Related papers: Finitary Boolean functions
It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction…
We study minimum-error identification of an unknown single-bit Boolean function given black-box (oracle) access with one allowed query. Rather than stopping at an abstract optimal measurement, we give a fully constructive solution: an…
Many quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given by a black box. As in the classical version of decision trees, different kinds of quantum query algorithms are possible: exact,…
The largest Hamming distance between a Boolean function in $n$ variables and the set of all affine Boolean functions in $n$ variables is known as the covering radius $\rho_n$ of the $[2^n,n+1]$ Reed-Muller code. This number determines how…
This paper introduces a novel quantum algorithm that is able to classify a hierarchy of classes of imbalanced Boolean functions. The fundamental characteristic of imbalanced Boolean functions is that the proportion of elements in their…
We derive ultimate precision bounds for estimating parameters encoded in \emph{time-dependent} Hamiltonians in the presence of general Markovian noise, allowing for arbitrary adaptive protocols with fast controls and noiseless ancillas.…
Since the work of Polyanskiy, Poor and Verd\'u on the finite blocklength performance of capacity-achieving codes for discrete memoryless channels, many papers have attempted to find further results for more practically relevant channels.…
The well-known Kruskal-Katona theorem in combinatorics says that (under mild conditions) every monotone Boolean function $f: \{0,1\}^n \to \{0,1\}$ has a nontrivial "density increment." This means that the fraction of inputs of Hamming…
Kohn-Sham spin-density functional theory provides an efficient and accurate model to study electron-electron interaction effects in quantum dots, but its application to large systems is a challenge. An efficient algorithm for the…
We characterize the power of constant-depth Boolean circuits in generating uniform symmetric distributions. Let $f\colon\{0,1\}^m\to\{0,1\}^n$ be a Boolean function where each output bit of $f$ depends only on $O(1)$ input bits. Assume the…
We discuss stability for a class of learning algorithms with respect to noisy labels. The algorithms we consider are for regression, and they involve the minimization of regularized risk functionals, such as L(f) := 1/N sum_i…
The Friedgut-Kalai-Naor (FKN) theorem states that if $f$ is a Boolean function on the Boolean cube which is close to degree 1, then $f$ is close to a dictator, a function depending on a single coordinate. The author has extended the theorem…
Let $\mathcal{F}_{n}^*$ be the set of Boolean functions depending on all $n$ variables. We prove that for any $f\in \mathcal{F}_{n}^*$, $f|_{x_i=0}$ or $f|_{x_i=1}$ depends on the remaining $n-1$ variables, for some variable $x_i$. This…
Quantum annealing is a heuristic quantum algorithm which exploits quantum resources to minimize an objective function embedded as the energy levels of a programmable physical system. To take advantage of a potential quantum advantage, one…
The spectrum of a complex-valued function $f$ on $\mathbb{Z}_{q}^n$ is the set $\{|u|:u\in \mathbb{Z}_q^n~\mathrm{and}~\widehat{f}(u)\neq 0\}$, where $|u|$ is the Hamming weight of $u$ and $\widehat{f}$ is the Fourier transform of $f$. Let…
We study the extremal competitive ratio of Boolean function evaluation. We provide the first non-trivial lower and upper bounds for classes of Boolean functions which are not included in the class of monotone Boolean functions. For the…
We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, {\em if-then-else} and…
Variational hybrid quantum-classical optimization represents one of the most promising avenue to show the advantage of nowadays noisy intermediate-scale quantum computers in solving hard problems, such as finding the minimum-energy state of…
The Bernstein-Vazirani (BV) algorithm offers exceptional accuracy in finding the hidden bit string of a function. We explore how the algorithm performs in real-world situations where noise can potentially interfere with its performance. In…
It is generally considered that the signal output by a quantum circuit is attenuated exponentially fast in the number of gates. This letter explores how algorithms using mid-circuit measurements and classical conditioning as computational…