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This study explores quantum encodings for implementing logical gates via simple physical operations. It identifies specific encodings like GKP and cat qudit codes for transversal gate implementations, advancing quantum error correction.

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Area of Science:

  • Quantum Information Science
  • Quantum Error Correction
  • Quantum Computing

Background:

  • Implementing logical quantum gates using physical operations is crucial for fault-tolerant quantum computing.
  • Understanding which quantum encodings support transversal gate operations is a key challenge.

Purpose of the Study:

  • To develop a general framework for identifying quantum encodings that allow logical gates to be implemented by simple physical operations.
  • To explore transversal implementations of specific quantum error-correcting codes and their associated symmetry groups.

Main Methods:

  • Construction of a general form for encoding maps that facilitate transversal gate operations.
  • Analysis of specific quantum codes, including the ⟦5,1,3⟧ code, Steane code, GKP encoding, and cat qudit encoding.
  • Introduction of a novel two-mode bosonic code based on a constellation of 48 coherent states.

Main Results:

  • Demonstration that the ⟦5,1,3⟧ and Steane codes admit transversal implementations of the binary tetrahedral and binary octahedral groups, respectively.
  • Derivation of GKP and cat qudit encodings by selecting appropriate symmetry groups and simple physical implementations.
  • Characterization of a new two-mode bosonic code where Clifford gates correspond to passive Gaussian unitaries.

Conclusions:

  • The developed framework provides a systematic approach to finding quantum encodings for transversal gate implementations.
  • The findings offer practical insights into constructing efficient quantum error-correcting codes and implementing quantum gates.
  • This work contributes to the development of robust quantum computing architectures by simplifying gate operations.