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More global randomness from less random local gates (2410.24127v2)

Published 31 Oct 2024 in quant-ph, cond-mat.str-el, cs.IT, and math.IT

Abstract: Random circuits giving rise to unitary designs are key tools in quantum information science and many-body physics. In this work, we investigate a class of random quantum circuits with a specific gate structure. Within this framework, we prove that one-dimensional structured random circuits with non-Haar random local gates can exhibit substantially more global randomness compared to Haar random circuits with the same underlying circuit architecture. In particular, we derive all the exact eigenvalues and eigenvectors of the second-moment operators for these structured random circuits under a solvable condition, by establishing a link to the Kitaev chain, and show that their spectral gaps can exceed those of Haar random circuits. Our findings have applications in improving circuit depth bounds for randomized benchmarking and the generation of approximate unitary 2-designs from shallow random circuits.

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Summary

  • The paper shows that structured quantum circuits with non-Haar local gates produce enhanced global randomness, speeding up convergence to approximate unitary 2-designs.
  • The paper employs spectral analysis with exact diagonalization to demonstrate a larger spectral gap in both local and brick-wall circuit architectures compared to Haar random circuits.
  • The paper highlights practical benefits such as improved randomized benchmarking and reduced circuit depth for efficient quantum state characterization and error correction.

Insightful Overview of "More Global Randomness from Less Random Local Gates"

The paper entitled "More Global Randomness from Less Random Local Gates" presents an intriguing investigation into the randomness properties of structured random quantum circuits, challenging the conventional paradigm that gates constituting local random circuits must be Haar random to achieve optimal global randomness. In this context, structured circuits refer to those using non-Haar random local gates, which are interspersed with single-qudit Haar gates before and after each two-qudit operation.

Main Contributions

The authors introduce a framework wherein they analyze a specific class of structured random quantum circuits. These circuits are comprised of two-qudit gates sandwiched by single-qudit Haar random gates. They analyze two main architectures: local random circuits—where pairs of qudits are randomly selected to apply the gate—and brick-wall circuits, which entail successive layers of two-qudit gates across the chain.

  1. Enhanced Randomness with Structured Gates: The authors prove that structured random circuits can exhibit more global randomness than those composed of conventional Haar random gates. This implies that certain structured gates, specifically when the entangling power (eue_u) exceeds that of Haar random gates (eHe_H), contribute to a smaller second-largest eigenvalue of the second-moment operator. This convergence indicates a faster approach toward approximate unitary 2-designs, which are crucial for various quantum computational tasks.
  2. Spectral Analysis: The paper provides exact diagonalization results for the second-moment operators within structured random circuits. Under specific conditions, which align with free-fermionic models, these structured gates lead to a larger spectral gap than Haar random circuits, thereby enhancing randomness. This observation holds for both local and brick-wall circuit architectures.
  3. Practical Implications: Such random circuits have profound implications for improving the efficiency of randomized benchmarking and generating unitary 2-designs. They enable substantial reductions in circuit depth—a critical consideration for practical quantum computing implementations. These properties make structured gates viable candidates for realizing tasks that benefit from enhanced randomness properties, such as quantum state characterization and error correction.

Future Directions and Theoretical Implications

Among the theoretical implications, one notable insight is the potential to expand beyond the traditional reliance on Haar randomness for engineering quantum randomness. By leveraging structured randomness with carefully chosen two-qudit gates, the research opens avenues for quantum circuits that are both practically efficient and theoretically interesting. This challenges researchers to explore the design of new circuit architectures and gate compilations that exploit specific non-Haar structures to optimize performance for quantum information tasks.

Given these findings, future research could explore varying dimensionalities and optimize circuit architectures in higher dimensions, potentially enhancing practical implementations of quantum algorithms. Investigations might also extend into understanding the role of higher-moment operators and how their properties could further inform randomness paradigms in quantum circuit design.

In conclusion, this paper contributes significantly to the field of quantum information science by demonstrating that structured randomness offers an innovative avenue for developing efficient quantum processes. By understanding and harnessing the unique properties of these structured random circuits, researchers can design more adaptable and functional quantum computing systems.

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