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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

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 Õ(2^{n/2}) and Õ(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(np), after which queries can be made in time O(n) 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. This provides a notable improvement on the best known O(2^{0.773n}) classical running time established by Mucha et al. [Mucha et al., 2019]. We also study Pigeonhole Equal-Sums, a variant of Equal-Sums where the existence of a solution is guaranteed by the pigeonhole principle. For this problem we give faster classical and quantum algorithms with running time Õ(2^{n/2}) and Õ(2^{2n/5}), respectively.

Jonathan Allcock, Yassine Hamoudi, Antoine Joux, Felix Klingelhöfer, and Miklos Santha. Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{allcock_et_al:LIPIcs.ESA.2022.6, author = {Allcock, Jonathan and Hamoudi, Yassine and Joux, Antoine and Klingelh\"{o}fer, Felix and Santha, Miklos}, title = {{Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {6:1--6:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.6}, URN = {urn:nbn:de:0030-drops-169444}, doi = {10.4230/LIPIcs.ESA.2022.6}, annote = {Keywords: Quantum algorithm, classical algorithm, dynamic programming, representation technique, subset-sum, equal-sum, shifted-sum} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

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 [Brassard et al., 2002; Brassard et al., 2011], or that require prior information on the variance [Heinrich, 2002; Montanaro, 2015; Hamoudi and Magniez, 2019]. 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.

Yassine Hamoudi. Quantum Sub-Gaussian Mean Estimator. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 50:1-50:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hamoudi:LIPIcs.ESA.2021.50, author = {Hamoudi, Yassine}, title = {{Quantum Sub-Gaussian Mean Estimator}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {50:1--50:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.50}, URN = {urn:nbn:de:0030-drops-146318}, doi = {10.4230/LIPIcs.ESA.2021.50}, annote = {Keywords: Quantum algorithm, statistical analysis, mean estimator, sub-Gaussian estimator, (\epsilon,\delta)-approximation, lower bound} }

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**Published in:** LIPIcs, Volume 197, 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)

We study the problem of finding K collision pairs in a random function f : [N] → [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 T³ S ≥ Ω(K³N). Classically, the same question has only been settled recently by Dinur [Dinur, 2020], who showed that the Parallel Collision Search algorithm of van Oorschot and Wiener [Oorschot and Wiener, 1999] achieves the optimal time-space tradeoff of T² S = Θ(K² 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 [Zhandry, 2019] 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 T² S ≥ Ω(N³) for sorting N numbers on a quantum computer, which was first obtained by Klauck, Špalek and de Wolf [Klauck et al., 2007].

Yassine Hamoudi and Frédéric Magniez. Quantum Time-Space Tradeoff for Finding Multiple Collision Pairs. In 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 197, pp. 1:1-1:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hamoudi_et_al:LIPIcs.TQC.2021.1, author = {Hamoudi, Yassine and Magniez, Fr\'{e}d\'{e}ric}, title = {{Quantum Time-Space Tradeoff for Finding Multiple Collision Pairs}}, booktitle = {16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)}, pages = {1:1--1:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-198-6}, ISSN = {1868-8969}, year = {2021}, volume = {197}, editor = {Hsieh, Min-Hsiu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2021.1}, URN = {urn:nbn:de:0030-drops-139961}, doi = {10.4230/LIPIcs.TQC.2021.1}, annote = {Keywords: Quantum computing, query complexity, lower bound, time-space tradeoff} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

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 >= 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 = O~(n^{1-2/k}), whereas the best classical algorithm requires P M = Theta(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 O~(sqrt{n}/m^{1/4}) and 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 dependence is quadratic. Our algorithm subsumes a previous result of Montanaro [Montanaro, 2015]. This new paradigm is based on a refinement of the Amplitude Estimation algorithm of Brassard et al. [Brassard et al., 2002] 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. Finally, we develop a new technique called "variable-time amplitude estimation" that reduces the dependence of our algorithm on the sample preparation time.

Yassine Hamoudi and Frédéric Magniez. Quantum Chebyshev’s Inequality and Applications. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 69:1-69:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hamoudi_et_al:LIPIcs.ICALP.2019.69, author = {Hamoudi, Yassine and Magniez, Fr\'{e}d\'{e}ric}, title = {{Quantum Chebyshev’s Inequality and Applications}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {69:1--69:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.69}, URN = {urn:nbn:de:0030-drops-106458}, doi = {10.4230/LIPIcs.ICALP.2019.69}, annote = {Keywords: Quantum algorithms, approximation algorithms, sublinear-time algorithms, Monte Carlo method, streaming algorithms, subgraph counting} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

The Number On the Forehead (NOF) model is a multiparty communication game between k players that collaboratively want to evaluate a given function F : X_1 x *s x X_k - > Y on some input (x_1,...,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 (as if x_i is written on the forehead of player i). In the Simultaneous Message Passing (SMP) model, the players have the extra condition that they cannot speak to each other, but instead send information to a referee. The referee does not know the players' inputs, and cannot give any information back. At the end, the referee must be able to recover F(x_1,...,x_k) from what she obtained from the players.
A central open question in the simultaneous NOF model, called the log n barrier, is to find a function which is hard to compute when the number of players is polylog(n) or more (where the x_i's have size poly(n)). This has an important application in circuit complexity, as it could help to separate ACC^0 from other complexity classes [Håstad and Goldmann, 1991; Babai et al., 2004]. One of the candidates for breaking the log n barrier belongs to the family of composed functions. The input to these functions in the k-party NOF model is represented by a k x (t * n) boolean matrix M, whose row i is the number x_i on the forehead of player 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 (respectively), that output f o g(M) = f(g(B_1),...,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. [Babai et al., 1995; Babai et al., 2004] suggested to study the composed function Maj o Maj_t, with t large enough, for breaking the log n barrier (where Maj_t outputs 1 if at least kt/2 bits of the input block are set to 1).
So far, it was only known that block-width t = 1 is not enough for Maj o Maj_t to break the log n barrier in the simultaneous NOF model [Babai et al., 2004] (Chattopadhyay and Saks [Chattopadhyay and Saks, 2014] found an efficient protocol for t <= polyloglog(n), but it requires randomness to be simultaneous). In this paper, we extend this result to any constant block-width t > 1 by giving a deterministic simultaneous protocol of cost 2^O(2^t) log^(2^(t+1))(n) for any symmetric composed function f o g (which includes Maj o Maj_t) when there are more than 2^Omega(2^t) log n players.

Yassine Hamoudi. Simultaneous Multiparty Communication Protocols for Composed Functions. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hamoudi:LIPIcs.MFCS.2018.14, author = {Hamoudi, Yassine}, title = {{Simultaneous Multiparty Communication Protocols for Composed Functions}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {14:1--14:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.14}, URN = {urn:nbn:de:0030-drops-95969}, doi = {10.4230/LIPIcs.MFCS.2018.14}, annote = {Keywords: Communication complexity, Number On the Forehead model, Simultaneous Message Passing, Log n barrier, Symmetric Composed functions} }

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