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Matthew B Hastings1,2, Jeongwan Haah2

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|February 27, 2018
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Summary
This summary is machine-generated.

The conjecture that many noisy magic states are needed for quantum error correction is false. New quantum Reed-Muller codes achieve a significantly lower ratio of input to output magic states for the T gate.

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Area of Science:

  • Quantum Information Science
  • Quantum Error Correction
  • Quantum Computing Algorithms

Background:

  • Distillation of magic states is crucial for fault-tolerant quantum computation, particularly for implementing the T gate.
  • A key conjecture proposed a lower bound of Ω[log(1/ε)] noisy input magic states per output magic state for error rate ε.
  • Efficient magic state distillation is essential for reducing the overhead of quantum error correction.

Purpose of the Study:

  • To investigate the validity of the conjecture regarding the number of magic states required for T gate distillation.
  • To develop new quantum error-correcting codes that can improve the efficiency of magic state distillation.
  • To establish a new upper bound for the ratio of input to output magic states in distillation protocols.

Main Methods:

  • Construction of a novel family of quantum error-correcting codes by puncturing quantum Reed-Muller codes.
  • Analysis of the parameters of these codes, specifically ⟦∑[under i=w+1][over m](m/i),∑[under i=0][over w](m/i),∑[under i=w+1][over r+1](r+1/i)⟧ for integers m>2r, r>w≥0.
  • Demonstration that codes with m>νr admit a transversal logical gate at the νth level of the Clifford hierarchy.

Main Results:

  • The study disproves the conjecture, showing that the number of noisy input magic states required per output magic state is not Ω[log(1/ε)].
  • A new upper bound of O(log^{γ}(1/ε)) is established for the ratio of input to output magic states for T gate distillation, with γ=log(n/k)/log(d)<0.678.
  • The smallest code in the family achieving γ<1 requires approximately 2^58 qubits, demonstrating a significant improvement in distillation efficiency.

Conclusions:

  • The conjecture on the lower bound for magic state distillation is false, opening new avenues for efficient quantum computation.
  • The developed quantum Reed-Muller codes provide a practical method for reducing the resource overhead in quantum algorithms requiring the T gate.
  • This work advances the field of quantum error correction by offering more efficient protocols for magic state distillation.