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Weiqiang Yuan

Publications and source records attributed to Weiqiang Yuan.

9 recordsLinked to original sources
arXiv2026

Quantum Advantage in Tolerant Junta Testing

We establish the first super-polynomial quantum advantage for the tolerant junta testing problem in the adaptive setting. Specifically, we show that within a certain parameter regime, tolerant $k$-junta testing with high precision can be solved using $\mathrm{poly}(k)$ quantum queries, whereas any classical algorithm requires at least $k^{Ω(\log k)}$ queries. The problem of tolerant $k$-junta testing is as follows: given parameters $(k, ε_1, ε_2)$, with $0\le ε_1<ε_2 \le 1/2$, and black-box access to a Boolean function $f$ (defined on $n$ variables), distinguish whether $f$ is $ε_1$-close to some $k$-junta or $ε_2$-far from every $k$-junta. We show the quantum advantage for a range of parameters close to $1/2$, for example, $ε_1 = 1/2-1/k$ and $ε_2 = 1/2-1/(2k^2)$. The (non-adaptive) quantum tester we use was given by a recent work of Bao, Liu, Yao, Ye, and Zhang (SOSA 2026). We slightly adapt their analysis to show that it holds in the above parameter regime. On the other hand, our classical lower bound requires substantial new ideas. Inspired by the lower bound techniques of Chen and Patel (FOCS 2023), we introduce a new hard distribution of ``yes'' instances (i.e., instances with distance at most $ε_1$ to $k$-juntas) that is based on planting an ``approximate-junta'' as follows: we randomly pick $k$ out of $n$ coordinates, and for each fixing of the $k$ coordinates, the $2^{n-k}$ values in the restricted subcube are drawn randomly except for the set of points in an error-correcting code on which we place the same random bit. We show that this distribution is much closer to $k$-juntas than the uniform distribution, but on the other hand, they are indistinguishable with respect to any classical algorithm making $k^{o(\log k)}$ queries.

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arXiv2026

Pseudodeterministic Communication Complexity

We exhibit an $n$-bit partial function with randomized communication complexity $O(\log n)$ but such that any completion of this function into a total one requires randomized communication complexity $n^{Ω(1)}$. In particular, this shows an exponential separation between randomized and \emph{pseudodeterministic} communication protocols. Previously, Gavinsky (2025) showed an analogous separation in the weaker model of parity decision trees. We use lifting techniques to extend his proof idea to communication complexity.

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arXiv2025

Total Search Problems in $\mathsf{ZPP}$

We initiate a systematic study of ${\sf TFZPP}$, the class of total ${\sf NP}$ search problems solvable by polynomial time randomized algorithms. ${\sf TFZPP}$ contains a variety of important search problems such as $\text{Bertrand-Chebyshev}$ (finding a prime between $N$ and $2N$), refuter problems for many circuit lower bounds, and $\text{Lossy-Code}$. The $\text{Lossy-Code}$ problem has found prominence due to its fundamental connections to derandomization, catalytic computing, and the metamathematics of complexity theory, among other areas. While ${\sf TFZPP}$ collapses to ${\sf FP}$ under standard derandomization assumptions in the white-box setting, we are able to separate ${\sf TFZPP}$ from the major ${\sf TFNP}$ subclasses in the black-box setting. In fact, we are able to separate it from every uniform ${\sf TFNP}$ class assuming that ${\sf NP}$ is not in quasi-polynomial time. To do so, we extend the connection between proof complexity and black-box ${\sf TFNP}$ to randomized proof systems and randomized reductions. Next, we turn to developing a taxonomy of ${\sf TFZPP}$ problems. We highlight a problem called $\text{Nephew}$, originating from an infinity axiom in set theory. We show that $\text{Nephew}$ is in $\mathsf{PWPP}\cap \mathsf{TFZPP}$ and conjecture that it is not reducible to $\text{Lossy-Code}$. Intriguingly, except for some artificial examples, most other black-box ${\sf TFZPP}$ problems that we are aware of reduce to $\text{Lossy-Code}$.

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arXiv2025

Cell-Probe Lower Bounds via Semi-Random CSP Refutation: Simplified and the Odd-Locality Case

A recent work (Korten, Pitassi, and Impagliazzo, FOCS 2025) established an insightful connection between static data structure lower bounds, range avoidance of $\text{NC}^0$ circuits, and the refutation of pseudorandom CSP instances, leading to improvements to some longstanding lower bounds in the cell-probe/bit-probe models. Here, we improve these lower bounds in certain cases via a more streamlined reduction to XOR refutation, coupled with handling the odd-arity case. Our result can be viewed as a complete derandomization of the state-of-the-art semi-random $k$-XOR refutation analysis (Guruswami, Kothari and Manohar, STOC 2022, Hsieh, Kothari and Mohanty, SODA 2023), which complements the derandomization of the even-arity case obtained by Korten et al. As our main technical statement, we show that for any multi-output constant-depth circuit that substantially stretches its input, its output is very likely far from strings sampled from distributions with sufficient independence, and further this can be efficiently certified. Via suitable shifts in perspectives, this gives applications to cell-probe lower bounds and range avoidance algorithms for $\mathsf{NC}^0$ circuits.

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arXiv2025

Direct Sums for Parity Decision Trees

Direct sum theorems state that the cost of solving $k$ instances of a problem is at least $Ω(k)$ times the cost of solving a single instance. We prove the first such results in the randomised parity decision tree model. We show that a direct sum theorem holds whenever (1) the lower bound for parity decision trees is proved using the discrepancy method; or (2) the lower bound is proved relative to a product distribution.

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arXiv2022

The Exact Bipartite Matching Polytope Has Exponential Extension Complexity

Given a graph with edges colored red or blue and an integer $k$, the exact perfect matching problem asks if there exists a perfect matching with exactly $k$ red edges. There exists a randomized polylogarithmic-time parallel algorithm to solve this problem, dating back to the eighties, but no deterministic polynomial-time algorithm is known, even for bipartite graphs. In this paper we show that there is no sub-exponential sized linear program that can describe the convex hull of exact matchings in bipartite graphs. In fact, we prove something stronger, that there is no sub-exponential sized linear program to describe the convex hull of perfect matchings with an odd number of red edges.

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arXiv2022

Lower Bounds for Unambiguous Automata via Communication Complexity

We use results from communication complexity, both new and old ones, to prove lower bounds for unambiguous finite automata (UFAs). We show three results. $\textit{Complement:}$ There is a language $L$ recognised by an $n$-state UFA such that the complement language $\overline{L}$ requires NFAs with $n^{\tildeΩ(\log n)}$ states. This improves on a lower bound by Raskin. $\textit{Union:}$ There are languages $L_1$, $L_2$ recognised by $n$-state UFAs such that the union $L_1\cup L_2$ requires UFAs with $n^{\tildeΩ(\log n)}$ states. $\textit{Separation:}$ There is a language $L$ such that both $L$ and $\overline{L}$ are recognised by $n$-state NFAs but such that $L$ requires UFAs with $n^{Ω(\log n)}$ states. This refutes a conjecture by Colcombet.

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arXiv2020

Log-rank and lifting for AND-functions

Let $f: \{0,1\}^n \to \{0, 1\}$ be a boolean function, and let $f_\land (x, y) = f(x \land y)$ denote the AND-function of $f$, where $x \land y$ denotes bit-wise AND. We study the deterministic communication complexity of $f_\land$ and show that, up to a $\log n$ factor, it is bounded by a polynomial in the logarithm of the real rank of the communication matrix of $f_\land$. This comes within a $\log n$ factor of establishing the log-rank conjecturefor AND-functions with no assumptions on $f$. Our result stands in contrast with previous results on special cases of the log-rank conjecture, which needed significant restrictions on $f$ such as monotonicity or low $\mathbb{F}_2$-degree. Our techniques can also be used to prove (within a $\log n$ factor) a lifting theorem for AND-functions, stating that the deterministic communication complexity of $f_\land$ is polynomially-related to the AND-decision tree complexity of $f$. The results rely on a new structural result regarding boolean functions $f:\{0, 1\}^n \to \{0, 1\}$ with a sparse polynomial representation, which may be of independent interest. We show that if the polynomial computing $f$ has few monomials then the set system of the monomials has a small hitting set, of size poly-logarithmic in its sparsity. We also establish extensions of this result to multi-linear polynomials $f:\{0,1\}^n \to \mathbb{R}$ with a larger range.

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