This paper presents a method for constructing efficient approximate unitary designs using random Pauli rotations. The authors show that a random walk on the unitary or orthogonal group of dimension $2^n$ converges quickly to the Haar measure, with a spectral gap of $\Omega(1/t)$. This implies that the walk provides an $\varepsilon$-approximate unitary $t$-design in depth $\mathcal{O}(\mathfrak{n}t^2 + t \log 1/\varepsilon)$, where $\mathfrak{n} = \mathcal{O}(\log n)$ is the circuit depth to implement a single Pauli rotation. The proof uses quadratic Casimir operators of Lie algebras. The paper introduces the concept of approximate unitary $t$-designs, which are ensembles of unitaries that behave similarly to the Haar random ensemble up to the $t$-th moment. The authors show that the product of $k$ independent random Pauli rotations converges to a unitary $t$-design as $k$ increases. They provide a theorem that bounds the difference between the expected value of a random Pauli rotation and the Haar random ensemble. They also show that the difference between two mixed unitary channels is bounded by the diamond norm. The paper discusses the implications of these results for quantum circuit complexity and seed length. It shows that the results can be used to establish lower bounds for robust quantum circuit complexity. The authors also discuss the application of these results to orthogonal designs and other groups, and note that the results can be used to construct pseudorandom generators for halfspaces. The paper also provides a detailed analysis of the spectral gap bounds for different groups, showing that the best known spectral gap for the symmetric group is matched by the results for the special unitary and orthogonal groups. The authors also discuss the use of representation theory to analyze the spectral properties of unitary designs, and show that the results can be extended to other Lie groups. The paper concludes with a discussion of the implications of the results for quantum computing and quantum information theory, and notes that the results have potential applications in areas such as quantum error correction and quantum cryptography.This paper presents a method for constructing efficient approximate unitary designs using random Pauli rotations. The authors show that a random walk on the unitary or orthogonal group of dimension $2^n$ converges quickly to the Haar measure, with a spectral gap of $\Omega(1/t)$. This implies that the walk provides an $\varepsilon$-approximate unitary $t$-design in depth $\mathcal{O}(\mathfrak{n}t^2 + t \log 1/\varepsilon)$, where $\mathfrak{n} = \mathcal{O}(\log n)$ is the circuit depth to implement a single Pauli rotation. The proof uses quadratic Casimir operators of Lie algebras. The paper introduces the concept of approximate unitary $t$-designs, which are ensembles of unitaries that behave similarly to the Haar random ensemble up to the $t$-th moment. The authors show that the product of $k$ independent random Pauli rotations converges to a unitary $t$-design as $k$ increases. They provide a theorem that bounds the difference between the expected value of a random Pauli rotation and the Haar random ensemble. They also show that the difference between two mixed unitary channels is bounded by the diamond norm. The paper discusses the implications of these results for quantum circuit complexity and seed length. It shows that the results can be used to establish lower bounds for robust quantum circuit complexity. The authors also discuss the application of these results to orthogonal designs and other groups, and note that the results can be used to construct pseudorandom generators for halfspaces. The paper also provides a detailed analysis of the spectral gap bounds for different groups, showing that the best known spectral gap for the symmetric group is matched by the results for the special unitary and orthogonal groups. The authors also discuss the use of representation theory to analyze the spectral properties of unitary designs, and show that the results can be extended to other Lie groups. The paper concludes with a discussion of the implications of the results for quantum computing and quantum information theory, and notes that the results have potential applications in areas such as quantum error correction and quantum cryptography.