Jeroen Zuiddam

STOC Conference 2025 Conference Paper

Asymptotic Tensor Rank Is Characterized by Polynomials

• Matthias Christandl
• Koen Hoeberechts
• Harold Nieuwboer
• Péter Vrana
• Jeroen Zuiddam

STOC Conference 2025 Conference Paper

Computing Moment Polytopes of Tensors, with Applications in Algebraic Complexity and Quantum Information

• Maxim van den Berg
• Matthias Christandl
• Vladimir Lysikov
• Harold Nieuwboer
• Michael Walter 0005
• Jeroen Zuiddam

FOCS Conference 2021 Conference Paper

Amortized Circuit Complexity, Formal Complexity Measures, and Catalytic Algorithms

• Robert Robere
• Jeroen Zuiddam

We study the amortized circuit complexity of boolean functions. Given a circuit model $\mathcal{F}$ and a boolean function $f: \{0, 1\}^{n}\rightarrow\{0, 1\}$, the $\mathcal{F}$ -amortized circuit complexity is defined to be the size of the smallest circuit that outputs $m$ copies of $f$ (evaluated on the same input), divided by $m$, as $m\rightarrow\infty$. We prove a general duality theorem that characterizes the amortized circuit complexity in terms of “formal complexity measures”. More precisely, we prove that the amortized circuit complexity in any circuit model composed out of gates from a finite set is equal to the pointwise maximum of the family of “formal complexity measures” associated with $\mathcal{F}$. Our duality theorem captures many of the formal complexity measures that have been previously studied in the literature for proving lower bounds (such as formula complexity measures, submodular complexity measures, and branching program complexity measures), and thus gives a characterization of formal complexity measures in terms of circuit complexity. We also introduce and investigate a related notion of catalytic circuit complexity, which we show is “intermediate” between amortized circuit complexity and standard circuit complexity, and which we also characterize (now, as the best integer solution to a linear program). Finally, using our new duality theorem as a guide, we strengthen the known upper bounds for non-uniform catalytic space, introduced by Buhrman et. al [1] (this is related to, but not the same as, our notion of catalytic circuit size). Potechin [2] proved that for any boolean function $f: \{0, 1\}^{n}\rightarrow\{0, 1\}$, there is a catalytic branching program computing $m=2^{2^{n}-1}$ copies of $f$ with total size $O(mn)$ -that is, linear size per copy — refuting a conjecture of Girard, Koucký and McKenzie [3]. Potechin then asked if the number of copies $m$ can be reduced while retaining the amortized upper bound. We make progress on this question by showing that if $f$ has degree $d$ when represented as polynomial over $\mathbb{F}_{2}$, then there is a catalytic branching program computing $m=2^{\begin{pmatrix}n\\ \leq d\end{pmatrix}}$ copies of $f$ with total size $O(mn)$.

STOC Conference 2018 Conference Paper

Universal points in the asymptotic spectrum of tensors

• Matthias Christandl
• Péter Vrana
• Jeroen Zuiddam

The asymptotic restriction problem for tensors s and t is to find the smallest β ≥ 0 such that the n th tensor power of t can be obtained from the (β n + o ( n ))th tensor power of s by applying linear maps to the tensor legs — this is called restriction — when n goes to infinity. Applications include computing the arithmetic complexity of matrix multiplication in algebraic complexity theory, deciding the feasibility of an asymptotic transformation between pure quantum states via stochastic local operations and classical communication in quantum information theory, bounding the query complexity of certain properties in algebraic property testing, and bounding the size of combinatorial structures like tri-colored sum-free sets in additive combinatorics.