Quantum Error-Correcting Codes via Higher Grassmann Embeddings
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Solution Overview
Problem
Existing quantum error correcting codes, particularly those derived from classical Grassmann codes, face limitations in dimension and minimum distance, especially over large finite fields, necessitating improved methods for quantum error correction.
Innovation Solution
The construction of quantum error correcting codes from higher Grassmann codes involves a composition of diagonal and Segre embeddings into higher-dimensional projective spaces, entangling quantum data with redundancy qubits and performing stabilizer checks to create logical qubits, followed by syndrome measurement and decoding.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Quantity of substance
If classical Grassmann codes are used for quantum error correction, then the code construction is well-understood, but the dimension is limited
Solution Approach 1:
The patent applies dimensionality change by using higher-order Grassmann codes embedded in higher-dimensional projective spaces. Specifically, it composes diagonal embeddings followed by Segre embeddings into projective spaces of dimension greater than the classical case, thereby increasing the code dimension while maintaining a systematic construction approach based on algebraic geometry
2Quantity of substance
If higher-dimensional projective spaces are used, then code dimension increases, but the minimum distance may deteriorate
Solution Approach 1:
The patent employs parameter changes by carefully selecting the embedding order and projective space dimension to optimize both code dimension and minimum distance. The composition of diagonal and Segre embeddings with specific parameters allows achieving greater dimensions while maintaining asymptotically similar minimum distances to classical Grassmann codes over large finite fields
Data Source
AI summary
Systems and methods to construct Quantum Error Correcting codes from Higher Grassmann Codes. The present disclosure is directed to algebraic codes obtained from families of imbeddings of the Grassmannian, constructed as the composition of a diagonal imbedding followed by a Segre imbedding into various high dimensional projective spaces. As a result, a large family of new error correcting codes is obtained, and the parameters of such codes are determined.