Qudit Designs and Where to Find Them

TL;DR

This paper develops new methods for constructing and benchmarking unitary t-designs in qudit systems, overcoming limitations of existing group-based designs and expanding their applicability in quantum information processing.

Contribution

It introduces a technique for weighted state t-designs in arbitrary qudit dimensions, a Clifford character RB for benchmarking qudit Clifford groups, and bounds on circuit complexity for generating approximate designs.

Findings

Weighted state t-designs extend shadow tomography to qudits

Clifford character RB benchmarks non-prime-power qudit dimensions

Bounds on circuit complexity for approximate unitary-designs

Abstract

Unitary t-designs are some of the most versatile tools in quantum information theory. Their applications range from randomized benchmarking and shadow tomography, to more fundamental ones such as emulating quantum chaos and establishing exponential separations between classical and quantum query complexity. While unitary designs originating from a group structure, such as the Clifford group, have proven to be incredibly useful for qubit systems, unfortunately, this is no longer true for qudits. In fact, the classification of finite-group representations rules out the existence of unitary 2-designs for arbitrary qudit dimensions. This severely limits the applicability of standard quantum information primitives when it comes to qudit systems. We overcome these limitations with a three-fold contribution. First, we introduce a general technique to construct families of weighted state t-designs in arbitrary qudit dimensions. These weighted state-designs generalize classical shadow tomography protocol from qubits to qudits. Second, we introduce a Clifford character RB that allows us to benchmark the qudit Clifford group in any dimension, including non-prime-power dimensions. And third, we establish bounds on the quantum circuit complexity of generating approximate unitary-designs from native gates in existing quantum hardware such as high-spin and cavity-QED qudits. Our work further highlights the analogy between spin and optical coherent states by proving that spin-GKP codewords form a state 2-design while spin coherent states do not; in direct analogy with the optical case. This work is structured as a pedagogical and self-contained introduction to unitary designs and their applications to qudit systems.