arXiv subjects

Aleksandrs Belovs

Publications and source records attributed to Aleksandrs Belovs.

At least 19 recordsLinked to original sources

arXiv2026

Tight Quantum Lower Bound for k-Distinctness

In this paper, we introduce a new quantum query lower bound framework. It is inspired by Zhandry's compressed oracle technique, but it also subsumes the polynomial method as a special case. Compared to Zhandry's technique, our approach has two key differences. First, we do not use any oracles (except for the standard input oracle), and define ``knowledge'' directly through the expansion of the state of the algorithm in the Fourier basis. Second, we allow arbitrary probability distributions of inputs. We show how this framework behaves on the problem of finding equal elements in the input string. In particular, we demonstrate its power by proving a first tight quantum query lower bound for the k-Distinctness problem.

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arXiv2026

Space-Efficient Quantum Error Reduction without log Factors

Given an algorithm that outputs the correct answer with bounded error, say $1/3$, it is sometimes desirable to reduce this error to some arbitrarily small $\varepsilon$ -- e.g., if one wants to call the algorithm many times as a subroutine. The usual method, for both quantum and randomized algorithms, is majority voting, which incurs a multiplicative overhead of $O(\log\frac{1}{\varepsilon})$ from calling the algorithm this many times. Transducers are a recently introduced model of quantum computation, and it is possible to reduce the ``error'' of a transducer arbitrarily with only constant overhead, using a construction analogous to majority voting called purification. Even error-free transducers map to bounded-error quantum algorithms, so this does not let you reduce algorithmic error for free, but it does allow bounded-error quantum algorithms to be composed without incurring log factors. In this paper, we present a new highly simplified purifier, that can be understood as a weighted walk on a line similar to a random walk interpretation of majority voting. Our purifier has much smaller space and time complexity than the previous one. Indeed, it only uses one additional counter, and only performs two increment and two decrement operations on each iteration. It also has quadratically better dependence on the soundness-completeness gap of the algorithm being purified. We prove that our purifier has optimal query complexity, even down to the constant, which matters when one composes quantum algorithms to super-constant depth. Purifiers can be seen as a way of turning a ``Monte Carlo'' quantum algorithm into a ``Las Vegas'' quantum algorithm -- a process for which there is no classical analogue. Our simplified construction sheds light on this strange quantum phenomenon, and could have implications for the complexity of composed quantum algorithms.

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arXiv2024

Taming Quantum Time Complexity

Quantum query complexity has several nice properties with respect to composition. First, bounded-error quantum query algorithms can be composed without incurring log factors through error reduction (exactness). Second, through careful accounting (thriftiness), the total query complexity is smaller if subroutines are mostly run on cheaper inputs -- a property that is much less obvious in quantum algorithms than in their classical counterparts. While these properties were previously seen through the model of span programs (alternatively, the dual adversary bound), a recent work by two of the authors (Belovs, Yolcu 2023) showed how to achieve these benefits without converting to span programs, by defining quantum Las Vegas query complexity. Independently, recent works, including by one of the authors (Jeffery 2022), have worked towards bringing thriftiness to the more practically significant setting of quantum time complexity. In this work, we show how to achieve both exactness and thriftiness in the setting of time complexity. We generalize the quantum subroutine composition results of Jeffery 2022 so that, in particular, no error reduction is needed. We give a time complexity version of the well-known result in quantum query complexity, $Q(f\circ g)=O(Q(f)\cdot Q(g))$, without log factors. We achieve this by employing a novel approach to the design of quantum algorithms based on what we call transducers, and which we think is of large independent interest. While a span program is a completely different computational model, a transducer is a direct generalisation of a quantum algorithm, which allows for much greater transparency and control. Transducers naturally characterize general state conversion, rather than only decision problems; provide a very simple treatment of other quantum primitives such as quantum walks; and lend themselves well to time complexity analysis.

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arXiv2024

Tight Quantum Lower Bound for Approximate Counting with Quantum States

We prove tight lower bounds for the following variant of the counting problem considered by Aaronson, Kothari, Kretschmer, and Thaler (2020). The task is to distinguish whether an input set $x\subseteq [n]$ has size either $k$ or $k'=(1+\varepsilon)k$. We assume the algorithm has access to * the membership oracle, which, for each $i\in [n]$, can answer whether $i\in x$, or not; and \item the uniform superposition $|ψ_x\rangle = \sum_{i\in x} |i\rangle/\sqrt{|x|}$ over the elements of $x$. Moreover, we consider three different ways how the algorithm can access this state: - the algorithm can have copies of the state $|ψ_x\rangle$; - the algorithm can execute the reflecting oracle which reflects about the state $|ψ_x\rangle$; - the algorithm can execute the state-generating oracle (or its inverse) which performs the transformation $|0\rangle\mapsto|ψ_x\rangle$. Without the second type of resources (the ones related to $|ψ_x\rangle$), the problem is well-understood. The study of the problem with the second type of resources was recently initiated by Aaronson et al. We completely resolve the problem for all values of $1/k \le \varepsilon\le 1$, giving tight trade-offs between all types of resources available to the algorithm. We also demonstrate that our lower bounds are tight. Thus, we close the main open problems from Aaronson et al. The lower bounds are proven using variants of the adversary bound from Belovs (2015) and employing representation theory of the symmetric group applied to the $S_n$-modules $\mathbb{C}^{\binom{[n]}k}$ and $\mathbb{C}^{\binom{[n]}k}\otimes \mathbb{C}$.

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arXiv2024

Quantum algorithms for Hopcroft's problem

In this work we study quantum algorithms for Hopcroft's problem which is a fundamental problem in computational geometry. Given $n$ points and $n$ lines in the plane, the task is to determine whether there is a point-line incidence. The classical complexity of this problem is well-studied, with the best known algorithm running in $O(n^{4/3})$ time, with matching lower bounds in some restricted settings. Our results are two different quantum algorithms with time complexity $\widetilde O(n^{5/6})$. The first algorithm is based on partition trees and the quantum backtracking algorithm. The second algorithm uses a quantum walk together with a history-independent dynamic data structure for storing line arrangement which supports efficient point location queries. In the setting where the number of points and lines differ, the quantum walk-based algorithm is asymptotically faster. The quantum speedups for the aforementioned data structures may be useful for other geometric problems.

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arXiv2024

Global Phase Helps in Quantum Search: Yet Another Look at the Welded Tree Problem

Up to now, relatively few exponential quantum speed-ups have been achieved. Out of them, the welded tree problem (Childs, Cleve, Deotto, Farhi, Gutmann, and Spielman'2003) is one of the unusual examples, as the exponential speed-up is attained by a quantum walk. In this paper, we give a very short proof of the optimal linear hitting time for this problem by a discrete-time quantum walk, which is based on a simple modification of the electric quantum walk framework. The same technique can be applied to other 1-dimensional hierarchical graphs, yielding results similar to (Balasubramanian, Li, and Harrow'2023).

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arXiv2023

On the quantum time complexity of divide and conquer

We initiate a systematic study of the time complexity of quantum divide and conquer algorithms for classical problems. We establish generic conditions under which search and minimization problems with classical divide and conquer algorithms are amenable to quantum speedup and apply these theorems to an array of problems involving strings, integers, and geometric objects. They include LONGEST DISTINCT SUBSTRING, KLEE'S COVERAGE, several optimization problems on stock transactions, and k-INCREASING SUBSEQUENCE. For most of these results, our quantum time upper bound matches the quantum query lower bound for the problem, up to polylogarithmic factors.

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arXiv2023

An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree

While it is known that there is at most a polynomial separation between quantum query complexity and the polynomial degree for total functions, the precise relationship between the two is not clear for partial functions. In this paper, we demonstrate an exponential separation between exact polynomial degree and approximate quantum query complexity for a partial Boolean function. For an unbounded alphabet size, we have a constant versus polynomial separation.

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arXiv2023

A Direct Reduction from the Polynomial to the Adversary Method

The polynomial and the adversary methods are the two main tools for proving lower bounds on query complexity of quantum algorithms. Both methods have found a large number of applications, some problems more suitable for one method, some for the other. It is known though that the adversary method, in its general negative-weighted version, is tight for bounded-error quantum algorithms, whereas the polynomial method is not. By the tightness of the former, for any polynomial lower bound, there ought to exist a corresponding adversary lower bound. However, direct reduction was not known. In this paper, we give a simple and direct reduction from the polynomial method (in the form of a dual polynomial) to the adversary method. This shows that any lower bound in the form of a dual polynomial is actually an adversary lower bound of a specific form.

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arXiv2023

One-Way Ticket to Las Vegas and the Quantum Adversary

We propose a new definition of quantum Las Vegas query complexity. We show that it is exactly equal to the quantum adversary bound. This is achieved by a new and very simple way of transforming a feasible solution to the adversary optimisation problem into a quantum query algorithm. This allows us to generalise the bound to include unidirectional access, multiple input oracles, and input oracles that are not unitary. As an application, we demonstrate a separation between unidirectional and bidirectional access to an input oracle for a rather natural unitary permutation inversion problem.

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arXiv2020

Quantum Communication Complexity of Distribution Testing

The classical communication complexity of testing closeness of discrete distributions has recently been studied by Andoni, Malkin and Nosatzki (ICALP'19). In this problem, two players each receive $t$ samples from one distribution over $[n]$, and the goal is to decide whether their two distributions are equal, or are $ε$-far apart in the $l_1$-distance. In the present paper we show that the quantum communication complexity of this problem is $\tilde{O}(n/(tε^2))$ qubits when the distributions have low $l_2$-norm, which gives a quadratic improvement over the classical communication complexity obtained by Andoni, Malkin and Nosatzki. We also obtain a matching lower bound by using the pattern matrix method. Let us stress that the samples received by each of the parties are classical, and it is only communication between them that is quantum. Our results thus give one setting where quantum protocols overcome classical protocols for a testing problem with purely classical samples.

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arXiv2020

The quantum query complexity of composition with a relation

The negative weight adversary method, $\mathrm{ADV}^\pm(g)$, is known to characterize the bounded-error quantum query complexity of any Boolean function $g$, and also obeys a perfect composition theorem $\mathrm{ADV}^\pm(f \circ g^n) = \mathrm{ADV}^\pm(f) \mathrm{ADV}^\pm(g)$. Belovs gave a modified version of the negative weight adversary method, $\mathrm{ADV}_{rel}^\pm(f)$, that characterizes the bounded-error quantum query complexity of a relation $f \subseteq \{0,1\}^n \times [K]$, provided the relation is efficiently verifiable. A relation is efficiently verifiable if $\mathrm{ADV}^\pm(f_a) = o(\mathrm{ADV}_{rel}^\pm(f))$ for every $a \in [K]$, where $f_a$ is the Boolean function defined as $f_a(x) = 1$ if and only if $(x,a) \in f$. In this note we show a perfect composition theorem for the composition of a relation $f$ with a Boolean function $g$ \[ \mathrm{ADV}_{rel}^\pm(f \circ g^n) = \mathrm{ADV}_{rel}^\pm(f) \mathrm{ADV}^\pm(g) \enspace . \] For an efficiently verifiable relation $f$ this means $Q(f \circ g^n) = Θ( \mathrm{ADV}_{rel}^\pm(f) \mathrm{ADV}^\pm(g) )$.

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arXiv2020

Quantum Coupon Collector

We study how efficiently a $k$-element set $S\subseteq[n]$ can be learned from a uniform superposition $|S\rangle$ of its elements. One can think of $|S\rangle=\sum_{i\in S}|i\rangle/\sqrt{|S|}$ as the quantum version of a uniformly random sample over $S$, as in the classical analysis of the ``coupon collector problem.'' We show that if $k$ is close to $n$, then we can learn $S$ using asymptotically fewer quantum samples than random samples. In particular, if there are $n-k=O(1)$ missing elements then $O(k)$ copies of $|S\rangle$ suffice, in contrast to the $Θ(k\log k)$ random samples needed by a classical coupon collector. On the other hand, if $n-k=Ω(k)$, then $Ω(k\log k)$ quantum samples are~necessary. More generally, we give tight bounds on the number of quantum samples needed for every $k$ and $n$, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can additionally reflect through $|S\rangle$. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case.

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arXiv2019

Testing convexity of functions over finite domains

We establish new upper and lower bounds on the number of queries required to test convexity of functions over various discrete domains. 1. We provide a simplified version of the non-adaptive convexity tester on the line. We re-prove the upper bound $O(\frac{\log(εn)}ε)$ in the usual uniform model, and prove an $O(\frac{\log n}ε)$ upper bound in the distribution-free setting. 2. We show a tight lower bound of $Ω(\frac{\log(εn)}ε)$ queries for testing convexity of functions $f: [n] \rightarrow \mathbb{R}$ on the line. This lower bound applies to both adaptive and non-adaptive algorithms, and matches the upper bound from item 1, showing that adaptivity does not help in this setting. 3. Moving to higher dimensions, we consider the case of a stripe $[3] \times [n]$. We construct an \emph{adaptive} tester for convexity of functions $f\colon [3] \times [n] \to \mathbb R$ with query complexity $O(\log^2 n)$. We also show that any \emph{non-adaptive} tester must use $Ω(\sqrt{n})$ queries in this setting. Thus, adaptivity yields an exponential improvement for this problem. 4. For functions $f\colon [n]^d \to \mathbb R$ over domains of dimension $d \geq 2$, we show a non-adaptive query lower bound $Ω((\frac{n}{d})^{\frac{d}{2}})$.

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arXiv2019

Quantum Algorithms for Classical Probability Distributions

We study quantum algorithms working on classical probability distributions. We formulate four different models for accessing a classical probability distribution on a quantum computer, which are derived from previous work on the topic, and study their mutual relationships. Additionally, we prove that quantum query complexity of distinguishing two probability distributions is given by their inverse Hellinger distance, which gives a quadratic improvement over classical query complexity for any pair of distributions. The results are obtained by using the adversary method for state-generating input oracles and for distinguishing probability distributions on input strings.

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arXiv2019

Quantum Algorithm for Distribution-Free Junta Testing

Inspired by a recent classical distribution-free junta tester by Chen, Liu, Serverdio, Sheng, and Xie (STOC'18), we construct a quantum tester for the same problem with complexity $O(k/\varepsilon)$, which constitutes a quadratic improvement. We also prove that there is no efficient quantum algorithm for this problem using quantum examples as opposed to quantum membership queries. This result was obtained independently from the $\tilde O(k/\varepsilon)$ algorithm for this problem by Bshouty.

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arXiv2018

Adaptive Lower Bound for Testing Monotonicity on the Line

In the property testing model, the task is to distinguish objects possessing some property from the objects that are far from it. One of such properties is monotonicity, when the objects are functions from one poset to another. This is an active area of research. In this paper we study query complexity of $ε$-testing monotonicity of a function $f\colon [n]\to[r]$. All our lower bounds are for adaptive two-sided testers. * We prove a nearly tight lower bound for this problem in terms of $r$. The bound is $Ω(\frac{\log r}{\log \log r})$ when $ε= 1/2$. No previous satisfactory lower bound in terms of $r$ was known. * We completely characterise query complexity of this problem in terms of $n$ for smaller values of $ε$. The complexity is $Θ(ε^{-1} \log (εn))$. Apart from giving the lower bound, this improves on the best known upper bound. Finally, we give an alternative proof of the $Ω(ε^{-1}d\log n - ε^{-1}\logε^{-1})$ lower bound for testing monotonicity on the hypergrid $[n]^d$ due to Chakrabarty and Seshadhri (RANDOM'13).

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arXiv2018

Quantum Lower Bounds for Tripartite Versions of the Hidden Shift and the Set Equality Problems

In this paper, we study quantum query complexity of the following rather natural tripartite generalisations (in the spirit of the 3-sum problem) of the hidden shift and the set equality problems, which we call the 3-shift-sum and the 3-matching-sum problems. The 3-shift-sum problem is as follows: given a table of $3\times n$ elements, is it possible to circularly shift its rows so that the sum of the elements in each column becomes zero? It is promised that, if this is not the case, then no 3 elements in the table sum up to zero. The 3-matching-sum problem is defined similarly, but it is allowed to arbitrarily permute elements within each row. For these problems, we prove lower bounds of $Ω(n^{1/3})$ and $Ω(\sqrt n)$, respectively. The second lower bound is tight. The lower bounds are proven by a novel application of the dual learning graph framework and by using representation-theoretic tools.

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