A Strong Direct Sum Theorem for Distributional Query Complexity

TL;DR

This work resolves a central question in distributional query complexity by proving a strong direct sum theorem for the k-fold direct product $f^{\otimes k}$ under a product distribution $\mu^k$. The authors show that solving all $k$ instances essentially requires depth $\tilde{\Omega}(\varepsilon^2 k)\cdot$ the single-copy depth at error $\Theta\left(\tfrac{\varepsilon}{k}\right)$, thereby establishing a blowup in both query complexity and error that matches the naive strategy up to polylog factors. A key technical advance is adapting Impagliazzo's Hardcore Theorem to query complexity via a resilience lemma that shows the hardcore of $f^{\otimes k}$ remains dense under arbitrary input partitions, enabling a threshold version of the direct sum theorem. The work also derives a strong XOR lemma as a corollary, via an equivalence between direct sum theorems and XOR lemmas, and discusses tightness and the necessity of linear dependence on the error parameter. Collectively, these results clarify the landscape of distributional direct sum phenomena, bridge gaps with randomized models, and introduce robust hardness tools with potential impact on related areas such as derandomization and property testing.

Abstract

Consider the expected query complexity of computing the $k$-fold direct product $f^{\otimes k}$ of a function $f$ to error $\varepsilon$ with respect to a distribution $μ^k$. One strategy is to sequentially compute each of the $k$ copies to error $\varepsilon/k$ with respect to $μ$ and apply the union bound. We prove a strong direct sum theorem showing that this naive strategy is essentially optimal. In particular, computing a direct product necessitates a blowup in both query complexity and error. Strong direct sum theorems contrast with results that only show a blowup in query complexity or error but not both. There has been a long line of such results for distributional query complexity, dating back to (Impagliazzo, Raz, Wigderson 1994) and (Nisan, Rudich, Saks 1994), but a strong direct sum theorem had been elusive. A key idea in our work is the first use of the Hardcore Theorem (Impagliazzo 1995) in the context of query complexity. We prove a new "resilience lemma" that accompanies it, showing that the hardcore of $f^{\otimes k}$ is likely to remain dense under arbitrary partitions of the input space.

Figures (4)

• Figure 1: Illustration of a hardcore density. The tree $T:(\{\pm 1\}^n)^3\to\{\pm 1\}^3$ seeks to compute a function $f^{\otimes 3}$. The tuple of squares at the top of the figure illustrates the set of all inputs to the function while the strings in the support of the hardcore measure are shaded gray. The tuple at the bottom of the figure illustrates the set of inputs reaching the leaf $\ell$. Each block is the subcube consistent with the path $\pi$ and the shaded region denotes the fragment of $H$ which is contained in the corresponding subcube.
• Figure 2: An illustration of our resilience lemma (\ref{['lem:resilience lemma']}). This lemma shows that all trees resemble the one on the right, with $\mathrm{Dens}_H(\boldsymbol{\ell})$ tightly concentrated around its mean of $\delta k$. This allows us to rule out bad trees such as those on the left where all of the hardness is concentrated on a small fraction of the leaves.
• Figure 3: Illustration of a stacked decision tree for a function $f^{\otimes k}$. The decision tree consists of $k$ depth-$d$ decision trees, $T_1,\ldots, T_k$, stacked on top of each other. For an input $X \in (\{\pm 1\}^n)^k$, the output $T(X)$ is computed sequentially, first by computing $T_1(X)$, then $T_2(X)$, and so on. The final output is $T(X)\coloneqq (T_1(X),\ldots, T_k(X))$.
• Figure 4: Illustration of a fair decision tree. For every block $i\in [k]$ and path $\pi$, the input block $X^{(i)}$ is queried at most $d$ times.

Theorems & Definitions (56)

• Theorem 1: Strong direct sum theorem for distributional query complexity; special case of \ref{['thm:main-formal']}
• Theorem 2: Strong direct sum theorem
• Theorem 3: Strong XOR lemma
• Claim 4.1: Linear dependence on $\gamma$ is necessary
• Definition 5.1: Hardcore measure for query complexity
• Theorem 4: Hardcore Theorem for query complexity
• Remark 5.2
• Definition 5.3: Hardcore density at $\ell$
• Definition 5.4: Hardcore advantage at $\ell$
• Lemma 5.5: Accuracy in terms of hardcore density and advantage at leaves