Related papers: Direct sum theorems beyond query complexity
We show the following hold, unconditionally unless otherwise stated, relative to a random oracle: - There are NP search problems solvable by quantum polynomial-time machines but not classical probabilistic polynomial-time machines. - There…
We define a new query measure we call quantum distinguishing complexity, denoted QD(f) for a Boolean function f. Unlike a quantum query algorithm, which must output a state close to |0> on a 0-input and a state close to |1> on a 1-input, a…
Traditional statistical analysis requires that the analysis process and data are independent. By contrast, the new field of adaptive data analysis hopes to understand and provide algorithms and accuracy guarantees for research as it is…
Prefix aggregation operation (also called scan), and its particular case, prefix summation, is an important parallel primitive and enjoys a lot of attention in the research literature. It is also used in many algorithms as one of the steps.…
The direct product problem is a fundamental question in complexity theory which seeks to understand how the difficulty of computing a function on each of k independent inputs scales with k. We prove the following direct product theorem…
In this paper, we study the lower complexity bounds for finite-sum optimization problems, where the objective is the average of $n$ individual component functions. We consider Proximal Incremental First-order (PIFO) algorithms which have…
We consider the problem of unconstrained minimization of a smooth function in the derivative-free setting using. In particular, we propose and study a simplified variant of the direct search method (of direction type), which we call…
`What more than its truth do we know if we have a proof of a theorem in a given formal system?' We examine Kreisel's question in the particular context of program termination proofs, with an eye to deriving complexity bounds on program…
We give a comprehensive characterization of the computational power of shallow quantum circuits combined with classical computation. Specifically, for classes of search problems, we show that the following statements hold, relative to a…
Simon's problem is one of the most important problems demonstrating the power of quantum computers, which achieves a large separation between quantum and classical query complexities. However, Simon's discussion on his problem was limited…
We propose a new family of fairness definitions for classification problems that combine some of the best properties of both statistical and individual notions of fairness. We posit not only a distribution over individuals, but also a…
PARITY is the problem of determining the parity of a string $f$ of $n$ bits given access to an oracle that responds to a query $x\in\{0,1,...,n-1\}$ with the $x^{\rm th}$ bit of the string, $f(x)$. Classically, $n$ queries are required to…
There exists a huge number of numerical methods that iteratively construct approximations to the solution $y(x)$ of an ordinary differential equation (ODE) $y'(x)=f(x,y)$ starting from an initial value $y_0=y(x_0)$ and using a finite…
This paper shows that, if we could examine the entire history of a hidden variable, then we could efficiently solve problems that are believed to be intractable even for quantum computers. In particular, under any hidden-variable theory…
We show how to efficiently simulate the sending of a message M to a receiver who has partial information about the message, so that the expected number of bits communicated in the simulation is close to the amount of additional information…
We study the fine-grained complexity of conjunctive queries with grouping and aggregation. For common aggregate functions (e.g., min, max, count, sum), such a query can be phrased as an ordinary conjunctive query over a database annotated…
We achieve essentially the largest possible separation between quantum and classical query complexities. We do so using a property-testing problem called Forrelation, where one needs to decide whether one Boolean function is highly…
Naively storing a counter up to value $n$ would require $\Omega(\log n)$ bits of memory. Nelson and Yu [NY22], following work of [Morris78], showed that if the query answers need only be $(1+\epsilon)$-approximate with probability at least…
An average-case variant of the $k$-SUM conjecture asserts that finding $k$ numbers that sum to 0 in a list of $r$ random numbers, each of the order $r^k$, cannot be done in much less than $r^{\lceil k/2 \rceil}$ time. On the other hand, in…
We consider fast kernel summations in high dimensions: given a large set of points in $d$ dimensions (with $d \gg 3$) and a pair-potential function (the {\em kernel} function), we compute a weighted sum of all pairwise kernel interactions…