亚线性深度群设计的不可行定理
No-go theorems for sublinear-depth group designs
AI总结：
本文针对具有不变态的群，证明亚线性深度局域电路无法构成近似k-设计，并给出深度下界及与精确实现的恒定间隔。
摘要：
构造能够在各种群上高效逼近 Haar 测度的电路系综是量子信息论中长期存在的基本问题。近期研究表明，可以在量子比特数的对数深度下获得酉群上的近似设计，但正交群上不存在亚线性深度的近似设计。在此，我们针对任何具有不变态 $G^{\otimes k} \lvert\Psi\rangle= \lvert\Psi\rangle$ 的群 $G$，推导出 $G$ 中任何元素系综的第 $k$ 阶矩算子与 $G$ 上 Haar 测度的第 $k$ 阶矩算子之间的金刚石距离的下界。然后，我们利用这一界限证明，对于许多感兴趣的群，由具有局部门的一维亚线性深度电路组成的 $G$ 的任何子集都不能构成 $G$ 上的近似 $k$-设计。更一般地，在 $D$ 维晶格上，我们的结果表明此类群设计所需的深度至少按 $n^{1/D}$ 缩放。此外，对于我们考虑的大多数群，我们发现此类系综可以高概率地通过单次恒定深度测量与 $k$-设计区分开来。在其他例子中，我们展示了 (a) 无法近似随机 matchgate 电路甚至二阶矩的最大深度和门数，与 (b) 精确实现 matchgate Haar 分布所需的深度和门数之间存在恒定间隔。我们还排除了 Clifford 群上亚线性深度 $8$-设计的存在性。最后，我们放宽了使用局部门这一假设，并证明了使用由 Pauli 字符串生成的亚线性数量门的任何电路都不可能获得 $G$ 上的近似设计。
英文摘要：
Constructing ensembles of circuits which efficiently approximate the Haar measure over various groups is a long-standing and fundamental problem in quantum information theory. Recently it was shown that one can obtain approximate designs over the unitary group with depths scaling logarithmically in the number of qubits, but that no sublinear-depth approximate designs exist over the orthogonal group. Here we derive, for any group $G$ possessing an invariant state $G^{\otimes k} \lvertΨ\rangle= \lvertΨ\rangle$, a lower bound on the diamond distance between the $k$\textsuperscript{th} moment operator of any ensemble of elements of $G$, and that of the Haar measure over $G$. We then use this bound to prove that for many groups of interest, no subset of $G$ consisting of sublinear-depth one-dimensional circuits with local gates can form an approximate $k$-design over $G$. More generally, on a $D$-dimensional lattice, our results imply that such group designs require depths scaling at least as $n^{1/D}$. Moreover, for most of the groups we consider we find that such ensembles can, with high probability, be distinguished from $k$-designs by a single shot of a constant-depth measurement. Among other examples, we show that there is a constant separation between (a) the maximum depth and gate count for which no circuit can approximate even the second moment of random matchgate circuits, and (b) the depth and gate count required to implement the matchgate Haar distribution exactly. We furthermore rule out the existence of sublinear-depth $8$-designs over the Clifford group. Finally, we relax the assumption of working with local gates, and prove the impossibility of obtaining approximate designs over $G$ using any circuit comprised of a sublinear number of gates generated by Pauli strings.
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