[论文解读] On the Power of Random Bases in Fourier Sampling: Hidden Subgroup Problem in the Heisenberg Group

Jaikumar Radhakrishnan, Martin Roetteler+1 more|arXiv (Cornell University)|Mar 11, 2005

Quantum Computing Algorithms and Architecture参考文献 13被引用 17

一句话总结

本文证明了随机强傅里叶采样——在量子傅里叶变换后于随机正交基上测量——可高效求解有限海森堡群中的隐藏子群问题（HSP）。当 $ r(G) = \Omega(1) $ 时，证明了 $ O((\log|G|/r(G))^{2}) $ 次迭代已足够，且展示了海森堡群满足 $ r({\cal H}_p) = \Omega(1) $，从而得到一种具有经典后处理的多项式时间量子算法，这是首个在缺乏已知显式基的情况下实现此类结果的例子。

ABSTRACT

The hidden subgroup problem (HSP) provides a unified framework to study problems of group-theoretical nature in quantum computing such as order finding and the discrete logarithm problem. While it is known that Fourier sampling provides an efficient solution in the abelian case, not much is known for general non-abelian groups. Recently, some authors raised the question as to whether post-processing the Fourier spectrum by measuring in a random orthonormal basis helps for solving the HSP. Several negative results on the shortcomings of this random strong method are known. In this paper however, we show that the random strong method can be quite powerful under certain conditions on the group G. We define a parameter r(G) for a group G and show that O((\log |G| / r(G))^2) iterations of the random strong method give enough classical information to identify a hidden subgroup in G. We illustrate the power of the random strong method via a concrete example of the HSP over finite Heisenberg groups. We show that r(G) = Ω(1) for these groups; hence the HSP can be solved using polynomially many random strong Fourier samplings followed by a possibly exponential classical post-processing without further queries. The quantum part of our algorithm consists of a polynomial computation followed by measuring in a random orthonormal basis. This gives the first example of a group where random representation bases do help in solving the HSP and for which no explicit representation bases are known that solve the problem with (\log G)^O(1) Fourier samplings. As an interesting by-product of our work, we get an algorithm for solving the state identification problem for a set of nearly orthogonal pure quantum states.

研究动机与目标

• 研究随机傅里叶基上的强采样是否能克服非交换HSP中弱傅里叶采样的局限性。
• 确定随机强傅里叶采样在何种条件下可有效求解HSP。
• 提出一个新的群论参数 $ r(G) $，用于量化随机强采样的有效性。
• 证明尽管缺乏已知显式基，海森堡群仍可通过随机强采样实现高效求解。
• 作为副产品，提供一种用于近似正交量子态识别的量子算法。

提出的方法

• 引入一个新的群参数 $ r(G) $，用于衡量通过随机强傅里叶采样区分子群的能力。
• 基于投影算子的范数与秩条件，定义 $ r(G; H_1, H_2; \rho) $，其中 $ \rho $ 为不可约表示。
• 通过刻画其正规核心族并利用表示理论，分析海森堡群 $ {\cal H}_p $，计算 $ r({\cal H}_p) $。
• 通过有界子群相关投影算子的算子范数与秩，证明 $ r({\cal H}_p) = \Omega(1) $。
• 利用参数 $ r(G) $ 限制所需随机强傅里叶采样迭代次数：$ O((\log|G|/r(G))^{2}) $。
• 在 $ O(\log^3 p) $ 个门内实现 $ {\cal H}_p $ 上的量子傅里叶变换，并使用 $ \tilde{O}(p^2) $ 个门在随机正交基上进行测量。

实验结果

研究问题

• RQ1 在缺乏已知有效显式基的非阿贝尔群中，随机强傅里叶采样能否求解HSP？
• RQ2 何种群论条件可确保随机强采样提供求解HSP所需的足够信息？
• RQ3 参数 $ r(G) $ 与在随机基测量下隐藏子群的可区分性之间有何关系？
• RQ4 海森堡群是否允许使用随机强傅里叶采样实现多项式时间量子算法？
• RQ5 随机强方法能否用于求解近似正交量子态的识别问题？

主要发现

• 海森堡群 $ {\cal H}_p $ 满足 $ r({\cal H}_p) = \Omega(1) $，证明了随机强傅里叶采样在该群上是有效的。
• 通过 $ O(\log p) $ 次随机强傅里叶采样迭代，可求解 $ {\cal H}_p $ 上的HSP。
• 量子电路需要 $ O(\log^4 p) $ 个基本门，其中QFT在 $ O(\log^3 p) $ 个门内实现。
• 经典后处理耗时 $ \tilde{O}(p^4) $，主要由对子群的概率分布计算主导。
• 该算法在随机基选择下成功概率至少为 $ 2/3 $，无需额外查询。
• 副产品是一种利用随机强测量高效识别近似正交量子态的算法。