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**Published in:** LIPIcs, Volume 266, 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)

We study variable time search, a form of quantum search where queries to different items take different time. Our first result is a new quantum algorithm that performs variable time search with complexity O(√Tlog n) where T = ∑_{i = 1}ⁿ t_i² with t_i denoting the time to check the i^th item. Our second result is a quantum lower bound of Ω(√{Tlog T}). Both the algorithm and the lower bound improve over previously known results by a factor of √{log T} but the algorithm is also substantially simpler than the previously known quantum algorithms.

Andris Ambainis, Martins Kokainis, and Jevgēnijs Vihrovs. Improved Algorithm and Lower Bound for Variable Time Quantum Search. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ambainis_et_al:LIPIcs.TQC.2023.7, author = {Ambainis, Andris and Kokainis, Martins and Vihrovs, Jevg\={e}nijs}, title = {{Improved Algorithm and Lower Bound for Variable Time Quantum Search}}, booktitle = {18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)}, pages = {7:1--7:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-283-9}, ISSN = {1868-8969}, year = {2023}, volume = {266}, editor = {Fawzi, Omar and Walter, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2023.7}, URN = {urn:nbn:de:0030-drops-183177}, doi = {10.4230/LIPIcs.TQC.2023.7}, annote = {Keywords: quantum search, amplitude amplification} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Motivated by the quantum speedup for dynamic programming on the Boolean hypercube by Ambainis et al. (2019), we investigate which graphs admit a similar quantum advantage. In this paper, we examine a generalization of the Boolean hypercube graph, the n-dimensional lattice graph Q(D,n) with vertices in {0,1,…,D}ⁿ. We study the complexity of the following problem: given a subgraph G of Q(D,n) via query access to the edges, determine whether there is a path from 0ⁿ to Dⁿ. While the classical query complexity is Θ̃((D+1)ⁿ), we show a quantum algorithm with complexity Õ(T_Dⁿ), where T_D < D+1. The first few values of T_D are T₁ ≈ 1.817, T₂ ≈ 2.660, T₃ ≈ 3.529, T₄ ≈ 4.421, T₅ ≈ 5.332. We also prove that T_D ≥ (D+1)/e (here, e ≈ 2.718 is the Euler’s number), thus for general D, this algorithm does not provide, for example, a speedup, polynomial in the size of the lattice.
While the presented quantum algorithm is a natural generalization of the known quantum algorithm for D = 1 by Ambainis et al., the analysis of complexity is rather complicated. For the precise analysis, we use the saddle-point method, which is a common tool in analytic combinatorics, but has not been widely used in this field.
We then show an implementation of this algorithm with time and space complexity poly(n)^{log n} T_Dⁿ in the QRAM model, and apply it to the Set Multicover problem. In this problem, m subsets of [n] are given, and the task is to find the smallest number of these subsets that cover each element of [n] at least D times. While the time complexity of the best known classical algorithm is O(m(D+1)ⁿ), the time complexity of our quantum algorithm is poly(m,n)^{log n} T_Dⁿ.

Adam Glos, Martins Kokainis, Ryuhei Mori, and Jevgēnijs Vihrovs. Quantum Speedups for Dynamic Programming on n-Dimensional Lattice Graphs. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 50:1-50:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{glos_et_al:LIPIcs.MFCS.2021.50, author = {Glos, Adam and Kokainis, Martins and Mori, Ryuhei and Vihrovs, Jevg\={e}nijs}, title = {{Quantum Speedups for Dynamic Programming on n-Dimensional Lattice Graphs}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {50:1--50:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.50}, URN = {urn:nbn:de:0030-drops-144901}, doi = {10.4230/LIPIcs.MFCS.2021.50}, annote = {Keywords: Quantum query complexity, Dynamic programming, Lattice graphs} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

We show that all known classical adversary lower bounds on randomized query complexity are equivalent for total functions, and are equal to the fractional block sensitivity fbs(f). That includes the Kolmogorov complexity bound of Laplante and Magniez and the earlier relational adversary bound of Aaronson. For partial functions, we show unbounded separations between fbs(f) and other adversary bounds, as well as between the relational and Kolmogorov complexity bounds.
We also show that, for partial functions, fractional block sensitivity cannot give lower bounds larger than sqrt(n * bs(f)), where n is the number of variables and bs(f) is the block sensitivity. Then we exhibit a partial function f that matches this upper bound, fbs(f) = Omega(sqrt(n * bs(f))).

Andris Ambainis, Martins Kokainis, Krisjanis Prusis, and Jevgenijs Vihrovs. All Classical Adversary Methods are Equivalent for Total Functions. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ambainis_et_al:LIPIcs.STACS.2018.8, author = {Ambainis, Andris and Kokainis, Martins and Prusis, Krisjanis and Vihrovs, Jevgenijs}, title = {{All Classical Adversary Methods are Equivalent for Total Functions}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {8:1--8:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.8}, URN = {urn:nbn:de:0030-drops-84953}, doi = {10.4230/LIPIcs.STACS.2018.8}, annote = {Keywords: Randomized Query Complexity, Lower Bounds, Adversary Bounds, Fractional Block Sensitivity} }

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**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

We show a nearly quadratic separation between deterministic communication complexity and the logarithm of the partition number, which is essentially optimal. This improves upon a recent power 1.5 separation of Göös, Pitassi, and Watson (FOCS 2015). In query complexity, we establish a nearly quadratic separation between deterministic (and even randomized) query complexity and subcube partition complexity, which is also essentially optimal. We also establish a nearly power 1.5 separation between quantum query complexity and subcube partition complexity, the first superlinear separation between the two measures. Lastly, we show a quadratic separation between quantum query complexity and one-sided subcube partition complexity.
Our query complexity separations use the recent cheat sheet framework of Aaronson, Ben-David, and Kothari. Our query functions are built up in stages by alternating function composition with the cheat sheet construction. The communication complexity separation follows from "lifting" the query separation to communication complexity.

Andris Ambainis, Martins Kokainis, and Robin Kothari. Nearly Optimal Separations Between Communication (or Query) Complexity and Partitions. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ambainis_et_al:LIPIcs.CCC.2016.4, author = {Ambainis, Andris and Kokainis, Martins and Kothari, Robin}, title = {{Nearly Optimal Separations Between Communication (or Query) Complexity and Partitions}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {4:1--4:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.4}, URN = {urn:nbn:de:0030-drops-58471}, doi = {10.4230/LIPIcs.CCC.2016.4}, annote = {Keywords: Query Complexity, Communication Complexity, Subcube Partition Complexity, Partition Bound} }

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**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

We show an equivalence between 1-query quantum algorithms and representations by degree-2 polynomials. Namely, a partial Boolean function f is computable by a 1-query quantum algorithm with error bounded by epsilon<1/2 iff f can be approximated by a degree-2 polynomial with error bounded by epsilon'<1/2. This result holds for two different notions of approximation by a polynomial: the standard definition of Nisan and Szegedy and the approximation by block-multilinear polynomials recently introduced by Aaronson and Ambainis [Aaronson/Ambainis, STOC 2015]. The proof uses Grothendieck's inequality to relate two matrix norms, with one norm corresponding to polynomial approximations and the other norm corresponding to quantum algorithms.
We also show two results for polynomials of higher degree. First, there is a total Boolean function which requires ~Omega(n) quantum queries but can be represented by a block-multilinear polynomial of degree ~O(sqrt(n)). Thus, in the general case (for an arbitrary number of queries), block-multilinear polynomials are not equivalent to quantum algorithms.
Second, for any constant degree k, the two notions of approximation by a polynomial (the standard and the block-multilinear) are equivalent. As a consequence, we solve an open problem from [Aaronson/Ambainis, STOC 2015], showing that one can estimate the value of any bounded degree-k polynomial p:{0,1}^n -> [-1,1] with O(n^{1-1/(2k)) queries.

Scott Aaronson, Andris Ambainis, Janis Iraids, Martins Kokainis, and Juris Smotrovs. Polynomials, Quantum Query Complexity, and Grothendieck's Inequality. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 25:1-25:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{aaronson_et_al:LIPIcs.CCC.2016.25, author = {Aaronson, Scott and Ambainis, Andris and Iraids, Janis and Kokainis, Martins and Smotrovs, Juris}, title = {{Polynomials, Quantum Query Complexity, and Grothendieck's Inequality}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {25:1--25:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.25}, URN = {urn:nbn:de:0030-drops-58394}, doi = {10.4230/LIPIcs.CCC.2016.25}, annote = {Keywords: quantum algorithms, Boolean functions, approximation by polynomials, Grothendieck's inequality} }

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