Uniformly random unitaries, i.e. unitaries drawn from the Haar measure, have many useful properties, but cannot be implemented efficiently. This has motivated a long line of research into random unitaries that "look" sufficiently Haar random while also being efficient to implement. Two different notions of derandomisation have emerged: $t$-designs are random unitaries that information-theoretically reproduce the first $t$ moments of the Haar measure, and pseudorandom unitaries (PRUs) are random unitaries that are computationally indistinguishable from Haar random. In this work, we take a unified approach to constructing $t$-designs and PRUs. For this, we introduce and analyse the "$PFC$ ensemble", the product of a random computational basis permutation $P$, a random binary phase operator $F$, and a random Clifford unitary $C$. We show that this ensemble reproduces exponentially high moments of the Haar measure. We can then derandomise the $PFC$ ensemble to show the following: (1) Linear-depth $t$-designs. We give the first construction of a (diamond-error) approximate $t$-design with circuit depth linear in $t$. This follows from the $PFC$ ensemble by replacing the random phase and permutation operators with their $2t$-wise independent counterparts. (2) Non-adaptive PRUs. We give the first construction of PRUs with non-adaptive security, i.e. we construct unitaries that are indistinguishable from Haar random to polynomial-time distinguishers that query the unitary in parallel on an arbitary state. This follows from the $PFC$ ensemble by replacing the random phase and permutation operators with their pseudorandom counterparts. (3) Adaptive pseudorandom isometries. We show that if one considers isometries (rather than unitaries) from $n$ to $n + \omega(\log n)$ qubits, a small modification of our PRU construction achieves general adaptive security.

翻译：均匀随机酉矩阵（即从Haar测度中抽取的酉矩阵）具有许多有用性质，但无法高效实现。这促使学界长期研究那些既“看似”充分Haar随机又能高效实现的随机酉矩阵。两种不同的去随机化概念应运而生：t-设计是指信息论上再现Haar测度前t阶矩的随机酉矩阵，而伪随机酉矩阵（PRU）是指计算上无法与Haar随机区分的随机酉矩阵。本文采用统一方法构造t-设计与PRU。为此，我们引入并分析了“PFC系综”——随机计算基置换P、随机二进制相位算子F与随机Clifford酉矩阵C的乘积。我们证明该系综能再现Haar测度的指数高阶矩。通过去随机化PFC系综，我们得到以下结果：(1) 线性深度t-设计。我们首次构造了电路深度与t呈线性关系的（钻石误差）近似t-设计，这通过将PFC系综中的随机相位与置换算子替换为2t-独立对应项实现。(2) 非自适应PRU。我们首次构造了具备非自适应安全性的PRU，即能抵御对任意态并行查询酉矩阵的多项式时间区分者，这通过将PFC系综中的随机相位与置换算子替换为伪随机对应项实现。(3) 自适应伪随机等距。我们证明若考虑从n量子比特到n+ω(log n)量子比特的等距（而非酉矩阵），对PRU构造稍作修改即可实现完全自适应安全性。