Non-Haar random circuits form unitary designs as fast as Haar random circuits (2504.07390v1)

Abstract: The unitary design formation in random circuits has attracted considerable attention due to its wide range of practical applications and relevance to fundamental physics. While the formation rates in Haar random circuits have been extensively studied in previous works, it remains an open question how these rates are affected by the choice of local randomizers. In this work, we prove that the circuit depths required for general non-Haar random circuits to form unitary designs are upper bounded by those for the corresponding Haar random circuits, up to a constant factor independent of the system size. This result is derived in a broad range of circuit structures, including one- and higher-dimensional lattices, geometrically non-local configurations, and even extremely shallow circuits with patchwork architectures. We provide specific applications of these results in randomized benchmarking and random circuit sampling, and also discuss their implications for quantum many-body physics. Our work lays the foundation for flexible and robust randomness generation in real-world experiments, and offers new insights into chaotic dynamics in complex quantum systems.

Non-Haar Random Circuits Achieve Unitary Designs Efficiently

The paper "Non-Haar random circuits form unitary designs as fast as Haar random circuits" addresses a fundamental problem in quantum information science: the formation rate of unitary designs in random circuits. The authors explore how various circuit structures, particularly non-Haar random circuits, achieve unitary $t$-designs—a mathematical framework that mimics Haar randomness up to the $t$-th moment—at depths comparable to those in Haar random circuits.

Key Contributions

This paper confirms a longstanding hypothesis: the depth of non-Haar random circuits needed to form unitary designs is efficiently bounded by the depth of Haar random circuits, up to a constant factor independent of system size. The authors' findings span vast classes of circuit topologies, indicating their result's robustness. Specific circuit architectures examined include:

• Single-Layer-Connected Circuits: For circuits where a single layer is sufficient to connect the graph of operations, the depth required to achieve an $\varepsilon$-approximate unitary $t$-design equals the corresponding depth for Haar random circuits multiplied by a constant determined by the minimum spectral gap of the local two-qudit gates.
• Multilayer-Connected Circuits: Fixed-architecture circuits, such as brickwork circuits, where connectivity requires multiple layers, also maintain their unitary design formation rates when local gates are non-Haar random. The prolongation in depth is proportional to the power of the architecture's connection depth.
• Patchwork Circuits: These notable circuits, composed of unitary designs on small patches stitched together, achieve these designs in logarithmic depth. The authors build upon prior methods, namely the detectability lemma, revealing that patchwork architectures similarly produce unitary designs even with non-Haar local gates.

Numerical and Theoretical Implications

The authors emphasize the significance of their results with several implications:

• Global Randomness and Chaos: They establish that phenomena associated with quantum chaos, such as scrambling and complexity growth, persist regardless of local randomizer choices. This universality implies that chaotic properties in quantum many-body systems are robust, offering new perspectives for experimentalists conducting research under non-ideal conditions.
• Anticoncentration in Quantum Advantage: The results hint that anticoncentration—a critical component in arguments for quantum supremacy—can be established within shallow-depth circuits of non-Haar random gates. This directly impacts the ongoing development of quantum algorithms designed to outperform classical counterparts.
• Randomized Benchmarking: The work outlines enhanced flexibility for randomized benchmarking protocols by showing that systemic randomness generation is largely invariant with respect to variations in local gate choices. This provides assurance for practical quantum computing efforts aiming to quantify errors and gate fidelities.

Future Directions

The pathways introduced by this paper open several intriguing research avenues. Notably, it prompts further investigation into whether these principles hold across different quantum systems, including those with symmetry constraints or involving continuous-variable quantum information. Additionally, an understanding of cases where non-Haar random circuits may outpace their Haar counterparts in generating scrambling or complexity could further refine our approach to randomness in quantum systems.

In summary, this paper contributes significant theoretical insights into the behavior of non-Haar random circuits in quantum computing, forging new connections between fundamental physics and practical applications in the rapidly growing field of quantum information science.