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 $\leq$ relation. In this setting, there is an unknown monotone function $f: \{0,1,\ldots, n-1\}^k \to \{0,1,\ldots, n-1\}^k$ 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 $Ω\left( \frac{k \cdot \log^2{n}}{\log{k}} \right)$ for all $n,k \geq 2$ . This unifies and improves upon two prior results: a lower bound of $Ω(\log^2{n})$ from [EPRY 2019] and a lower bound of $Ω\left( \frac{k \cdot \log{n}}{\log{k}}\right)$ from [BPR 2024], respectively.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 $\leq$ relation. In this setting, there is an unknown monotone function $f: \{0,1,\ldots, n-1\}^k \to \{0,1,\ldots, n-1\}^k$ 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 $Ω\left( \frac{k \cdot \log^2{n}}{\log{k}} \right)$ for all $n,k \geq 2$. This unifies and improves upon two prior results: a lower bound of $Ω(\log^2{n})$ from [EPRY 2019] and a lower bound of $Ω\left( \frac{k \cdot \log{n}}{\log{k}}\right)$ from [BPR 2024], respectively.

arXiv:2502.16679Leibniz international proceedings in informaticsJan 1, 20250 citations

Quadratically Tight Relations for Randomized Query Complexity

Rahul Jain, Hartmut Klauck, Srijita Kundu +4 more

Theory of Computing SystemsAug 10, 20191 citation

Quantum Query Complexity of Finding a Tarski Fixed Point on a High-Dimensional Grid

Tongyang Li, Weiran Ma, Ziyi Yang +1 more

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 $Ω(k\log n)$ 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 $Ω(\log n)$ , respectively. For $n, k\geq 2$ , our bound improves the best previous classical lower bound when $n < k$ and is within a factor of $\log n / \log k$ compared to the known classical lower bound when $n \geq k$ . 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 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 $Ω(k\log n)$ 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 $Ω(\log n)$, respectively. For $n, k\geq 2$, our bound improves the best previous classical lower bound when $n < k$ and is within a factor of $\log n / \log k$ compared to the known classical lower bound when $n \geq k$. 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.

quant-pharXiv:2609.03802Sep 4, 20260 citations

The Quantum Query Complexity of Finding a Tarski Fixed Point on the 2D Grid

Reed Phillips

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 $\mathcal{L}^2_n$ 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.

arXiv:2604.08223Apr 9, 20260 citations

Bootstrapping Classical and Quantum Query Lower Bounds for Finding Tarski Fixed Points

Tongyang Li, Weiran Ma, Ziyi Yang +1 more

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 $Ω(\log m)$ when the width is multiplied by $m$ and the dimension is doubled. Applying these theorems, we obtain three consequences: (i) an $Ω(k\log^2 n)$ lower bound for $k\geq 4$ , 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 $Q_ε(\text{Tarski}(n,k))=Ω(kn^{δ(α)})$ whenever $k>n^α$ , and in particular, an $Ω(kn^{0.0072})$ lower bound for $k>n^{2/203}$ , the first lower bound with simultaneous linear dependence on the dimension and polynomial dependence on the width; and (iii) a superpolynomial lower bound of $(k/\log k)^{Ω(\log k)}$ for exponentially wide grids, where $n=2^{Θ(k)}$ . 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.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 $Ω(\log m)$ when the width is multiplied by $m$ and the dimension is doubled. Applying these theorems, we obtain three consequences: (i) an $Ω(k\log^2 n)$ lower bound for $k\geq 4$, 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 $Q_ε(\text{Tarski}(n,k))=Ω(kn^{δ(α)})$ whenever $k>n^α$, and in particular, an $Ω(kn^{0.0072})$ lower bound for $k>n^{2/203}$, the first lower bound with simultaneous linear dependence on the dimension and polynomial dependence on the width; and (iii) a superpolynomial lower bound of $(k/\log k)^{Ω(\log k)}$ for exponentially wide grids, where $n=2^{Θ(k)}$. 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.

quant-pharXiv:2610.16209Oct 13, 20260 citations

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

\leq

relation. In this setting, there is an unknown monotone function

f: \{0,1,\ldots, n-1\}^k \to \{0,1,\ldots, 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\left( \frac{k \cdot \log^2{n}}{\log{k}} \right)

for all

n,k \geq 2

. This unifies and improves upon two prior results: a lower bound of

Ω(\log^2{n})

from [EPRY 2019] and a lower bound of

Ω\left( \frac{k \cdot \log{n}}{\log{k}}\right)

from [BPR 2024], respectively.

Ω(k\log n)

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(\log n)

, respectively. For

n, k\geq 2

, our bound improves the best previous classical lower bound when

n < k

and is within a factor of

\log n / \log k

compared to the known classical lower bound when

n \geq 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.

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)

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

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.

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(\log m)

when the width is multiplied by

m

and the dimension is doubled. Applying these theorems, we obtain three consequences: (i) an

Ω(k\log^2 n)

lower bound for

k\geq 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

Q_ε(\text{Tarski}(n,k))=Ω(kn^{δ(α)})

whenever

k>n^α

, and in particular, an

Ω(kn^{0.0072})

lower bound for

k>n^{2/203}

, 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/\log k)^{Ω(\log k)}

for exponentially wide grids, where

n=2^{Θ(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.