[论文解读] The Flow of Information in Interactive Quantum Protocols: the Cost of Forgetting

Mathieu Laurière, Dave Touchette|arXiv (Cornell University)|Jan 9, 2017

Quantum Computing Algorithms and Architecture参考文献 13被引用 3

一句话总结

本文為具有經典輸入的協作協定提供了量子資訊成本（QIC）的新操作性特徵化，將其定義為經典資訊傳輸成本與擦除量子資訊成本之和。論文證明，擦除資訊對於實現量子加速至關重要——具體而言，任何避免擦除資訊的協定無法在不相交問題（Disjointness）中實現次線性通訊，從而確立資訊擦除是互動協定中實現顯著量子優勢的必要條件。

ABSTRACT

In the context of two-party interactive quantum communication protocols, we study a recently defined notion of quantum information cost (QIC), which possesses most of the important properties of its classical analogue. Although this definition has the advantage to be valid for fully quantum inputs and tasks, its interpretation for classical tasks remained rather obscure. Also, the link between this new notion and other notions of information cost for quantum protocols that had previously appeared in the literature was not clear, if existent at all. We settle both these issues: for quantum communication with classical inputs, we provide an alternate characterization of QIC in terms of information about the input registers, avoiding any reference to the notion of a purification of the classical input state. We provide an exact operational interpretation of this alternative characterization as the sum of the cost of transmitting information about the classical inputs and the cost of forgetting information about these inputs. To obtain this characterization, we prove a general lemma, the Information Flow Lemma, assessing exactly the transfer of information in general interactive quantum processes. Furthermore, we clarify the link between QIC and IC of classical protocols by simulating quantumly classical protocols. Finally, we apply these concepts to argue that any quantum protocol that does not forget information solves Disjointness on n-bits in Omega (n) communication, completely losing the quadratic quantum speedup. This provides a specific sense in which forgetting information is a necessary feature of interactive quantum protocols. We also apply these concepts to prove that QIC at zero-error is exactly n for the Inner Product function, and n (1 - o(1)) for a random Boolean function on n+n bits.

研究动机与目标

• 透過消除對輸入態純化之依賴，解決量子資訊成本（QIC）在經典任務中解釋上的模糊性。
• 以資訊傳輸與擦除成本為基礎，提供QIC的操作性、資訊理論解釋。
• 釐清QIC與先前量子協定中資訊成本概念（包括經典IC及其他量子變體）之間的關係。
• 確立在有界輪次協定中，擦除資訊是實現量子加速的必要條件。
• 推導出基本函數（如內積函數與隨機布林函數）的QIC緊緻下界。

提出的方法

• 提出資訊流引理，作為精確追蹤互動協定中量子資訊傳輸的一般工具。
• 利用輸入寄存器與協定通訊記錄之間的互資訊，定義QIC的新特徵化，避免對經典輸入進行純化。
• 將QIC操作化為經典資訊成本（CIC）與擦除成本（CRIC）之和，並以馮·諾伊曼熵的變化來量化。
• 利用馮·諾伊曼熵的等距不變性，將輸出寄存器的熵與輸入資訊內容關聯起來。
• 應用此框架以量子方式模擬經典協定，同時保持經典資訊成本不變，進而實現經典與量子設定之間的比較。
• 運用Rényi熵與集中不等式，證明隨機布林函數QIC的高機率下界。

实验结果

研究问题

• RQ1 能否在不依賴輸入純化概念的情況下特徵化量子資訊成本（QIC），特別是針對經典輸入？
• RQ2 QIC在資訊傳輸與擦除方面的操作意義為何？
• RQ3 QIC與經典資訊成本（IC）及其他先前的量子資訊成本概念有何關係？
• RQ4 在有界輪次協定中，擦除資訊是否為實現量子加速的必要條件？
• RQ5 在零誤差情況下，內積函數與隨機布林函數的QIC確切數值為何？

主要发现

• 在均勻輸入分佈下，內積函數的QIC精確為n個量子位元（qubits）（零誤差）。
• 在n+n位元的隨機布林函數上，於均勻輸入分佈下，QIC以高機率為n(1−o(1))。
• 任何不擦除資訊的量子協定無法以次線性通訊解決不相交問題（Disjointness），暗示擦除資訊是實現量子加速的必要條件。
• 協定的QIC在操作上等價於經典資訊成本（CIC）與擦除成本（CRIC）之和。
• QIC與經典IC幾乎等價，因為量子模擬能保持經典IC，進而實現直接比較。
• 當誤差ε→0時，隨機函數的經典IC極限趨近於n(1−o(1))，且在相同條件下，QIC亦有類似下界。