arXiv subjects

Nikhil S. Mande

Publications and source records attributed to Nikhil S. Mande.

At least 19 recordsLinked to original sources

arXiv2026

Optimal Quantum-Classical Separations for Exact Learning

We study exact learning with membership queries for concept classes $\mathcal C\subseteq\{0,1\}^N$, focusing on the relationships among their deterministic, randomized, and quantum query complexities, denoted $\mathsf{D}(\mathcal C)$, $\mathsf{R}(\mathcal C)$, and $\mathsf{Q}(\mathcal C)$, respectively. The two canonical quantum speedups in this model are witnessed by Grover search and Bernstein-Vazirani, leading to the longstanding conjecture $$ \mathsf{R}(\mathcal C)=O(\mathsf{Q}(\mathcal C)^2+\mathsf{Q}(\mathcal C)\log N). $$ We first refute this conjecture by constructing concept classes $\mathcal C$ and $\mathcal C'$ satisfying \[ \mathsf{R}(\mathcal C)=Ω\!\left(\frac{\mathsf{Q}(\mathcal C)^3\log N}{\log \mathsf{Q}(\mathcal C)}\right) \qquad\text{and}\qquad \mathsf{D}(\mathcal C')=Ω(\mathsf{Q}(\mathcal C')^3\log N). \] The first bound matches the upper bound of Arunachalam et al.~[Quantum'21] up to constant factors, while the second matches the upper bound of Servedio and Gortler~[SICOMP'04]. In particular, this shows that the saving in the randomized upper bound of Arunachalam et al. fundamentally relies on randomness. Apart from characterizing the optimal relationship between classical and quantum query complexity, our results are the first to show that quantum speedups for learning can go beyond the Grover and Bernstein-Vazirani paradigms.

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arXiv2026

The Role of Symmetry in Quantum Query-to-Communication Simulation

Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean function f : {-1,1}^n to {-1,1} and G in {AND_2, XOR_2}, the bounded-error quantum communication complexity of the composed function f o G equals O(Q(f) log n), where Q(f) denotes the bounded-error quantum query complexity of f. This is achieved by Alice running the optimal quantum query algorithm for f, using a round of O(log n) qubits of communication to implement each query. This is in contrast with the classical setting, where it is easy to show that R^{cc}(f o G) is at most 2R(f), where R^{cc} and R denote bounded-error communication and query complexity, respectively. We show that the O(log n) overhead is required for some functions in the quantum setting, and thus the BCW simulation is tight. We note here that prior to our work, the possibility of Q^{cc}(f o G) = O(Q(f)), for all f and all G in {AND_2, XOR_2}, had not been ruled out. More specifically, we show the following. - We show that the log n overhead is *not* required when f is symmetric, generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing'05). - In order to prove the above, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function. - In view of our first result above, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive, which is a weaker notion of symmetry. We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2. - We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model.

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arXiv2026

Complexity of learning matchings and half graphs via edge queries

The problem of learning or reconstructing an unknown graph from a known family via partial-information queries arises as a mathematical model in various contexts. The most basic type of access to the graph is via \emph{edge queries}, where an algorithm may query the presence/absence of an edge between a pair of vertices of its choosing, at unit cost. While more powerful query models have been extensively studied in the context of graph reconstruction, the basic model of edge queries seems to have not attracted as much attention. In this paper we study the edge query complexity of learning a hidden bipartite graph, or equivalently its bipartite adjacency matrix, in the classical as well as quantum settings. We focus on learning matchings and half graphs, which are graphs whose bipartite adjacency matrices are a row/column permutation of the identity matrix and the lower triangular matrix with all entries on and below the principal diagonal being 1, respectively. - For matchings of size $n$, we show a tight deterministic bound of $n(n-1)/2$ and an asymptotically tight randomized bound of $Θ(n^2)$. A quantum bound of $Θ(n^{1.5})$ was shown in a recent work of van Apeldoorn et al.~[ICALP'21]. - For half graphs whose bipartite adjacency matrix is a column-permutation of the $n \times n$ lower triangular matrix, we give tight $Θ(n \log n)$ bounds in both deterministic and randomized settings, and an $Ω(n)$ quantum lower bound. \item - For general half graphs, we observe that the problem is equivalent to a natural generalization of the famous nuts-and-bolts problem, leading to a tight $Θ(n \log n)$ randomized bound. - We also present a simple quicksort-style method that instantiates to a $O(n \log^2 n)$ randomized algorithm and a tight $O(n \log n)$ quantum algorithm.

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arXiv2026

Quantum Search With Generalized Wildcards

In the search with wildcards problem [Ambainis, Montanaro, Quantum Inf.~Comput.'14], one's goal is to learn an unknown bit-string $x \in \{-1,1\}^n$. An algorithm may, at unit cost, test equality of any subset of the hidden string with a string of its choice. Ambainis and Montanaro showed a quantum algorithm of cost $O(\sqrt{n} \log n)$ and a near-matching lower bound of $Ω(\sqrt{n})$. Belovs [Comput.~Comp.'15] subsequently showed a tight $O(\sqrt{n})$ upper bound. We consider a natural generalization of this problem, parametrized by a subset $\cal{Q} \subseteq 2^{[n]}$, where an algorithm may test whether $x_S = b$ for an arbitrary $S \in \cal{Q}$ and $b \in \{-1,1\}^S$ of its choice, at unit cost. We show near-tight bounds when $\cal{Q}$ is any of the following collections: bounded-size sets, contiguous blocks, prefixes, and only the full set. All of these results are derived using a framework that we develop. Using symmetries of the task at hand we show that the quantum query complexity of learning $x$ is characterized, up to a constant factor, by an optimization program, which is succinctly described as follows: `maximize over all odd functions $f : \{-1,1\}^n \to \mathbb{R}$ the ratio of the maximum value of $f$ to the maximum (over $T \in \cal{Q}$) standard deviation of $f$ on a subcube whose free variables are exactly $T$.' To the best of our knowledge, ours is the first work to use the primal version of the negative-weight adversary bound (which is a maximization program typically used to show lower bounds) to show new quantum query upper bounds without explicitly resorting to SDP duality.

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arXiv2026

Subcube Stifling

We introduce the subcube stifling number, a new combinatorial measure of total Boolean functions. This measure is the largest integer $k$ such that, for every set $S$ of at most $k$ input variables and every assignment $b \in \{0,1\}^S$, there is a fixing of the variables outside $S$ under which the resulting function on the free variables $S$ is the point indicator $\mathbb{I}[x_S=b]$. Equivalently, for every small set of coordinates, the function can isolate any prescribed point of the corresponding Boolean cube by suitably fixing all remaining coordinates. This measure is inspired by the stifling number of Chattopadhyay et al.~(ITCS'23); whereas their measure asks for restrictions realizing every constant function, ours asks for restrictions realizing every point indicator. Our results are as follows. 1) We show that the subcube stifling number gives rise to an approximate-degree composition theorem. In particular, if a Boolean function $f$ has approximate degree $O(\sqrt{μ(f)})$, then for every Boolean function $g$, approximate degree composes tightly. This motivates the study of the subcube stifling number, and in particular the search for functions whose approximate degree is $O(\sqrt{μ(f)})$. 2) We show that a random Boolean function on $n$ input bits has subcube stifling number $Θ(\log(n))$ with high probability. 3) We show that indicators of linear codes over $\mathbb{F}_2$ whose minimum distance and dual distance are both linear have high subcube stifling number. 4) We prove that the functions arising from this linear-code construction do not have approximate degree $O(\sqrt{μ(f)})$; in fact, they have approximate degree $Ω(μ(f))$. The main question left open is whether there exists a Boolean function $f$ with approximate degree $Θ(\sqrt{μ(f)})$. A positive answer would yield new instances of tight approximate-degree composition.

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arXiv2026

Instance complexity of Boolean functions

In the area of query complexity of Boolean functions, the most widely studied cost measure of an algorithm is the worst-case number of queries made by it on an input. Motivated by the most natural cost measure studied in online algorithms, the competitive ratio, we consider a different cost measure for query algorithms for Boolean functions that captures the ratio of the cost of the algorithm and the cost of an optimal algorithm that knows the input in advance. The cost of an algorithm is its largest cost over all inputs. Grossman, Komargodski and Naor [ITCS'20] introduced this measure for Boolean functions, and dubbed it instance complexity. Grossman et al. showed, among other results, that monotone Boolean functions with instance complexity 1 are precisely those that depend on one or two variables. We complement the above-mentioned result of Grossman et al. by completely characterizing the instance complexity of symmetric Boolean functions. As a corollary we conclude that the only symmetric Boolean functions with instance complexity 1 are the Parity function and its complement. We also study the instance complexity of some graph properties like Connectivity and k-clique containment. In all the Boolean functions we study above, and those studied by Grossman et al., the instance complexity turns out to be the ratio of query complexity to minimum certificate complexity. It is a natural question to ask if this is the correct bound for all Boolean functions. We show a negative answer in a very strong sense, by analyzing the instance complexity of the Greater-Than and Odd-Max-Bit functions. We show that the above-mentioned ratio is linear in the input size for both of these functions, while we exhibit algorithms for which the instance complexity is a constant.

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arXiv2026

Tight Bounds for Quantum Phase Estimation and Related Problems

Phase estimation, due to Kitaev [arXiv'95], is one of the most fundamental subroutines in quantum computing. In the basic scenario, one is given black-box access to a unitary $U$, and an eigenstate $\lvert ψ\rangle$ of $U$ with unknown eigenvalue $e^{iθ}$, and the task is to estimate the eigenphase $θ$ within $\pmδ$, with high probability. The cost of an algorithm for us is the number of applications of $U$ and $U^{-1}$. We tightly characterize the cost of several variants of phase estimation where we are no longer given an eigenstate, but are required to estimate the maximum eigenphase of $U$, aided by advice in the form of states (or a unitary preparing those states) which are promised to have at least a certain overlap $γ$ with the top eigenspace. We give algorithms and nearly matching lower bounds for all ranges of parameters. We show that a small number of copies of the advice state (or of an advice-preparing unitary) are not significantly better than having no advice at all. We also show that having lots of advice (applications of the advice-preparing unitary) does not significantly reduce cost, and neither does knowledge of the eigenbasis of $U$. We immediately obtain a lower bound on the complexity of the Unitary recurrence time problem, resolving an open question of She and Yuen~[ITCS'23]. Lastly, we study how efficiently one can reduce the error probability in the basic phase-estimation scenario. We show that a phase-estimation algorithm with precision $δ$ and error probability $ε$ has cost $Ω\left(\frac{1}δ\log\frac{1}ε\right)$, matching an easy upper bound. This contrasts with some other scenarios in quantum computing (e.g., search) where error-probability reduction costs only a factor $O(\sqrt{\log(1/ε)})$. Our lower bound uses a variant of the polynomial method with trigonometric polynomials.

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arXiv2025

Improved Quantum Query Upper Bounds Based on Classical Decision Trees

Given a classical query algorithm as a decision tree, when does there exist a quantum query algorithm with a speed-up over the classical one? We provide a general construction based on the structure of the underlying decision tree, and prove that this can give us an up-to-quadratic quantum speed-up. In particular, we obtain a bounded-error quantum query algorithm of cost $O(\sqrt{s})$ to compute a Boolean function (more generally, a relation) that can be computed by a classical (even randomized) decision tree of size $s$. Lin and Lin [ToC'16] and Beigi and Taghavi [Quantum'20] showed results of a similar flavor, and gave upper bounds in terms of a quantity which we call the "guessing complexity" of a decision tree. We identify that the guessing complexity of a decision tree equals its rank, a notion introduced by Ehrenfeucht and Haussler [Inf. Comp.'89] in the context of learning theory. This answers a question posed by Lin and Lin, who asked whether the guessing complexity of a decision tree is related to any complexity-theoretic measure. We also show a polynomial separation between rank and randomized rank for the complete binary AND-OR tree. Beigi and Taghavi constructed span programs and dual adversary solutions for Boolean functions given classical decision trees computing them and an assignment of non-negative weights to its edges. We explore the effect of changing these weights on the resulting span program complexity and objective value of the dual adversary bound, and capture the best possible weighting scheme by an optimization program. We exhibit a solution to this program and argue its optimality from first principles. We also exhibit decision trees for which our bounds are asymptotically stronger than those of Lin and Lin, and Beigi and Taghavi. This answers a question of Beigi and Taghavi, who asked whether different weighting schemes could yield better upper bounds.

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arXiv2025

Hardness of Finding Kings and Strong Kings

A king in a directed graph is a vertex $v$ such that every other vertex is reachable from $v$ via a path of length at most $2$. It is well known that every tournament (a complete graph where each edge has a direction) has at least one king. Our contributions in this work are: - We show that the query complexity of determining existence of a king in arbitrary $n$-vertex digraphs is $Θ(n^2)$. This is in stark contrast to the case where the input is a tournament, where Shen, Sheng, and Wu [SICOMP'03] showed that a king can be found in $O(n^{3/2})$ queries. - In an attempt to increase the "fairness" in the definition of tournament winners, Ho and Chang [IPL'03] defined a strong king to be a king $k$ such that, for every $v$ that dominates $k$, the number of length-$2$ paths from $k$ to $v$ is strictly larger than the number of length-$2$ paths from $v$ to $k$. We show that the query complexity of finding a strong king in a tournament is $Θ(n^2)$. This answers a question of Biswas, Jayapaul, Raman, and Satti [DAM'22] in the negative. A key component in our proofs is the design of specific tournaments where every vertex is a king, and analyzing certain properties of these tournaments. We feel these constructions and properties are independently interesting and may lead to more interesting results about tournament solutions.

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arXiv2024

Lower bounds for quantum-inspired classical algorithms via communication complexity

Quantum-inspired classical algorithms provide us with a new way to understand the computational power of quantum computers for practically-relevant problems, especially in machine learning. In the past several years, numerous efficient algorithms for various tasks have been found, while an analysis of lower bounds is still missing. Using communication complexity, in this work we propose the first method to study lower bounds for these tasks. We mainly focus on lower bounds for solving linear regressions, supervised clustering, principal component analysis, recommendation systems, and Hamiltonian simulations. For those problems, we prove a quadratic lower bound in terms of the Frobenius norm of the underlying matrix. As quantum algorithms are linear in the Frobenius norm for those problems, our results mean that the quantum-classical separation is at least quadratic. As a generalisation, we extend our method to study lower bounds analysis of quantum query algorithms for matrix-related problems using quantum communication complexity. Some applications are given.

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arXiv2024

Query Complexity with Unknowns

We initiate the study of a new model of query complexity of Boolean functions where, in addition to 0 and 1, the oracle can answer queries with ``unknown''. The query algorithm is expected to output the function value if it can be conclusively determined by the partial information gathered, and it must output ``unknown'' if not. We formalize this model by using Kleene's strong logic of indeterminacy on three variables to capture unknowns. We call this model the `u-query model'. We relate the query complexity of functions in the new u-query model with their analogs in the standard query model. We show an explicit function that is exponentially harder in the u-query model than in the usual query model. We give sufficient conditions for a function to have u-query complexity asymptotically the same as its query complexity. Using u-query analogs of the combinatorial measures of sensitivity, block sensitivity, and certificate complexity, we show that deterministic, randomized, and quantum u-query complexities of all total Boolean functions are polynomially related to each other, just as in the usual query models.

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arXiv2024

Quantum Sabotage Complexity

Given a Boolean function $f:\{0,1\}^n\to\{0,1\}$, the goal in the usual query model is to compute $f$ on an unknown input $x \in \{0,1\}^n$ while minimizing the number of queries to $x$. One can also consider a "distinguishing" problem denoted by $f_{\mathsf{sab}}$: given an input $x \in f^{-1}(0)$ and an input $y \in f^{-1}(1)$, either all differing locations are replaced by a $*$, or all differing locations are replaced by $\dagger$, and an algorithm's goal is to identify which of these is the case while minimizing the number of queries. Ben-David and Kothari [ToC'18] introduced the notion of randomized sabotage complexity of a Boolean function to be the zero-error randomized query complexity of $f_{\mathsf{sab}}$. A natural follow-up question is to understand $\mathsf{Q}(f_{\mathsf{sab}})$, the quantum query complexity of $f_{\mathsf{sab}}$. In this paper, we initiate a systematic study of this. The following are our main results: $\bullet\;\;$ If we have additional query access to $x$ and $y$, then $\mathsf{Q}(f_{\mathsf{sab}})=O(\min\{\mathsf{Q}(f),\sqrt{n}\})$. $\bullet\;\;$ If an algorithm is also required to output a differing index of a 0-input and a 1-input, then $\mathsf{Q}(f_{\mathsf{sab}})=O(\min\{\mathsf{Q}(f)^{1.5},\sqrt{n}\})$. $\bullet\;\;$ $\mathsf{Q}(f_{\mathsf{sab}}) = Ω(\sqrt{\mathsf{fbs}(f)})$, where $\mathsf{fbs}(f)$ denotes the fractional block sensitivity of $f$. By known results, along with the results in the previous bullets, this implies that $\mathsf{Q}(f_{\mathsf{sab}})$ is polynomially related to $\mathsf{Q}(f)$. $\bullet\;\;$ The bound above is easily seen to be tight for standard functions such as And, Or, Majority and Parity. We show that when $f$ is the Indexing function, $\mathsf{Q}(f_{\mathsf{sab}})=Θ(\mathsf{fbs}(f))$, ruling out the possibility that $\mathsf{Q}(f_{\mathsf{sab}})=Θ(\sqrt{\mathsf{fbs}(f)})$ for all $f$.

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arXiv2024

On the communication complexity of finding a king in a tournament

A tournament is a complete directed graph. A king in a tournament is a vertex v such that every other vertex is reachable from v via a path of length at most 2. It is well known that every tournament has at least one king, one of which is a maximum out-degree vertex. The tasks of finding a king, a maximum out-degree vertex and a source in a tournament has been relatively well studied in the context of query complexity. We study the communication complexity of these tasks, where the edges are partitioned between two players. The following are our main results for n-vertex tournaments: 1) The deterministic communication complexity of finding whether a source exists is tilde{Theta}(log^2 n). 2) The deterministic and randomized communication complexities of finding a king are Theta(n). The quantum communication complexity is tilde{Theta}(sqrt{n}). 3) The deterministic, randomized and quantum communication complexities of finding a maximum out-degree vertex are Theta(n log n), tilde{Theta}(n) and tilde{Theta}(sqrt{n}), respectively. Our upper bounds hold for all partitions of edges, and the lower bounds for a specific partition of the edges. To show the first bullet above, we show, perhaps surprisingly, that finding a source in a tournament is equivalent to the well-studied Clique vs. Independent Set (CIS) problem on undirected graphs. Our bounds for finding a source then follow from known bounds on the complexity of the CIS problem. In view of this equivalence, we can view the task of finding a king in a tournament to be a natural generalization of CIS. One of our lower bounds uses a fooling-set based argument, and all our other lower bounds follow from carefully-constructed reductions from Set-Disjointness.

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arXiv2023

Tight Bounds for the Randomized and Quantum Communication Complexities of Equality with Small Error

We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability $ε$, getting optimal constant factors in the leading terms in a number of different models. In the randomized model, 1) we give a general technique to convert public-coin protocols to private-coin protocols by incurring a small multiplicative error, at a small additive cost. This is an improvement over Newman's theorem [Inf. Proc. Let.'91] in the dependence on the error parameter. 2) Using this we obtain a $(\log(n/ε^2)+4)$-cost private-coin communication protocol that computes the $n$-bit Equality function, to error $ε$. This improves upon the $\log(n/ε^3)+O(1)$ upper bound implied by Newman's theorem, and matches the best known lower bound, which follows from Alon [Comb. Prob. Comput.'09], up to an additive $\log\log(1/ε)+O(1)$. In the quantum model, 1) we exhibit a one-way protocol of cost $\log(n/ε)+4$, that uses only pure states and computes the $n$-bit Equality function to error $ε$. This bound was implicitly already shown by Nayak [PhD thesis'99]. 2) We show that any $ε$-error one-way protocol for $n$-bit Equality that uses only pure states communicates at least $\log(n/ε)-\log\log(1/ε)-O(1)$ qubits. 3) We exhibit a one-way protocol of cost $\log(\sqrt{n}/ε)+3$, that uses mixed states and computes the $n$-bit Equality function to error $ε$. This is also tight up to an additive $\log\log(1/ε)+O(1)$, which follows from Alon's result. 4) We study the number of EPR pairs required to be shared in an entanglement-assisted one-way protocol. Our upper bounds also yield upper bounds on the approximate rank and related measures of the Identity matrix.

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arXiv2023

Randomized and quantum query complexities of finding a king in a tournament

A tournament is a complete directed graph. It is well known that every tournament contains at least one vertex v such that every other vertex is reachable from v by a path of length at most 2. All such vertices v are called *kings* of the underlying tournament. Despite active recent research in the area, the best-known upper and lower bounds on the deterministic query complexity (with query access to directions of edges) of finding a king in a tournament on n vertices are from over 20 years ago, and the bounds do not match: the best-known lower bound is Omega(n^{4/3}) and the best-known upper bound is O(n^{3/2}) [Shen, Sheng, Wu, SICOMP'03]. Our contribution is to show essentially *tight* bounds (up to logarithmic factors) of Theta(n) and Theta(sqrt{n}) in the *randomized* and *quantum* query models, respectively. We also study the randomized and quantum query complexities of finding a maximum out-degree vertex in a tournament.

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arXiv2022

Lifting to Parity Decision Trees Via Stifling

We show that the deterministic decision tree complexity of a (partial) function or relation $f$ lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation $f \circ g$ as long as the gadget $g$ satisfies a property that we call stifling. We observe that several simple gadgets of constant size, like Indexing on 3 input bits, Inner Product on 4 input bits, Majority on 3 input bits and random functions, satisfy this property. It can be shown that existing randomized communication lifting theorems ([Göös, Pitassi, Watson. SICOMP'20], [Chattopadhyay et al. SICOMP'21]) imply PDT-size lifting. However there are two shortcomings of this approach: first they lift randomized decision tree complexity of $f$, which could be exponentially smaller than its deterministic counterpart when either $f$ is a partial function or even a total search problem. Second, the size of the gadgets in such lifting theorems are as large as logarithmic in the size of the input to $f$. Reducing the gadget size to a constant is an important open problem at the frontier of current research. Our result shows that even a random constant-size gadget does enable lifting to PDT size. Further, it also yields the first systematic way of turning lower bounds on the width of tree-like resolution proofs of the unsatisfiability of constant-width CNF formulas to lower bounds on the size of tree-like proofs in the resolution with parity system, i.e., $\textit{Res}$($\oplus$), of the unsatisfiability of closely related constant-width CNF formulas.

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arXiv2022

One-way communication complexity and non-adaptive decision trees

We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the Inner Product on a constant number of inputs. Let $IP$ denote Inner Product on $2b$ bits. - If $f$ is a total Boolean function that depends on all of its inputs, the bounded-error one-way quantum communication complexity of $f \circ IP$ equals $Ω(n(b-1))$. - If $f$ is a partial Boolean function, the deterministic one-way communication complexity of $f \circ IP$ is at least $Ω(b \cdot D_{dt}^{\rightarrow}(f))$, where $D_{dt}^{\rightarrow}(f)$ denotes the non-adaptive decision tree complexity of $f$. Montanaro and Osborne [arXiv'09] observed that the deterministic one-way communication complexity of $f \circ XOR_2$ equals the non-adaptive parity decision tree complexity of $f$. In contrast, we show the following with the gadget $AND_2$. - There exists a function for which even the quantum non-adaptive AND decision tree complexity of $f$ is exponentially large in the deterministic one-way communication complexity of $f \circ AND_2$. - For symmetric functions $f$, the non-adaptive AND decision tree complexity of $f$ is at most quadratic in the (even two-way) communication complexity of $f \circ AND_2$. In view of the first point, a lower bound on non-adaptive AND decision tree complexity of $f$ does not lift to a lower bound on one-way communication complexity of $f \circ AND_2$. In our final result we show that for all $f$, the deterministic one-way communication complexity of $F = f \circ AND_2$ is at most $(rank(M_{F}))(1 - Ω(1))$, where $M_{F}$ denotes the communication matrix of $F$. This shows that the rank upper bound on one-way communication complexity (which can be tight in general) is not tight for AND-composed functions.

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arXiv2021

Exact quantum query complexity of computing Hamming weight modulo powers of two and three

We study the problem of computing the Hamming weight of an $n$-bit string modulo $m$, for any positive integer $m \leq n$ whose only prime factors are 2 and 3. We show that the exact quantum query complexity of this problem is $\left\lceil n(1 - 1/m) \right\rceil$. The upper bound is via an iterative query algorithm whose core components are the well-known 1-query quantum algorithm (essentially due to Deutsch) to compute the Hamming weight a 2-bit string mod 2 (i.e., the parity of the input bits), and a new 2-query quantum algorithm to compute the Hamming weight of a 3-bit string mod 3. We show a matching lower bound (in fact for arbitrary moduli $m$) via a variant of the polynomial method [de Wolf, SIAM J. Comput., 32(3), 2003]. This bound is for the weaker task of deciding whether or not a given $n$-bit input has Hamming weight 0 modulo $m$, and it holds even in the stronger non-deterministic quantum query model where an algorithm must have positive acceptance probability iff its input evaluates to 1. For $m>2$ our lower bound exceeds $n/2$, beating the best lower bound provable using the general polynomial method [Theorem 4.3, Beals et al., J. ACM 48(4), 2001].

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