<p>Geometrically local quantum codes, comprised of qubits and checks embedded in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({{\mathbb{R}}}^{D}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>D</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> with local check operators, have been a subject of significant interest. A key challenge is identifying the optimal code construction that maximizes both code dimension and distance under the geometric constraints. In this work, we introduce a construction that can transform any good quantum LDPC code into an almost optimal geometrically local quantum code. Our approach hinges on a novel yet simple procedure that extracts a two-dimensional structure from an arbitrary three-term chain complex, building a connection between geometric operations and code constructions. We expect that this procedure will find broader applications in areas such as weight reduction and the geometric realization of chain complexes.</p>

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Almost optimal geometrically local quantum LDPC codes in any dimension

  • Xingjian Li,
  • Ting-Chun Lin,
  • Adam Wills,
  • Min-Hsiu Hsieh

摘要

Geometrically local quantum codes, comprised of qubits and checks embedded in \({{\mathbb{R}}}^{D}\) R D with local check operators, have been a subject of significant interest. A key challenge is identifying the optimal code construction that maximizes both code dimension and distance under the geometric constraints. In this work, we introduce a construction that can transform any good quantum LDPC code into an almost optimal geometrically local quantum code. Our approach hinges on a novel yet simple procedure that extracts a two-dimensional structure from an arbitrary three-term chain complex, building a connection between geometric operations and code constructions. We expect that this procedure will find broader applications in areas such as weight reduction and the geometric realization of chain complexes.