环境幺正算子无法实现浅群设计
Ambient unitaries don't enable shallow group designs
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中文总结 AI 辅助
该研究证明，即使使用环境幺正算子，局部近邻亚线性深度幺正系综也无法实现匹配门等群上的近似设计，线性深度设计构造在常数因子内最优。
中文摘要 AI 辅助
表征在幺正群的不同子集上构造设计的效率是量子信息理论的重要目标。目前已知近似幺正设计可在深度为系统大小对数级的电路中实现，但近期研究表明，基于匹配门（matchgate）、正交群、辛群的局部近邻亚线性深度一维电路系综无法形成其父群上的近似2-设计；类似地，亚线性深度的Clifford系综无法形成Clifford 4-设计。本文中我们证明，这种显著的指数级差异并非仅源于限制为仅包含子群内幺正算子的系综，而是在上述情形下，即使使用来自子群之外的“环境”幺正算子（可能作用于辅助量子比特），任何局部近邻亚线性深度幺正算子的系综都无法实现近似设计。这意味着，与从全幺正群采样的类似协议相比，涉及从这些子群采样的各类自然层析成像和基准测试方案会面临巨大的电路深度开销。我们进一步得出结论，在上述所有情形中，已知的线性深度设计构造在常数因子范围内是最优的。
英文摘要
Characterising the efficiency with which designs over various subsets of the unitary group may be constructed is an important goal of quantum information theory. While it is now known that approximate unitary designs can be realised in depth logarithmic in the system size, it has recently been shown that ensembles of local nearest-neighbour sublinear-depth one-dimensional circuits over the matchgate, orthogonal, and symplectic groups cannot form approximate 2-designs over their parent groups; similarly, sublinear-depth ensembles of Cliffords cannot form a Clifford 4-design. In this note we show that this remarkable exponential separation is not merely an artefact of restricting to ensembles consisting of unitaries from the subgroups themselves, but rather that no ensemble of local nearest-neighbour sublinear-depth unitaries can realise approximate designs in the aforementioned cases, even when employing "ambient" unitaries from beyond the subgroup itself (possibly acting on ancilla qubits). This implies that various natural tomography and benchmarking schemes which involves sampling from these groups suffer from a dramatic circuit depth overhead compared to similar protocols which involve sampling from the full unitary group. We additionally conclude that, in all of the above cases, the known linear-depth design constructions are up to constant factors optimal.
