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Non-Haar Random Circuits form Unitary Designs as Fast as Haar Random Circuits.

Toshihiro Yada1, Ryotaro Suzuki2, Yosuke Mitsuhashi3

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This study shows non-Haar random circuits form unitary designs faster than Haar random circuits. This finding impacts quantum many-body physics and real-world quantum experiments.

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Area of Science:

  • Quantum Information Science
  • Quantum Computing
  • Statistical Physics

Background:

  • Unitary design formation in random circuits is crucial for quantum technologies.
  • Previous research focused on Haar random circuits, leaving non-Haar circuits understudied.
  • Understanding the impact of local randomizers on design formation is an open question.

Purpose of the Study:

  • To investigate the formation rates of unitary designs in general non-Haar random circuits.
  • To compare the efficiency of non-Haar random circuits against Haar random circuits for design formation.
  • To establish theoretical bounds for circuit depths required for unitary designs in various circuit architectures.

Main Methods:

  • Theoretical analysis of unitary design formation in non-Haar random circuits.
  • Derivation of upper bounds for circuit depths across different dimensional lattices and circuit structures.
  • Application of findings to randomized benchmarking and random circuit sampling protocols.

Main Results:

  • General non-Haar random circuits form unitary designs with depths upper bounded by Haar random circuits, independent of system size.
  • This holds true for diverse circuit structures, including 1D/higher-D lattices, nonlocal configurations, and shallow patchwork circuits.
  • Provides a constant factor improvement over Haar random circuits for design formation.

Conclusions:

  • The study provides a theoretical foundation for flexible and robust randomness generation in quantum experiments.
  • Offers new insights into chaotic dynamics within complex quantum systems.
  • Demonstrates the potential for more efficient unitary design formation using non-Haar random circuits.