Multiqubit Clifford groups are unitary 3-designs (1510.02619v2)

Abstract: Unitary $t$-designs are a ubiquitous tool in many research areas, including randomized benchmarking, quantum process tomography, and scrambling. Despite the intensive efforts of many researchers, little is known about unitary $t$-designs with $t\geq3$ in the literature. We show that the multiqubit Clifford group in any even prime-power dimension is not only a unitary 2-design, but also a 3-design. Moreover, it is a minimal 3-design except for dimension~4. As an immediate consequence, any orbit of pure states of the multiqubit Clifford group forms a complex projective 3-design; in particular, the set of stabilizer states forms a 3-design. In addition, our study is helpful to studying higher moments of the Clifford group, which are useful in many research areas ranging from quantum information science to signal processing. Furthermore, we reveal a surprising connection between unitary 3-designs and the physics of discrete phase spaces and thereby offer a simple explanation of why no discrete Wigner function is covariant with respect to the multiqubit Clifford group, which is of intrinsic interest to studying quantum computation.

Analysis of "Multiqubit Clifford groups are unitary 3-designs"

The paper, "Multiqubit Clifford groups are unitary 3-designs," by Huangjun Zhu, explores the properties and implications of unitary $t$-designs within the context of multiqubit Clifford groups. This insightful investigation has implications for quantum information science, particularly in areas such as randomized benchmarking and quantum process tomography, as well as broad implications in fields like chaos and scrambling.

The primary result of this paper is the demonstration that multiqubit Clifford groups form unitary 3-designs. The importance of unitary $t$-designs lies in their ability to simulate Haar random unitaries, which are often crucial in quantum mechanics applications, in a more structured and predictable manner. While unitary 2-designs have been extensively studied, little was known about higher-order designs with $t \geq 3$. Zhu's contribution extends this knowledge, confirming that the multiqubit Clifford group not only serves as a unitary 2-design but also as a 3-design, making it a powerful tool in diverse quantum applications.

One of the key assertions in the paper is that except for dimension 4, multiqubit Clifford groups are minimal 3-designs. This means that no proper subgroup of these Clifford groups can replicate the 3-design property, underscoring their minimal yet comprehensive structure for this application. There are robust implications for this finding, particularly in simplifying computations in quantum mechanics involving quantum states and operations.

Moreover, the paper makes a compelling case on how these findings influence the understanding of discrete phase space physics. Zhu exposes a link between unitary 3-designs and the infeasibility of developing a Clifford-covariant discrete Wigner function in multiqubit systems, contrasting with systems having an odd local dimension where such a covariant function exists. This result sheds light on foundational differences between systems based on qubits and those based on odd local dimensions, providing insights that are critical for advancing quantum computation.

On the theoretical front, this paper sets the stage for further research into higher-order $t$-designs. Given the utility of these designs in realizing sophisticated quantum tasks like phase retrieval and quantum state discrimination, Zhu's work provides a framework for investigating unitary designs of even higher orders. The work also has ramifications for understanding multipartite entanglement in stabilizer tensor networks, which are relevant in exploring holographic duality—a significant aspect in theoretical physics.

Overall, the research presented in this paper represents a significant exploration of unitary 3-designs within quantum information science, providing a robust foundation for both current applications and future theoretical advancements. The work not only enhances our understanding of Clifford groups but also opens new avenues for exploring the complex interplay between quantum computation and discrete phase space physics. The conclusions drawn have a wide range of relevance, offering a potential pathway to more efficient and deeper insights into the nature of quantum systems.