arXiv subjects

Yassine Hamoudi

Publications and source records attributed to Yassine Hamoudi.

16 recordsLinked to original sources

arXiv2026

Dequantization Barriers for Guided Stoquastic Hamiltonians

We construct a probability distribution, induced by the Perron--Frobenius eigenvector of an exponentially large graph, that cannot be efficiently sampled by any classical algorithm, even when given the best-possible warm-start distribution. In the quantum setting, this problem can be viewed as preparing the ground state of a stoquastic Hamiltonian given a guiding state as input, and is known to be efficiently solvable on a quantum computer. We further show that even estimating the ground-state energy of that Hamiltonian remains classically hard. Our results suggest that no efficient classical algorithm can solve a broad class of stoquastic ground-state problems. Our graph is constructed from a class of high-degree, high-girth spectral expanders (such as the LPS Ramanujan graphs) to which self-similar trees are attached. This builds on and extends prior work of Gilyén, Hastings, and Vazirani (Quantum 2021, STOC 2021), which ruled out dequantization for a specific stoquastic adiabatic path algorithm. We strengthen their result by ruling out any classical algorithm for guided ground-state preparation or energy estimation, while also providing a derandomized construction.

quant-ph↗

arXiv2026

Quantum Advantage with Sign-Free Proofs: An Oracle Separation between $\mathsf{StoqMA}$ and $\mathsf{MA}$

Several landmark oracle results (FOCS 2000, FOCS 2025, STOC 2026) establish strict separations along the chain $\mathsf{MA} \subseteq \mathsf{QCMA} \subseteq \mathsf{QMA}$, showing that quantum verification and quantum witnesses can each provide additional power over classical proof systems. These separations, however, use substantial coherent quantum computation, often through the quantum Fourier transform. We prove that a minimal quantum resource already suffices to outperform every classical verifier. Namely, we show the separation $$\mathsf{MA} \subsetneq \mathsf{StoqMA}$$ relative to a classical oracle. The class $\mathsf{StoqMA}$, introduced by Bravyi, Bessen, and Terhal (arXiv 2006), restricts the verifier to classical reversible computation, followed by a single Hadamard-basis measurement. Its power is closely tied to the complexity of stoquastic Hamiltonians, a sign-problem-free and seemingly more tractable class of quantum Hamiltonians. Our separating problem is to determine whether a sparse high-girth graph, given via an adjacency-list oracle, has a large or small Perron--Frobenius eigenvalue. This amounts to a ground-energy estimation task for a related stoquastic Hamiltonian. Our construction builds on a line of work on graph exploration problems with exponential quantum speedups, from welded trees (STOC 2003) to recent constructions (STOC 2021, arXiv 2026).

quant-ph↗

arXiv2026

Quantum Sampling of Random Spanning Trees via Amortized Data Structures

We present a quantum algorithm that generates a uniform superposition over the spanning trees of a graph -- also known as a q-sample -- using only a sub-linear number of queries to the graph. This goes beyond previous work that focuses on generating classical samples, including a recent quantum algorithm by Apers, Gao, Ji, and Liu [ICALP 2025]. For an $n$-vertex, $m$-edge graph, our algorithm admits a pre-processing step of $\tilde{O}(\sqrt{mn} + m^{1-δ})$, after which each q-sample can be generated in $\tilde{O}(n^{1+2δ})$ for any $δ$. Alternatively, $k$ independent q-samples can be generated at a total cost of $\tilde{O}(\sqrt{kmn})$. We also prove a matching lower bound up to logarithmic factors, showing that our algorithm is essentially optimal. Further tradeoffs are given in the paper. In comparison, the optimal classical algorithms (Anari, Liu and Vuong [FOCS 2022]) need an $\tilde{O}(m)$ pre-processing step, after which each sample costs $\tilde{O}(n)$ operations. These results yield faster quantum walk-based algorithms for counting spanning trees or finding a marked one. Our result is obtained via quantum walk sampling over a sequence of slowly-changing Markov chains. Each chain is an isotropized up-down walk that mixes rapidly to the spanning tree distribution of the input graph. A key ingredient is an amortized data structure supporting fast implementation of the associated quantum walk operators throughout the sequence. This data structure maintains access to a spectral sparsifier and a leverage score sampler that evolve with the underlying graph.

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arXiv2025

Quantum speedup of non-linear Monte Carlo problems

The mean of a random variable can be understood as a linear functional on the space of probability distributions. Quantum computing is known to provide a quadratic speedup over classical Monte Carlo methods for mean estimation. In this paper, we investigate whether a similar quadratic speedup is achievable for estimating non-linear functionals of probability distributions. We propose a quantum-inside-quantum Monte Carlo algorithm that achieves such a speedup for a broad class of non-linear estimation problems, including nested conditional expectations and stochastic optimization. Our algorithm improves upon the direct application of the quantum multilevel Monte Carlo algorithm introduced by An et al. (2021). The existing lower bound indicates that our algorithm is optimal up polylogarithmic factors. A key innovation of our approach is a new sequence of multilevel Monte Carlo approximations specifically designed for quantum computing, which is central to the algorithm's improved performance.

quant-ph↗

arXiv2025

A Brief Introduction to Quantum Query Complexity

Quantum query complexity is a fundamental model for analyzing the computational power of quantum algorithms. It has played a key role in characterizing quantum speedups, from early breakthroughs such as Grover's and Simon's algorithms to more recent developments in quantum cryptography and complexity theory. This document provides a structured introduction to quantum query lower bounds, focusing on four major techniques: the hybrid method, the polynomial method, the recording method, and the adversary method. Each method is developed from first principles and illustrated through canonical problems. Additionally, the document discusses how the adversary method can be used to derive upper bounds, highlighting its dual role in quantum query complexity. The goal is to offer a self-contained exposition accessible to readers with a basic background in quantum computing, while also serving as an entry point for researchers interested in the study of quantum lower bounds.

quant-ph↗

arXiv2024

The NISQ Complexity of Collision Finding

Collision-resistant hashing, a fundamental primitive in modern cryptography, ensures that there is no efficient way to find distinct inputs that produce the same hash value. This property underpins the security of various cryptographic applications, making it crucial to understand its complexity. The complexity of this problem is well-understood in the classical setting and $Θ(N^{1/2})$ queries are needed to find a collision. However, the advent of quantum computing has introduced new challenges since quantum adversaries $\unicode{x2013}$ equipped with the power of quantum queries $\unicode{x2013}$ can find collisions much more efficiently. Brassard, Höyer and Tapp and Aaronson and Shi established that full-scale quantum adversaries require $Θ(N^{1/3})$ queries to find a collision, prompting a need for longer hash outputs, which impacts efficiency in terms of the key lengths needed for security. This paper explores the implications of quantum attacks in the Noisy-Intermediate Scale Quantum (NISQ) era. In this work, we investigate three different models for NISQ algorithms and achieve tight bounds for all of them: (1) A hybrid algorithm making adaptive quantum or classical queries but with a limited quantum query budget, or (2) A quantum algorithm with access to a noisy oracle, subject to a dephasing or depolarizing channel, or (3) A hybrid algorithm with an upper bound on its maximum quantum depth; i.e., a classical algorithm aided by low-depth quantum circuits. In fact, our results handle all regimes between NISQ and full-scale quantum computers. Previously, only results for the pre-image search problem were known for these models by Sun and Zheng, Rosmanis, Chen, Cotler, Huang and Li while nothing was known about the collision finding problem.

quant-ph↗

arXiv2023

Quantum Time-Space Tradeoff for Finding Multiple Collision Pairs

We study the problem of finding $K$ collision pairs in a random function $f : [N] \rightarrow [N]$ by using a quantum computer. We prove that the number of queries to the function in the quantum random oracle model must increase significantly when the size of the available memory is limited. Namely, we demonstrate that any algorithm using $S$ qubits of memory must perform a number $T$ of queries that satisfies the tradeoff $T^3 S \geq Ω(K^3 N)$. Classically, the same question has only been settled recently by Dinur [Eurocrypt'20], who showed that the Parallel Collision Search algorithm of van Oorschot and Wiener achieves the optimal time-space tradeoff of $T^2 S = Θ(K^2 N)$. Our result limits the extent to which quantum computing may decrease this tradeoff. Our method is based on a novel application of Zhandry's recording query technique [Crypto'19] for proving lower bounds in the exponentially small success probability regime. As a second application, we give a simpler proof of the time-space tradeoff $T^2 S \geq Ω(N^3)$ for sorting $N$ numbers on a quantum computer, which was first obtained by Klauck, Špalek and de Wolf [KŠW07].

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

Preparing Many Copies of a Quantum State in the Black-Box Model

We describe a simple quantum algorithm for preparing $K$ copies of an $N$-dimensional quantum state whose amplitudes are given by a quantum oracle. Our result extends a previous work of Grover, who showed how to prepare one copy in time $O(\sqrt{N})$. In comparison with the naive $O(K\sqrt{N})$ solution obtained by repeating this procedure~$K$ times, our algorithm achieves the optimal running time of $θ(\sqrt{KN})$. Our technique uses a refinement of the quantum rejection sampling method employed by Grover. As a direct application, we obtain a similar speed-up for obtaining $K$ independent samples from a distribution whose probability vector is given by a quantum oracle.

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arXiv2022

Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming

Subset-Sum is an NP-complete problem where one must decide if a multiset of $n$ integers contains a subset whose elements sum to a target value $m$. The best-known classical and quantum algorithms run in time $\tilde{O}(2^{n/2})$ and $\tilde{O}(2^{n/3})$, respectively, based on the well-known meet-in-the-middle technique. Here we introduce a novel classical dynamic-programming-based data structure with applications to Subset-Sum and a number of variants, including Equal-Sums (where one seeks two disjoint subsets with the same sum), 2-Subset-Sum (a relaxed version of Subset-Sum where each item in the input set can be used twice in the summation), and Shifted-Sums, a generalization of both of these variants, where one seeks two disjoint subsets whose sums differ by some specified value. Given any modulus $p$, our data structure can be constructed in time $O(n^2p)$, after which queries can be made in time $O(n^2)$ to the lists of subsets summing to any value modulo $p$. We use this data structure in combination with variable-time amplitude amplification and a new quantum pair finding algorithm, extending the quantum claw finding algorithm to the multiple solutions case, to give an $O(2^{0.504n})$ quantum algorithm for Shifted-Sums, an improvement on the best-known $O(2^{0.773n})$ classical running time. Incidentally, we obtain new $\tilde{O}(2^{n/2})$ and $\tilde{O}(2^{n/3})$ classical and quantum algorithms for Subset-Sum, not based on the seminal meet-in-the-middle method. We also study Pigeonhole Equal-Sums and Pigeonhole Modular Equal-Sums, where the existence of a solution is guaranteed by the pigeonhole principle. For the former problem, we give faster classical and quantum algorithms with running time $\tilde{O}(2^{n/2})$ and $\tilde{O}(2^{2n/5})$, respectively. For the more general modular problem, we give a classical algorithm that also runs in time $\tilde{O}(2^{n/2})$.

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.

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arXiv2021

Quantum Sub-Gaussian Mean Estimator

We present a new quantum algorithm for estimating the mean of a real-valued random variable obtained as the output of a quantum computation. Our estimator achieves a nearly-optimal quadratic speedup over the number of classical i.i.d. samples needed to estimate the mean of a heavy-tailed distribution with a sub-Gaussian error rate. This result subsumes (up to logarithmic factors) earlier works on the mean estimation problem that were not optimal for heavy-tailed distributions [BHMT02,BDGT11], or that require prior information on the variance [Hein02,Mon15,HM19]. As an application, we obtain new quantum algorithms for the $(ε,δ)$-approximation problem with an optimal dependence on the coefficient of variation of the input random variable.

quant-ph↗

arXiv2020

Quantum algorithms for hedging and the learning of Ising models

A paradigmatic algorithm for online learning is the Hedge algorithm by Freund and Schapire. An allocation into different strategies is chosen for multiple rounds and each round incurs corresponding losses for each strategy. The algorithm obtains a favorable guarantee for the total losses even in an adversarial situation. This work presents quantum algorithms for such online learning in an oracular setting. For $T$ time steps and $N$ strategies, we exhibit run times of about $O \left ({\rm poly} (T) \sqrt{N} \right)$ for estimating the losses and for betting on individual strategies by sampling. In addition, we discuss a quantum analogue of the Sparsitron, a machine learning algorithm based on the Hedge algorithm. The quantum algorithm inherits the provable learning guarantees from the classical algorithm and exhibits polynomial speedups. The speedups may find relevance in finance, for example for hedging risks, and machine learning, for example for learning generalized linear models or Ising models.

quant-ph↗

arXiv2020

Quantum and Classical Algorithms for Approximate Submodular Function Minimization

Submodular functions are set functions mapping every subset of some ground set of size $n$ into the real numbers and satisfying the diminishing returns property. Submodular minimization is an important field in discrete optimization theory due to its relevance for various branches of mathematics, computer science and economics. The currently fastest strongly polynomial algorithm for exact minimization [LSW15] runs in time $\widetilde{O}(n^3 \cdot \mathrm{EO} + n^4)$ where $\mathrm{EO}$ denotes the cost to evaluate the function on any set. For functions with range $[-1,1]$, the best $ε$-additive approximation algorithm [CLSW17] runs in time $\widetilde{O}(n^{5/3}/ε^{2} \cdot \mathrm{EO})$. In this paper we present a classical and a quantum algorithm for approximate submodular minimization. Our classical result improves on the algorithm of [CLSW17] and runs in time $\widetilde{O}(n^{3/2}/ε^2 \cdot \mathrm{EO})$. Our quantum algorithm is, up to our knowledge, the first attempt to use quantum computing for submodular optimization. The algorithm runs in time $\widetilde{O}(n^{5/4}/ε^{5/2} \cdot \log(1/ε) \cdot \mathrm{EO})$. The main ingredient of the quantum result is a new method for sampling with high probability $T$ independent elements from any discrete probability distribution of support size $n$ in time $O(\sqrt{Tn})$. Previous quantum algorithms for this problem were of complexity $O(T\sqrt{n})$.

cs.DS↗

arXiv2019

Quantum Chebyshev's Inequality and Applications

In this paper we provide new quantum algorithms with polynomial speed-up for a range of problems for which no such results were known, or we improve previous algorithms. First, we consider the approximation of the frequency moments $F_k$ of order $k \geq 3$ in the multi-pass streaming model with updates (turnstile model). We design a $P$-pass quantum streaming algorithm with memory $M$ satisfying a tradeoff of $P^2 M = \tilde{O}(n^{1-2/k})$, whereas the best classical algorithm requires $P M = Θ(n^{1-2/k})$. Then, we study the problem of estimating the number $m$ of edges and the number $t$ of triangles given query access to an $n$-vertex graph. We describe optimal quantum algorithms that perform $\tilde{O}(\sqrt{n}/m^{1/4})$ and $\tilde{O}(\sqrt{n}/t^{1/6} + m^{3/4}/\sqrt{t})$ queries respectively. This is a quadratic speed-up compared to the classical complexity of these problems. For this purpose we develop a new quantum paradigm that we call Quantum Chebyshev's inequality. Namely we demonstrate that, in a certain model of quantum sampling, one can approximate with relative error the mean of any random variable with a number of quantum samples that is linear in the ratio of the square root of the variance to the mean. Classically the dependency is quadratic. Our algorithm subsumes a previous result of Montanaro [Mon15]. This new paradigm is based on a refinement of the Amplitude Estimation algorithm of Brassard et al. [BHMT02] and of previous quantum algorithms for the mean estimation problem. We show that this speed-up is optimal, and we identify another common model of quantum sampling where it cannot be obtained. For our applications, we also adapt the variable-time amplitude amplification technique of Ambainis [Amb10] into a variable-time amplitude estimation algorithm.

quant-ph↗

arXiv2018

Simultaneous Multiparty Communication Complexity of Composed Functions

In the Number On the Forehead (NOF) multiparty communication model, $k$ players want to evaluate a function $F : X_1 \times\cdots\times X_k\rightarrow Y$ on some input $(x_1,\dots,x_k)$ by broadcasting bits according to a predetermined protocol. The input is distributed in such a way that each player $i$ sees all of it except $x_i$. In the simultaneous setting, the players cannot speak to each other but instead send information to a referee. The referee does not know the players' input, and cannot give any information back. At the end, the referee must be able to recover $F(x_1,\dots,x_k)$ from what she obtained. A central open question, called the $\log n$ barrier, is to find a function which is hard to compute for $polylog(n)$ or more players (where the $x_i$'s have size $poly(n)$) in the simultaneous NOF model. This has important applications in circuit complexity, as it could help to separate $ACC^0$ from other complexity classes. One of the candidates belongs to the family of composed functions. The input to these functions is represented by a $k\times (t\cdot n)$ boolean matrix $M$, whose row $i$ is the input $x_i$ and $t$ is a block-width parameter. A symmetric composed function acting on $M$ is specified by two symmetric $n$- and $kt$-variate functions $f$ and $g$, that output $f\circ g(M)=f(g(B_1),\dots,g(B_n))$ where $B_j$ is the $j$-th block of width $t$ of $M$. As the majority function $MAJ$ is conjectured to be outside of $ACC^0$, Babai et. al. suggested to study $MAJ\circ MAJ_t$, with $t$ large enough. So far, it was only known that $t=1$ is not enough for $MAJ\circ MAJ_t$ to break the $\log n$ barrier in the simultaneous deterministic NOF model. In this paper, we extend this result to any constant block-width $t>1$, by giving a protocol of cost $2^{O(2^t)}\log^{2^{t+1}}(n)$ for any symmetric composed function when there are $2^{Ω(2^t)}\log n$ players.

cs.CC↗