浅随机量子电路中的条件依赖与Scrooge集成
Conditional dependence and Scrooge ensembles in shallow random quantum circuits
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中文总结 AI 辅助
本文研究浅随机量子电路的条件依赖结构，推测其测量诱导的后测量态集成可由Scrooge集成近似，进而得出二维n量子比特浅随机量子电路在极小退极化噪声下可被经典高效模拟的结论。
中文摘要 AI 辅助
二维几何局域浅随机量子电路的输出态因光锥结构而不具有长程关联，但如果对部分量子比特进行测量，情况会发生改变：测量过程可诱导长程纠缠，进而在远距离量子比特间产生条件关联。本文研究这类电路中条件依赖的结构及其对量子优势的影响。对于量子比特的三分划ABC，我们考虑在B上特定测量结果的条件下，A上后测量态的集成，且该集成覆盖C上所有可能的测量结果。当电路深度超过某一恒定临界值d*时，我们推测该集成可被Haar集成的某种推广形式（称为Scrooge集成[Jozsa等人，Phys. Rev. A 49, 668 (1994)]）很好地近似；我们也提供了支持该推测的数值和解析证据。该推测描述了一种精确的情形：剩余未测量量子比特上的态仍保持其光锥结构，但也会产生一些源于测量的全局随机特征。一个推论是：二维中的n量子比特浅随机量子电路在存在极小退极化噪声率Ω(log(n)/n)时，可被经典高效模拟。
英文摘要
The output state of a 2D geometrically local shallow random quantum circuit does not have long range correlations due to its lightcone structure. But this changes if one measures a subset of the qubits: long-range entanglement can be induced by the measurement process, leading to conditional correlations between distant qubits. In this paper we investigate the structure of conditional dependence in these circuits and its consequences for quantum advantage. For a tripartition $ABC$ of the qubits, we consider the ensemble of post-measurement states on $A$ that is conditioned on a specific measurement outcome on $B$ and ranges over all possible measurement outcomes on $C$. For circuit depths exceeding a constant critical value $d^*$, we conjecture that this ensemble is well approximated by a certain generalization of the Haar ensemble, called the Scrooge ensemble~[Jozsa \textit{et al.}, \href{https://doi.org/10.1103/PhysRevA.49.668}{Phys. Rev. A \textbf{49}, 668 (1994)}]; we also provide supporting numerical and analytical evidence. Our conjecture describes a precise sense in which the state retains its lightcone structure on the remaining unmeasured qubits, but also develops some globally random features arising from the measurement. A consequence is that $n$-qubit shallow random quantum circuits in two dimensions are classically efficiently simulable in the presence of a tiny depolarizing noise rate $Ω(\log(n)/n)$.
