We propose a mechanism for reaching pseudorandom quantum states, computationally indistinguishable from Haar random, with shallow log-n depth quantum circuits, where n is the number of qudits. We argue that \(\log n\) depth 2-qubit-gate-based generic random quantum circuits that are claimed to provide a lower bound on the speed of information scrambling, cannot produce computationally pseudorandom quantum states. This conclusion is connected with the presence of polynomial (in n) tails in the stay probability of short Pauli strings that survive evolution through such shallow circuits. We show, however, that stay-probability-tails can be eliminated and pseudorandom quantum states can be accomplished with shallow \(\log n\) depth circuits built from a special universal family of “inflationary” quantum (IQ) gates. We prove that IQ-gates cannot be implemented with 2-qubit gates, but can be realized either as a subset of 2-qudit-gates in U(d²) with d ≥ 3 and d prime, or as special 3-qubit gates.

我们提出了一种利用浅 log -n深度量子电路达到伪随机量子态的机制，在计算上与哈尔随机态无法区分，其中n是量子数。我们认为，\(\log n\)深度的基于 2 量子位门的通用随机量子电路据称可以提供信息置乱速度的下限，但无法产生计算上的伪随机量子态。这一结论与短泡利弦的停留概率中存在多项式（以n为单位）尾部有关，这些短泡利弦通过如此浅的电路而在进化中幸存下来。然而，我们表明，可以消除停留概率尾部，并且可以通过由特殊的通用“膨胀”量子（IQ）门系列构建的浅\(\log n\)深度电路来实现伪随机量子态。我们证明 IQ 门不能用 2 量子位门来实现，但可以作为U ( d ² )中d  ≥ 3 且d为素数的 2 量子位门的子集来实现，或者作为特殊的 3 量子位门来实现。