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Decoupling with Random Quantum Circuits

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Abstract

Decoupling has become a central concept in quantum information theory, with applications including proving coding theorems, randomness extraction and the study of conditions for reaching thermal equilibrium. However, our understanding of the dynamics that lead to decoupling is limited. In fact, the only families of transformations that are known to lead to decoupling are (approximate) unitary two-designs, i.e., measures over the unitary group that behave like the Haar measure as far as the first two moments are concerned. Such families include for example random quantum circuits with O(n 2) gates, where n is the number of qubits in the system under consideration. In fact, all known constructions of decoupling circuits use Ω(n 2) gates. Here, we prove that random quantum circuits with O(n log2 n) gates satisfy an essentially optimal decoupling theorem. In addition, these circuits can be implemented in depth O(log3 n). This proves that decoupling can happen in a time that scales polylogarithmically in the number of particles in the system, provided all the particles are allowed to interact. Our proof does not proceed by showing that such circuits are approximate two-designs in the usual sense, but rather we directly analyze the decoupling property.

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Author information

Authors and Affiliations

  1. Département de Physique, Université de Sherbrooke, Sherbrooke, QC, Canada

    Winton Brown

  2. Institute for Theoretical Physics, ETH Zürich, Zurich, Switzerland

    Omar Fawzi

  3. LIP, UMR 5668 ENS Lyon-CNRS-UCBL-INRIA, Universite de Lyon, Lyon , France

    Omar Fawzi

Authors
  1. Winton Brown
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  2. Omar Fawzi
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Correspondence to Omar Fawzi.

Additional information

Communicated by A. Winter

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Brown, W., Fawzi, O. Decoupling with Random Quantum Circuits. Commun. Math. Phys. 340, 867–900 (2015). https://doi.org/10.1007/s00220-015-2470-1

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  • DOI: https://doi.org/10.1007/s00220-015-2470-1

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