Qudit unitary designs with applications to randomized benchmarking

Unitary designs are some of the most versatile tools in quantum information and computation, with applications ranging from randomized benchmarking and classical shadow tomography to more fundamental tasks such as minimal models of quantum chaos and decoupling techniques. Recent years have seen a flurry of activity in the search for optimal constructions of approximate unitary designs, primarily in the context of building them from local random quantum circuits. However, little is known about exact unitary designs in arbitrary qudit dimensions. In fact, a result of Bannai et. al. tells us that unitary t-groups do not exist for arbitrary dimensions and therefore severely limits the applicability of many quantum information primitives in qudit architectures. The rise of various qudit platforms such as photonics-based, cavity-QED systems, as well as high-spin nuclei makes this a timely problem. To circumvent these issues we introduce families of exact weighted 2- and 3-designs in arbitrary dimensions, and discuss their applications for natively benchmarking qudit platforms.

Along the way, we prove that the set of spin-coherent states, the finite-dimensional analog of optical coherent states, cannot form a state 2-design for d>2, simply due to the representation theory of SU(2). This complements the well-known result that optical coherent states cannot form a 2-design and provides a unifying picture of designs and coherent states in all dimensions. The analogy between spin and optical coherent states has been particularly useful for quantum error correction, such as the development of spin-GKP codes, in analogy to optical GKP codes.

Finally, we prove bounds on the length of circuits needed to generate t-designs for both spin-qudits and cavity-QED systems, using random sequences of SNAPs and displacements, the native gate set in these architectures. We also introduce a notion of fractional t-designs inspired from fractional calculus to quantify approximate designs.