Related papers: Constructive Separations from Gate Elimination
In a recent breakthrough, Chen, Hirahara and Ren prove that $\mathsf{S_2E}/_1 \not\subset \mathsf{SIZE}[2^n/n]$ by giving a single-valued $\mathsf{FS_2P}$ algorithm for the Range Avoidance Problem ($\mathsf{Avoid}$) that works for…
In quantum control theory, a question of fundamental and practical interest is how an arbitrary unitary transformation can be decomposed into minimum number of elementary rotations for implementation, subject to various physical…
We consider the multiplicative complexity of Boolean functions with multiple bits of output, studying how large a multiplicative complexity is necessary and sufficient to provide a desired nonlinearity. For so-called $\Sigma\Pi\Sigma$…
We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$, SoS requires degree $\Omega(s^{1-\epsilon})$…
Obtaining a non-trivial (super-linear) lower bound for computation of the Fourier transform in the linear circuit model has been a long standing open problem for over 40 years. An early result by Morgenstern from 1973, provides an $\Omega(n…
Gate camouflaging is a technique for obfuscating the function of a circuit against reverse engineering attacks. However, if an adversary has pre-existing knowledge about the set of functions that are viable for an application, random…
Decomposable Negation Normal Forms (DNNFs) are Boolean circuits in negation normal form where the subcircuits leading into each AND gate are defined on disjoint sets of variables. We prove a strongly exponential lower bound on the size of…
Proving that there are problems in $\mathsf{P}^\mathsf{NP}$ that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only…
Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in…
The efficient decomposition of multi-controlled gates is a significant factor in quantum compiling, both in circuit depth and T-gate count. Recent work has demonstrated that qudits have the potential to reduce resource requirements from…
Generalized counting constraint satisfaction problems include Holant problems with planarity restrictions; polynomial-time algorithms for such problems include matchgates and matchcircuits, which are based on Pfaffians. In particular, they…
Optimal implementation of quantum gates is crucial for designing a quantum computer. We consider the matrix representation of an arbitrary multiqubit gate. By ordering the basis vectors using the Gray code, we construct the quantum circuit…
We show that most arithmetic circuit lower bounds and relations between lower bounds naturally fit into the representation-theoretic framework suggested by geometric complexity theory (GCT), including: the partial derivatives technique…
We prove lower bounds on complexity measures, such as the approximate degree of a Boolean function and the approximate rank of a Boolean matrix, using quantum arguments. We prove these lower bounds using a quantum query algorithm for the…
The paper discusses the gate complexity of reversible circuits consisting of NOT, CNOT and 2-CNOT gates. The Shannon gate complexity function $L(n, q)$ for a reversible circuit, implementing a Boolean transformation $f\colon \mathbb Z_2^n…
We revisit known constructions of efficient learning algorithms from various notions of constructive circuit lower bounds such as distinguishers breaking pseudorandom generators or efficient witnessing algorithms which find errors of small…
Ref. 1 asked whether deleting gates from a random quantum circuit architecture can ever make the architecture a better approximate $t$-design. We show that it can. In particular, we construct a family of architectures such that the…
We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against…
Gate camouflaging is a known security enhancement technique that tries to thwart reverse engineering by hiding the functions of gates or the connections between them. A number of works on SAT-based attacks have shown that it is often…
We prove lower bounds on the number of product gates in bilinear and quadratic circuits that compute the product of two $n \cross n$ matrices over finite fields. In particular we obtain the following results: 1. We show that the number of…