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Quantifying Nonlocality: How Outperforming Local Quantum Codes Is Expensive.

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Implementing quantum low-density parity-check (LDPC) codes requires understanding the trade-offs between code parameters and interaction types. This study quantifies the number of long-range interactions needed for quantum LDPC codes with specific dimensions and distances.

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Area of Science:

  • Quantum Information Science
  • Quantum Error Correction
  • Theoretical Computer Science

Background:

  • Quantum low-density parity-check (LDPC) codes offer a path to efficient, scalable quantum circuit construction.
  • Previous work established limitations on quantum LDPC codes using only local interactions for dimension (k) and distance (d).

Purpose of the Study:

  • To investigate the number of long-range interactions necessary for implementing quantum LDPC codes with specific parameters (k, d).
  • To establish quantitative bounds on code parameters based on limited long-range connectivity.

Main Methods:

  • Analysis of 2D quantum LDPC codes.
  • Derivation of lower bounds on the number and length of interactions required.
  • Application to a stacked architecture model.

Main Results:

  • A quantum LDPC code with distance d∝n^{1/2+ϵ} requires Ω(n^{1/2+ϵ}) long-range interactions of length Ω[over ˜](n^{ϵ}).
  • A code with k∝n and d∝n^{α} necessitates Ω[over ˜](n) interactions of length Ω[over ˜](n^{α/2}).
  • Limited long-range connectivity in a stacked architecture imposes bounds on achievable distance and code dimension.

Conclusions:

  • The number of long-range interactions is a critical factor in the practical implementation of quantum LDPC codes.
  • Understanding these interaction requirements is essential for designing efficient and scalable quantum error correction schemes.
  • The findings provide quantitative insights into the resource costs associated with achieving desired code performance.