact maximum-likelihood decoding algorithm with polynomial computational complexity for repetition codes under circuit-level noise.

The decoding challenge

Quantum error correction relies on interpreting "syndrome" measurements. These are signals that detect where errors have occurred and determine the most likely error pattern that caused them.

Current experiments predominantly use minimum-weight perfect matching algorithms, which find the shortest error path but don't necessarily identify the most likely logical error.

"This suboptimality masked hardware performance and contributed to unaccounted error floors," the researchers explained.

The researchers' background in statistical physics made them realize a profound connection between quantum error correction and spin-glass models.

"While the QEC community knew planar graphs simplified MLE decoding, few recognized MLD could also be efficient for such topologies. This insight directly motivated our work," they explained.

PLANAR works by translating the quantum error correction problem into calculating the partition function of a spin glass, which is a well-studied problem in statistical physics. The important insight is that for repetition codes, this results in what's called a planar graph, where edges don't cross each other.

"This allows us to exploit the exact, efficient solutions for planar spin glasses. For non-planar graphs, such solutions are computationally intractable," explained the researchers.

The algorithm relies on the Kac-Ward formalism, a mathematical method that allows exact computation of partition functions for planar spin glass systems in polynomial time, making it possible to calculate exact maximum-likelihood probabilities.

Why repetition codes matter

The focus on repetition codes isn't arbitrary.

While other quantum error-correcting codes, like surface codes, have only achieved small distances and relatively high error rates in experiments, repetition codes are unique in demonstrating both large distances and extremely low error rates.

"The repetition code scales to large distances while achieving ultra-low error rates (down to 10^-10), unlike surface codes limited to small distances on current hardware," the researchers explain.

These capabilities make repetition codes crucial for benchmarking hardware, identifying cosmic-ray events in quantum computers, and laying the groundwork for scalable fault-tolerant quantum computing.

PLANAR was validated by the researchers through comprehensive testing involving synthetic data, Google's quantum experimental results, and their in-house 72-qubit quantum chip. In every case, the algorithm demonstrated better results than existing methods.

For synthetic noise models, PLANAR uncovered exact error thresholds: 6.7% for depolarizing noise and 2.0% for superconductor SI1000 noise, values that had never been precisely calculated before.

When applied to Google's quantum memory experiments, PLANAR not only achieved 25% lower logical error rates but also improved the suppression factor from approximately 8.11 to 8.28, indicating how quickly errors decrease as the quantum code size increases.

Perhaps most impressively, when the team tested PLANAR on their own 72-qubit chip (deliberately operated without reset gates to create challenging conditions with unknown error models), the algorithm still achieved up to 8.76% improvement in logical error rates compared to standard decoders.

Promising results and future work

The impact of the research goes well beyond the technical breakthrough itself.

The algorithm's efficient polynomial complexity allows it to scale with system size, which is essential as quantum computers become more powerful.

"Future directions include extending to other planar codes and integrating PLANAR into real-time error correction for fault-tolerant quantum computation as it is both accurate (theoretically optimal) and fast (scales polynomially)," the researchers noted.

The researchers have proven PLANAR's effectiveness on surface codes under particular noise conditions and plan to adapt it for non-planar graphs with finite genus, opening the door to broader use in different quantum error-correcting codes.

Demonstrated improvements in diverse hardware and under multiple noise models suggest the algorithm will play a crucial role in enabling practical quantum computing.

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