Adaptively Secure Unitary Designs with Constant Non-Clifford Cost

Randomness is a fundamental resource in quantum information, with crucial applications in cryptography, algorithms, and error correction. A central challenge is to construct unitary k designs that closely approximate Haar-random unitaries while minimizing the costly use of non-Clifford operations. In this Letter, we present a protocol able to generate unitary k designs on n qubits, secure against any adversarial quantum measurement, with a system-size-independent number of non-Clifford gates. Our construction applies a k design only to a subsystem of size Θ(k), independent of n. This "seed" design is then "diluted" across the entire n-qubit system by sandwiching it between two random Clifford operators. The resulting ensemble forms an ϵ-approximate unitary k design on n qubits. We prove that this construction achieves full quantum security against adaptive adversaries using only O[over ˜](k^{2}logϵ^{-1}) non-Clifford gates. If one requires security only against polynomial-time adaptive adversaries, the non-Clifford cost decreases to O[over ˜](k+log^{1+c}ϵ^{-1}). This is optimal, since we show that at least Ω(k) non-Clifford gates are required in this setting. Compared to existing approaches, our method significantly reduces non-Clifford overhead while strengthening security guarantees to adaptive security as well as removing artificial assumptions between n and k. These results make high-order unitary designs practically attainable in near-term fault-tolerant quantum architectures.