论文标题
具有系统大小的独立数量的非克利福德门的有效统一设计
Efficient unitary designs with a system-size independent number of non-Clifford gates
论文作者
论文摘要
许多量子信息协议需要实施随机单位。因为需要指数级的资源来生产从整个$ n $ qubit Group中汲取的HAAR随机型单位，因此经常求助于$ t $ designs。统一$ t $ - 设计模仿最高$ t $的时刻。众所周知，Clifford操作最多可以实施$ 3 $ - 设计。在这项工作中，我们量化了打破此障碍所需的非克利福德资源。我们发现，将$ O（t^{4} \ log^{2}（t）（t）\ log（1/\ varepsilon））注入$ O（T^{4} \ log^{2}（1/\ Varepsilon））$许多非clifford gates中的多项式clifford电路以获得$ \ varepsilon $ -Approximate $ t $ -Design是足够的。令人惊讶的是，所需的非克利福德门的数量与系统大小无关 - 渐近地，非克利福德门的密度被允许趋于零。我们还在随机的克利福德电路到克利福德集团均匀分布的$ t $ the时刻的融合时间中得出了新的界限。我们的证明利用了Clifford组的最近开发的Schur-Weyl二元性变体，并在平均操作员的限制频谱差距上界定。
Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full $n$-qubit group, one often resorts to $t$-designs. Unitary $t$-designs mimic the Haar-measure up to $t$-th moments. It is known that Clifford operations can implement at most $3$-designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject $O(t^{4}\log^{2}(t)\log(1/\varepsilon))$ many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an $\varepsilon$-approximate $t$-design. Strikingly, the number of non-Clifford gates required is independent of the system size -- asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the $t$-th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators.
