Pseudorandom unitaries (PRUs) are ensembles of efficiently implementable unitary operators that cannot be distinguished from Haar random unitaries by any quantum polynomial-time algorithm with query access to the unitary. We present a simple PRU construction that is a concatenation of a random Clifford unitary, a pseudorandom binary phase operator, and a pseudorandom permutation operator. We prove that this PRU construction is secure against non-adaptive distinguishers assuming the existence of quantum-secure one-way functions. This means that no efficient quantum query algorithm that is allowed a single application of $U^{\otimes \mathrm{poly}(n)}$ can distinguish whether an $n$-qubit unitary $U$ was drawn from the Haar measure or our PRU ensemble. We conjecture that our PRU construction remains secure against adaptive distinguishers, i.e. secure against distinguishers that can query the unitary polynomially many times in sequence, not just in parallel.

翻译：伪随机酉算子是一种可通过有效算法实现的酉算子族，任何具有量子多项式时间查询能力的算法都无法将其与哈尔随机酉算子区分。我们提出一种简单的伪随机酉算子构造方案，该方案由随机克利福德酉算子、伪随机二进制相位算子及伪随机置换算子串联而成。我们证明，在量子安全单向函数存在的前提下，该伪随机酉算子构造对非适应性区分器具有安全性。这意味着，任何允许单次使用$U^{\otimes \mathrm{poly}(n)}$的高效量子查询算法都无法区分$n$量子比特酉算子$U$是来自哈尔测度还是我们的伪随机酉算子族。我们推测该伪随机酉算子构造对适应性区分器（即允许顺序进行多项式次查询而非并行查询的区分器）仍保持安全性。