arXiv subjects

Arjan Cornelissen

Publications and source records attributed to Arjan Cornelissen.

At least 19 recordsLinked to original sources

arXiv2026

Quantum algorithms for path and cycle containment problems

The quantum query complexity of subgraph-containment problems, which ask whether a given subgraph $H$ is present in an input graph $G$, has been the subject of considerable study. However, even for relatively simple subgraphs, such as paths and cycles, a complete understanding of their query complexities remains elusive. In this work, we consider several variants of path- and cycle-containment problems in the adjacency matrix model, where we search for paths or cycles of constant length $k$. We compare the settings where the graphs are directed or undirected, where the goal is to detect or find the existence of a path/cycle, and where the path/cycle we are looking for has length exactly $k$, or at most $k$. We also consider several promise versions of these problems, where we suppose that the input graph has a certain structure. We characterize the relative difficulty of these variants of the path/cycle-containment problems, by relating them to one another using randomized reductions, and grouping them into equivalence classes. When we restrict our attention to path-containment problems, we get a dichotomy result. Some of the path-containment problems can be solved using a linear number of queries, and all the others are equivalent to one another (and additionally to several cycle-containment problems) under randomized reductions, up to constant overhead. For the latter equivalence class, we prove a novel quantum-walk-based algorithm that achieves query complexity $\widetilde{O}(n^{3/2-α_k})$, where $α_k \in Θ(c^{-k})$ and $c = \sqrt{3+\sqrt{17}}/2 \approx 1.33$, beating the previous best upper bound $O(n^{3/2})$ on its query complexity. We also provide a conditional lower bound based on the graph-collision problem, which implies that this equivalence class does not admit linear-query quantum algorithms unless graph collision admits an $O(\sqrt{n})$ query 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.

cs.CC↗

arXiv2026

Quantum algorithms through graph composition

In this work, we unify several quantum algorithmic frameworks for boolean functions that are based on the quantum adversary bound. First, we show that the $st$-connectivity framework subsumes the (adaptive/extended) learning graph framework, and the weighted-decision-tree framework. Additionally, we show that every randomized algorithm can be turned into an $st$-connectivity problem as well, with the same complexity up to constants. This situates the $st$-connectivity framework in between randomized and quantum algorithms, indicating that it's an intermediate computational model between classical and quantum. We also introduce a generalization of the $st$-connectivity framework, the \textit{graph composition framework}, and show that it subsumes part of the quantum divide and conquer framework, and it is itself subsumed by the multidimensional quantum walk framework. Second, we investigate these frameworks' power by investigating the most efficient algorithms they can produce, in terms of the number of queries to the input. We show that the weighted-decision-tree framework's power is polynomially related to deterministic query complexity, showing that the quantum speed-ups that can be obtained with this framework are at most quadratic. Finally, we turn our attention to time-efficient implementations of the algorithms constructed through the $st$-connectivity and graph composition frameworks. To that end, we convert instances to the two-subspace phase estimation framework, and we show how we can implement these as transducers. This has the added benefit of removing the effective spectral gap lemma from the construction, significantly simplifying the analysis. We showcase the techniques developed in this work to give improved algorithms for various string search problems.

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arXiv2025

Randomized and quantum approximate matrix multiplication

The complexity of matrix multiplication is a central topic in computer science. While the focus has traditionally been on exact algorithms, a long line of literature also considers randomized algorithms, which return an approximate solution in faster time. In this work, we adopt a unifying perspective that frames these randomized algorithms in terms of mean estimation. Using it, we first give refined analyses of classical algorithms based on random walks by Cohen-Lewis (`99), and based on sketching by Sarlós (`06) and Drineas-Kannan-Mahoney (`06). We then propose an improvement on Cohen-Lewis that yields a single classical algorithm that is faster than all the other approaches, if we assume no use of (exact) fast matrix multiplication as a subroutine. Second, we demonstrate a quantum speedup on top of these algorithms by using the recent quantum multivariate mean estimation algorithm by Cornelissen-Hamoudi-Jerbi (`22).

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arXiv2025

Quantum walks through generalized graph composition

In this work, we generalize the recently-introduced graph composition framework to the non-boolean setting. A quantum algorithm in this framework is represented by a hypergraph, where each hyperedge is adjacent to multiple vertices. The input and output to the quantum algorithm is represented by a set of boundary vertices, and the hyperedges act like switches, connecting the input vertex to the output that the algorithm computes. Apart from generalizing the graph composition framework, our new proposed framework unifies the quantum divide and conquer framework, the decision-tree framework, and the unified quantum walk search framework. For the decision trees, we additionally construct a quantum algorithm from an improved weighting scheme in the non-boolean case. For quantum walk search, we show how our techniques naturally allow for amortization of the subroutines' costs. Previous work showed how one can speed up ``detection'' of marked vertices by amortizing the costs of the quantum walk. In this work, we extend these results to the setting of ``finding'' such marked vertices, albeit in some restricted settings. Along the way, we provide a novel analysis of irreducible, reversible Markov processes, by linear-algebraically connecting its effective resistance to the random walk operator. This significantly simplifies the algorithmic implementation of the quantum walk search algorithm, achieves an amortization speed-up for quantum walks over Johnson graphs, avoids the need for quantum fast-forwarding, and removes the log-factors from the query complexity statements.

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arXiv2025

Do you know what q-means?

Clustering is one of the most important tools for analysis of large datasets, and perhaps the most popular clustering algorithm is Lloyd's algorithm for $k$-means. This algorithm takes $n$ vectors $V=[v_1,\dots,v_n]\in\mathbb{R}^{d\times n}$ and outputs $k$ centroids $c_1,\dots,c_k\in\mathbb{R}^d$; these partition the vectors into clusters based on which centroid is closest to a particular vector. We present a classical $\varepsilon$-$k$-means algorithm that performs an approximate version of one iteration of Lloyd's algorithm with time complexity $\tilde{O}\big(\frac{\|V\|_F^2}{n}\frac{k^{2}d}{\varepsilon^2}(k + \log{n})\big)$, exponentially improving the dependence on the data size $n$ and matching that of the "$q$-means" quantum algorithm originally proposed by Kerenidis, Landman, Luongo, and Prakash (NeurIPS'19). Moreover, we propose an improved $q$-means quantum algorithm with time complexity $\tilde{O}\big(\frac{\|V\|_F}{\sqrt{n}}\frac{k^{3/2}d}{\varepsilon}(\sqrt{k}+\sqrt{d})(\sqrt{k} + \log{n})\big)$ that quadratically improves the runtime of our classical $\varepsilon$-$k$-means algorithm in several parameters. Our quantum algorithm does not rely on quantum linear algebra primitives of prior work, but instead only uses QRAM to prepare simple states based on the current iteration's clusters and multivariate quantum amplitude estimation. Finally, we provide classical and quantum query lower bounds, showing that our algorithms are optimal in most parameters.

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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

How to compute the volume in low dimension?

Estimating the volume of a convex body is a canonical problem in theoretical computer science. Its study has led to major advances in randomized algorithms, Markov chain theory, and computational geometry. In particular, determining the query complexity of volume estimation to a membership oracle has been a longstanding open question. Most of the previous work focuses on the high-dimensional limit. In this work, we tightly characterize the deterministic, randomized and quantum query complexity of this problem in the high-precision limit, i.e., when the dimension is constant.

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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 $f$.

quant-ph↗

arXiv2023

A Sublinear-Time Quantum Algorithm for Approximating Partition Functions

We present a novel quantum algorithm for estimating Gibbs partition functions in sublinear time with respect to the logarithm of the size of the state space. This is the first speed-up of this type to be obtained over the seminal nearly-linear time algorithm of Štefankovič, Vempala and Vigoda [JACM, 2009]. Our result also preserves the quadratic speed-up in precision and spectral gap achieved in previous work by exploiting the properties of quantum Markov chains. As an application, we obtain new polynomial improvements over the best-known algorithms for computing the partition function of the Ising model, counting the number of $k$-colorings, matchings or independent sets of a graph, and estimating the volume of a convex body. Our approach relies on developing new variants of the quantum phase and amplitude estimation algorithms that return nearly unbiased estimates with low variance and without destroying their initial quantum state. We extend these subroutines into a nearly unbiased quantum mean estimator that reduces the variance quadratically faster than the classical empirical mean. No such estimator was known to exist prior to our work. These properties, which are of general interest, lead to better convergence guarantees within the paradigm of simulated annealing for computing partition functions.

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arXiv2022

Quantum policy gradient algorithms

Understanding the power and limitations of quantum access to data in machine learning tasks is primordial to assess the potential of quantum computing in artificial intelligence. Previous works have already shown that speed-ups in learning are possible when given quantum access to reinforcement learning environments. Yet, the applicability of quantum algorithms in this setting remains very limited, notably in environments with large state and action spaces. In this work, we design quantum algorithms to train state-of-the-art reinforcement learning policies by exploiting quantum interactions with an environment. However, these algorithms only offer full quadratic speed-ups in sample complexity over their classical analogs when the trained policies satisfy some regularity conditions. Interestingly, we find that reinforcement learning policies derived from parametrized quantum circuits are well-behaved with respect to these conditions, which showcases the benefit of a fully-quantum reinforcement learning framework.

quant-ph↗

arXiv2022

Near-Optimal Quantum Algorithms for Multivariate Mean Estimation

We propose the first near-optimal quantum algorithm for estimating in Euclidean norm the mean of a vector-valued random variable with finite mean and covariance. Our result aims at extending the theory of multivariate sub-Gaussian estimators to the quantum setting. Unlike classically, where any univariate estimator can be turned into a multivariate estimator with at most a logarithmic overhead in the dimension, no similar result can be proved in the quantum setting. Indeed, Heinrich ruled out the existence of a quantum advantage for the mean estimation problem when the sample complexity is smaller than the dimension. Our main result is to show that, outside this low-precision regime, there is a quantum estimator that outperforms any classical estimator. Our approach is substantially more involved than in the univariate setting, where most quantum estimators rely only on phase estimation. We exploit a variety of additional algorithmic techniques such as amplitude amplification, the Bernstein-Vazirani algorithm, and quantum singular value transformation. Our analysis also uses concentration inequalities for multivariate truncated statistics. We develop our quantum estimators in two different input models that showed up in the literature before. The first one provides coherent access to the binary representation of the random variable and it encompasses the classical setting. In the second model, the random variable is directly encoded into the phases of quantum registers. This model arises naturally in many quantum algorithms but it is often incomparable to having classical samples. We adapt our techniques to these two settings and we show that the second model is strictly weaker for solving the mean estimation problem. Finally, we describe several applications of our algorithms, notably in measuring the expectation values of commuting observables and in the field of machine learning.

quant-ph↗

arXiv2022

Quantum tomography using state-preparation unitaries

We describe algorithms to obtain an approximate classical description of a $d$-dimensional quantum state when given access to a unitary (and its inverse) that prepares it. For pure states we characterize the query complexity for $\ell_q$-norm error up to logarithmic factors. As a special case, we show that it takes $\widetildeΘ(d/\varepsilon)$ applications of the unitaries to obtain an $\varepsilon$-$\ell_2$-approximation of the state. For mixed states we consider a similar model, where the unitary prepares a purification of the state. In this model we give an efficient algorithm for obtaining Schatten $q$-norm estimates of a rank-$r$ mixed state, giving query upper bounds that are close to optimal. In particular, we show that a trace-norm ($q=1$) estimate can be obtained with $\widetilde{\mathcal{O}}(dr/\varepsilon)$ queries. This improves (assuming our stronger input model) the $\varepsilon$-dependence over the algorithm of Haah et al.\ (2017) that uses a joint measurement on $\widetilde{\mathcal{O}}(dr/\varepsilon^2)$ copies of the state. To our knowledge, the most sample-efficient results for pure-state tomography come from setting the rank to $1$ in generic mixed-state tomography algorithms, which can be computationally demanding. We describe sample-optimal algorithms for pure states that are easy and fast to implement. Along the way we show that an $\ell_\infty$-norm estimate of a normalized vector induces a (slightly worse) $\ell_q$-norm estimate for that vector, without losing a dimension-dependent factor in the precision. We also develop an unbiased and symmetric version of phase estimation, where the probability distribution of the estimate is centered around the true value. Finally, we give an efficient method for estimating multiple expectation values, improving over the recent result by Huggins et al.\ (2021) when the measurement operators do not fully overlap.

quant-ph↗

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].

quant-ph↗

arXiv2021

Quantum algorithms for multivariate Monte Carlo estimation

We consider the problem of estimating the expected outcomes of Monte Carlo processes whose outputs are described by multidimensional random variables. We tightly characterize the quantum query complexity of this problem for various choices of oracle access to the Monte Carlo process and various normalization conditions on its output variables and the estimates to be made. More specifically, we design quantum algorithms that provide a quadratic speed-up in query complexity with respect to the precision of the resulting estimates and show that these complexities are asymptotically optimal in each of our scenarios. Interestingly, we find that these speed-ups with respect to the precision come at the cost of a (up to exponential) slowdown with respect to the dimension of the Monte Carlo random variables, for most of the parameter settings we consider. We also describe several applications of our algorithms, notably in the field of machine learning, and discuss the implications of our results.

quant-ph↗

arXiv2021

Scalable Benchmarks for Gate-Based Quantum Computers

In the near-term "NISQ"-era of noisy, intermediate-scale, quantum hardware and beyond, reliably determining the quality of quantum devices becomes increasingly important: users need to be able to compare them with one another, and make an estimate whether they are capable of performing a given task ahead of time. In this work, we develop and release an advanced quantum benchmarking framework in order to help assess the state of the art of current quantum devices. Our testing framework measures the performance of universal quantum devices in a hardware-agnostic way, with metrics that are aimed to facilitate an intuitive understanding of which device is likely to outperform others on a given task. This is achieved through six structured tests that allow for an immediate, visual assessment of how devices compare. Each test is designed with scalability in mind, making this framework not only suitable for testing the performance of present-day quantum devices, but also of those released in the foreseeable future. The series of tests are motivated by real-life scenarios, and therefore emphasise the interplay between various relevant characteristics of quantum devices, such as qubit count, connectivity, and gate and measurement fidelity. We present the benchmark results of twenty-one different quantum devices from IBM, Rigetti and IonQ.

quant-ph↗

arXiv2020

Span programs and quantum time complexity

Span programs are an important model of quantum computation due to their tight correspondence with quantum query complexity. For any decision problem $f$, the minimum complexity of a span program for $f$ is equal, up to a constant factor, to the quantum query complexity of $f$. Moreover, this correspondence is constructive. A span program for $f$ with complexity $C$ can be compiled into a bounded error quantum algorithm for $f$ with query complexity $O(C)$, and vice versa. In this work, we prove an analogous connection for quantum time complexity. In particular, we show how to convert a quantum algorithm for $f$ with time complexity $T$ into a span program for $f$ such that it compiles back into a quantum algorithm for $f$ with time complexity $\widetilde{O}(T)$. While the query complexity of quantum algorithms obtained from span programs is well-understood, it is not generally clear how to implement certain query-independent operations in a time-efficient manner. We show that for span programs derived from algorithms with a time-efficient implementation, we can preserve the time efficiency when implementing the span program. This means in particular that span programs not only fully capture quantum query complexity, but also quantum time complexity. One practical advantage of being able to convert quantum algorithms to span programs in a way that preserves time complexity is that span programs compose very nicely. We demonstrate this by improving Ambainis's variable-time quantum search result using our construction through a span program composition for the OR function.

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