傅里叶层级中的预言机分离

中文摘要 AI 辅助

傅里叶层级 $\mathrm{FH}_0\subseteq\mathrm{FH}_1\subseteq\mathrm{FH}_2\subseteq\cdots$ 由 Shi 提出（TCS 2005），通过量子计算使用的哈达玛层数来衡量其复杂度。在两层之间，电路可以置换基态并附加相位，但不能产生叠加；这些层是其唯一的干涉来源。第一层恰好是 $\mathrm{BPP}$，而第二层已经能解决 Simon 问题，并通过相位估计分解整数。Shi 猜想每增加一层都会严格提升计算能力，并作为第一步，要求给出相邻层级之间的预言机分离。据我们所知，该问题在每一层 $k\ge2$ 上都是开放的。我们证明对于每个常数 $k\ge2$，存在一个预言机，使得 $\mathrm{FH}_k\subsetneq\mathrm{FH}_{k+1}$。该分离问题基于 Forrelation（Aaronson 和 Ambainis，STOC 2015）：更高层级用常数次查询即可解决，而在第 $k$ 层，即使电路进行指数多次查询仍然困难。这适用于电路访问预言机的两种常见方式：相位预言机和标准预言机（后者将答案写入寄存器）。两者不可互换：相对于预言机，在相同层数下标准预言机严格更强。我们还相对于预言机将整个层级的并集与 $\mathrm{BQP}$ 分离。下界依赖于层级的一个结构性质：哈达玛层数限制了电路查询预言机的自适应性。使用相位预言机时，具有 $k$ 层的电路恰好可被仅进行 $k-1$ 轮并行查询的算法精确模拟，从而可应用已知的此类算法的下界。标准预言机允许电路根据早期答案进行分支，这种情况需要单独论证。

英文摘要

The Fourier hierarchy $\mathrm{FH}_0\subseteq\mathrm{FH}_1\subseteq\mathrm{FH}_2\subseteq\cdots$, introduced by Shi (TCS 2005), measures a quantum computation by the number of Hadamard layers it uses. Between two layers the circuit may permute basis states and attach phases, but it may not create superposition; the layers are its only source of interference. The first level is exactly $\mathrm{BPP}$, while the second already solves Simon's problem and, through phase estimation, factors integers. Shi conjectured that every additional layer strictly increases computational power, and asked, as a first step, for oracle separations between consecutive levels. To our knowledge, the question was open at every level $k\ge2$. We prove that for every constant $k\ge2$ there is an oracle relative to which $\mathrm{FH}_k\subsetneq\mathrm{FH}_{k+1}$. The separating problem is built from Forrelation (Aaronson and Ambainis, STOC 2015): the level above solves it with a constant number of queries, whereas at level $k$ it stays hard even for circuits making exponentially many queries. This holds for both of the usual ways of giving a circuit access to an oracle, the phase oracle and the standard oracle, which writes its answer into a register. The two are not interchangeable: relative to an oracle, the standard oracle is strictly more powerful at the same number of layers. We also separate the union of all the levels from $\mathrm{BQP}$ relative to an oracle. The lower bounds rest on a structural property of the hierarchy: the number of Hadamard layers limits how adaptively a circuit can query its oracle. With a phase oracle, a circuit with $k$ layers is reproduced exactly by an algorithm making only $k-1$ rounds of parallel queries, which brings known lower bounds for such algorithms to bear. The standard oracle lets a circuit branch on earlier answers, and that case needs a separate argument.

发表机构

• Virginia Tech（弗吉尼亚理工大学）

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