Tarski's theorem states that every monotone function from a complete lattice to itself has a fixed point. We specifically consider the two-dimensional lattice $\mathcal{L}^2_n$ on points $\{1, \ldots, n\}^2$ and where $(x_1, y_1) \leq (x_2, y_2)$ if $x_1 \leq x_2$ and $y_1 \leq y_2$. We show that the quantum query complexity of finding a fixed point given query access to a monotone function on $\mathcal{L}^2_n$ is $Ω((\log n)^2)$, matching the classical deterministic upper bound. The proof consists of two main parts: a lower bound on the quantum query complexity of a composition of a class of functions including ordered search, and an extremely close relationship between finding Tarski fixed points and nested ordered search.
Tarski's theorem states that every monotone function from a complete lattice to itself has a fixed point. We specifically consider the two-dimensional lattice Ln2 on points {1,…,n}2 and where (x1,y1)≤(x2,y2) if x1≤x2 and y1≤y2. We show that the quantum query complexity of finding a fixed point given query access to a monotone function on Ln2 is Ω((logn)2), matching the classical deterministic upper bound. The proof consists of two main parts: a lower bound on the quantum query complexity of a composition of a class of functions including ordered search, and an extremely close relationship between finding Tarski fixed points and nested ordered search.
The Knaster-Tarski fixed-point theorem states that every monotone function over a complete lattice has a fixed point. Beyond its fundamental role in order theory, the theorem and its algorithmic variants have found broad applications in areas such as economics, game theory, and programming languages. While the query complexity of finding a Tarski fixed point has been extensively studied in classical models, comparatively little is known in the quantum setting.
We prove an Ω(klogn) quantum query lower bound for finding a fixed point of a monotone function on [n]k, using the nonnegative spectral adversary method. In the two extremal regimes n=2 and k=1, our quantum lower bound matches the previous classical lower bounds Ω(k) and Ω(logn), respectively. For n,k≥2, our bound improves the best previous classical lower bound when n<k and is within a factor of logn/logk compared to the known classical lower bound when n≥k. To construct the adversary matrix, we develop the Tree--Filtration Adversary Method. Besides yielding our lower bound, the method offers a more transparent combinatorial interpretation of the nonnegative spectral adversary method. When the hard instances of a problem admit a tree-like organization and suggest an intuition analogous to classical decision-tree lower bounds, our method provide a promising approach to establishing quantum complexity lower bounds.
The Tarski fixed-point problem asks for a fixed point of a monotone map on a k-dimensional grid lattice of width n and has applications in game theory, economics, and verification. Substantial gaps remain between the known upper and lower bounds on its quantum query complexity.
We prove two hardness bootstrapping theorems for quantum query lower bounds. Starting from a hard instance family whose instances have pair-wise incomparable unique fixed points, and a lower bound established by the nonnegative adversary method, we bootstrap this lower bound by (i) a factor of Ω(t) when the dimension is multiplied by t and the width remains unchanged, and (ii) a factor of Ω(logm) when the width is multiplied by m and the dimension is doubled. Applying these theorems, we obtain three consequences: (i) an Ω(klog2n) lower bound for k≥4, improving previously best-known classical and quantum lower bounds; (ii) for every fixed constant α>0, there exists a constant δ(α)>0 such that Qε(Tarski(n,k))=Ω(knδ(α)) whenever k>nα, and in particular, an Ω(kn0.0072) lower bound for k>n2/203, the first lower bound with simultaneous linear dependence on the dimension and polynomial dependence on the width; and (iii) a superpolynomial lower bound of (k/logk)Ω(logk) for exponentially wide grids, where n=2Θ(k). The last result rules out a route to polynomial-time algorithms for simple stochastic games and ARRIVAL through generic polynomial-query black-box Tarski solvers.
Tarski's theorem states that every monotone function from a complete lattice to itself has a fixed point. We analyze the query complexity of finding such a fixed point on the k-dimensional grid of side length n under the ≤ relation. In this setting, there is an unknown monotone function f:{0,1,…,n−1}k→{0,1,…,n−1}k and an algorithm must query a vertex v to learn f(v). The goal is to find a fixed point of f using as few oracle queries as possible. We show that the randomized query complexity of this problem is Ω(logkk⋅log2n) for all n,k≥2. This unifies and improves upon two prior results: a lower bound of Ω(log2n) from [EPRY 2019] and a lower bound of Ω(logkk⋅logn) from [BPR 2024], respectively.
The Six-Dimensional Tarski Problem is Ω(log3n)-Hard
We study the deterministic query complexity of finding a fixed point of a monotone function f:[n]d→[n]d, where querying x returns f(x). For fixed d≥2, earlier work established a lower bound of Ω((logn)2) and an upper bound of Od((logn)⌈(d+2)/3⌉) for the query complexity. Very recently, several papers obtained improved lower bounds using AI-assisted proofs. In this paper, we further improve these lower bounds to Tarski(n,d)=Ωd((logn)2+⌊log3(d/2)⌋). In particular, we obtain the optimal Θ((logn)3) deterministic query complexity in six dimensions.
Quantum query complexity enjoys a variety of pleasing joint computation properties: for example, a composition theorem asserting Q(f∘g)=Θ(Q(f)Q(g)) for all Boolean functions f and g; a direct sum theorem asserting that computing k copies of a function (or search problem) costs Ω(k) times as much as the cost of computing one copy; and a direct product theorem asserting that for Boolean functions, even succeeding at the direct sum problem with exponentially small probability still requires Ω(k) times the cost of computing one copy to bounded error.
However, all of these results are strictly for worst-case quantum query complexity. For example, if we have a fixed distribution μ over inputs, the direct sum theorem says nothing about the quantum query complexity of computing k copies of f when the input comes from the product distribution μk instead of being worst-case. (Note that while a standard Yao-type minimax theorem guarantees a hard distribution for the direct sum problem, there's no guarantee that this hard distribution is a product distribution.) A similar problem occurs for the direct product theorem and the composition theorem: none of these results respect distributions.
In this work, we give distributional joint computation lower bounds for the composition, direct sum, and direct product problems. Along the way, we introduce some new tools for handling quantum query lower bounds, including (a) a new ``multiplicative'' variant of the γ2 norm (which we use in place of the multiplicative adversary method for proving the direct product theorem), and (b) a new ``Shaltiel-free'' measure of quantum query complexity, which we show characterizes the composition behavior of distributional quantum query complexity and satisfies pleasing properties.
We study the quantum query complexity of two problems. First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most k. We call this the Dyckk,n problem. We prove a lower bound of Ω(ckn), showing that the complexity of this problem increases exponentially in k. Here n is the length of the word. When k is a constant, this is interesting as a representative example of star-free languages for which a surprising O~(n) query quantum algorithm was recently constructed by Aaronson et al. Their proof does not give rise to a general algorithm. When k is not a constant, Dyckk,n is not context-free. We give an algorithm with O(n(logn)0.5k) quantum queries for Dyckk,n for all k. This is better than the trival upper bound n for k=o(loglognlog(n)). Second, we consider connectivity problems on grid graphs in 2 dimensions, if some of the edges of the grid may be missing. By embedding the "balanced parentheses" problem into the grid, we show a lower bound of Ω(n1.5−ε) for the directed 2D grid and Ω(n2−ε) for the undirected 2D grid. The directed problem is interesting as a black-box model for a class of classical dynamic programming strategies including the one that is usually used for the well-known edit distance problem. We also show a generalization of this result to more than 2 dimensions.