Structured Unitaries Demonstrate Exponentially Large State Complexity and Near-Orthogonality in Random Circuits

The complexity of quantum states and circuits represents a fundamental challenge with far-reaching implications for fields ranging from particle physics to machine learning, and researchers are continually seeking ways to characterise this complexity. Oxana Shaya from Leibniz Universität Hannover, Zoë Holmes from Ecole Polytechnique Fédérale de Lausanne, Christoph Hirche from Leibniz Universität Hannover, and Armando Angrisani et al. investigate this challenge by examining quantum systems governed not by standard, random transformations, but by more structured rules derived from mathematical groups known as symplectic and special orthogonal groups. Their work reveals that these structured systems surprisingly exhibit a level of complexity comparable to that of completely random systems, generating states that are both incredibly intricate and remarkably distinct from one another. This discovery demonstrates that complexity does not necessarily require complete randomness, and opens new avenues for understanding and potentially harnessing the power of quantum systems with tailored properties.

Understanding random states and circuits is central to advancements in quantum information science, with implications for many-body physics, high-energy physics and quantum learning theory. Traditionally, researchers model the behaviour of typical states and circuits by sampling unitary transformations from the Haar measure on the unitary group. This work departs from this standard approach, instead studying structured unitaries drawn from other compact connected groups, namely the symplectic and special orthogonal groups. By leveraging the concentration of measure phenomenon, the research establishes that random quantum states generated using these structured unitaries typically exhibit a.

Structured Unitaries Generate High Quantum Complexity

Scientists have demonstrated that quantum states generated using specific types of structured unitaries, symplectic, special orthogonal, and special unitary, exhibit remarkably high complexity, comparable to that of states created with Haar-random unitaries. This research moves beyond modelling quantum behaviour with the full unitary group and explores the properties of these classical compact and connected groups, revealing surprising parallels in their ability to generate complex states. The team discovered that most states evolved under unitaries within these groups possess exponentially large strong state complexity, indicating they are computationally challenging to produce and nearly orthogonal to one another. This finding extends to k-designs over these groups, ensembles of unitaries that mimic the statistical properties of Haar-random unitaries up to a certain order, further solidifying the connection between these structured approaches and the standard model.

The results have broad implications for understanding quantum chaos, quantum gravity, and condensed matter physics, as the complexity of random states plays a crucial role in modelling these phenomena. Moreover, this discovery aligns with previous research suggesting that efficiently testing the physicality of quantum states is difficult, reinforcing the idea that high complexity is a natural characteristic of many quantum systems. Furthermore, scientists investigated the average-case complexity of learning the output distributions of quantum circuits built from these structured unitaries. Their analysis demonstrates that learning these distributions is computationally challenging, requiring significant resources even with limited access to the circuit. This suggests that these structured groups offer a viable alternative to the full unitary group without sacrificing the complexity needed for advanced quantum computations and modelling. The team’s findings open new avenues for exploring quantum information processing and understanding the fundamental properties of complex quantum systems.

Structured Unitaries Maintain High Quantum Complexity

This research demonstrates that random states created with symplectic or special orthogonal unitaries exhibit a surprisingly large strong state complexity, comparable to that achieved using Haar-random unitaries, and are, on average, nearly orthogonal to one another. The study also indicates that learning circuits built from gates within these structured groups is computationally challenging. These findings suggest that restricting the types of unitary transformations used does not necessarily lead to a simplification of quantum system complexity; instead, certain structured subgroups can maintain a level of complexity similar to that of the broader unitary group. The authors acknowledge that their analysis relies on specific mathematical assumptions and focuses on average-case behaviour, meaning that individual instances may differ. Future work could explore the implications of these findings for specific quantum algorithms and investigate whether these structured groups offer advantages in terms of implementation or resource requirements.