Algebraic Geometry Codes Achieve Block Length with Hermitian Codes, Requiring One-third Fewer Qubits for Quantum Interferometry

The challenge of efficiently processing quantum information drives innovation in error correction and algorithm design, and recent work explores a surprising connection between decoding and optimisation. Andi Gu and Stephen P. Jordan from Google Quantum AI demonstrate this connection using a class of error-correcting codes known as Hermitian codes, which offer potential advantages over traditional Reed-Solomon codes by requiring fewer qubits for implementation. Their research establishes a duality between decoding these codes and solving a specific optimisation problem, termed Hermitian Optimal Polynomial Intersection, revealing that this problem is fundamentally a polynomial regression task on a Hermitian curve. By comparing the performance of this new approach to established classical algorithms, the team finds a significant parameter range where their method achieves a better approximation, suggesting that the benefits of this duality extend beyond existing techniques and offer a promising pathway for advancing quantum computation.

These codes are built by evaluating functions on a curve, utilizing the Riemann-Roch theorem to determine the code’s capabilities, which relates the complexity of the curve to the code’s dimension. Information is encoded by considering points on the curve and their associated functions, creating a structure that allows for the detection and correction of errors. Key characteristics of these codes, including length, dimension, and minimum distance, are determined by the geometry of the curve used in their construction. Hermitian codes represent a special case, built on a specific type of curve known as a Hermitian curve, particularly well-suited for coding due to its inherent properties. A crucial property of Hermitian codes is their duality, meaning the dual code also possesses the same structure, simplifying the decoding process and enabling efficient algorithms for faster, more reliable error correction.

Hermitian Codes Enable Efficient Quantum Decoding

Scientists have developed Decoded Interferometry (DQI), establishing a connection between decoding problems and optimization problems, extending previous work with Reed-Solomon codes to encompass Hermitian codes. This leverages the unique properties of Hermitian codes, achieving longer code lengths using a smaller alphabet size compared to Reed-Solomon codes, requiring fewer qubits per field element for quantum implementations. They pioneered a method to view the HOPI problem as approximate list recovery for Hermitian codes, enabling comparisons with established classical algorithms. Experiments using simulations assessed DQI’s performance against Prange’s algorithm, algebraic list recovery methods, and simulated annealing, revealing a large parameter regime where DQI efficiently achieves better approximations. Researchers demonstrate that DQI efficiently solves optimization problems linked to these codes, achieving improved performance compared to established classical algorithms. Experiments reveal that DQI outperforms algorithms like Prange’s information set decoding, algebraic list decoding methods, and simulated annealing across a significant parameter regime.

Specifically, the team established a quantum advantage by leveraging the semicircle law and comparing its performance against these classical benchmarks, suggesting the speedup isn’t limited to Reed-Solomon codes, but extends to a broader range of polynomial regression problems on algebraic varieties. The team focused on Hermitian codes defined over finite fields, achieving a code length with good distance and rate properties. Crucially, the duality property of Hermitian codes ensures efficient decoding for both the code and its dual. Measurements confirm that DQI can efficiently handle list recovery, even when input list sizes approach the field size, a limitation often encountered with classical algorithms. This advancement opens new avenues for solving complex optimization problems with potential applications in cryptography and other fields reliant on efficient polynomial reconstruction.

Hermitian Codes Unlock Quantum List Recovery

This research demonstrates a powerful connection between decoding problems and optimization problems, formalized through Decoded Interferometry (DQI). Extending previous work on Reed-Solomon codes, scientists have successfully applied DQI to Hermitian codes, a class of algebraic geometry codes known for their efficient use of qubits. These findings suggest that the quantum advantage initially observed with Reed-Solomon codes isn’t limited to a specific code family, but stems from DQI’s ability to exploit the underlying algebraic structure of structured codes. The research quantitatively links performance gains to the dual distance of the Hermitian code, reinforcing the idea that efficient decoding of these structures is key to DQI’s power.