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**Published in:** LIPIcs, Volume 158, 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)

We study how efficiently a k-element set S⊆[n] can be learned from a uniform superposition |S> of its elements. One can think of |S>=∑_{i∈S}|i>/√|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> 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>. 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.

Srinivasan Arunachalam, Aleksandrs Belovs, Andrew M. Childs, Robin Kothari, Ansis Rosmanis, and Ronald de Wolf. Quantum Coupon Collector. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 158, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{arunachalam_et_al:LIPIcs.TQC.2020.10, author = {Arunachalam, Srinivasan and Belovs, Aleksandrs and Childs, Andrew M. and Kothari, Robin and Rosmanis, Ansis and de Wolf, Ronald}, title = {{Quantum Coupon Collector}}, booktitle = {15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)}, pages = {10:1--10:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-146-7}, ISSN = {1868-8969}, year = {2020}, volume = {158}, editor = {Flammia, Steven T.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2020.10}, URN = {urn:nbn:de:0030-drops-120692}, doi = {10.4230/LIPIcs.TQC.2020.10}, annote = {Keywords: Quantum algorithms, Adversary method, Coupon collector, Quantum learning theory} }

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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

The index erasure problem is a quantum state generation problem that asks a quantum computer to prepare a uniform superposition over the image of an injective function given by an oracle. We prove a tight Ω(√n) lower bound on the quantum query complexity of the non-coherent case of the problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state. This resolves an open question of Ambainis, Magnin, Roetteler, and Roland (CCC 2011), who gave a tight bound for the coherent case, the case where the ancillary memory must return to its initial state.
To prove our main result, we first extend the so-called automorphism principle (Høyer et al. STOC 2007) to the general adversary method for state conversion problems (Lee et al. STOC 2011), which allows one to exploit the symmetries of these problems to lower bound their quantum query complexity. Using this method, we establish a strong connection between the quantum query complexity of non-coherent symmetric state generation problems and the well-known Krein parameters of association schemes. Krein parameters are usually hard to determine, nevertheless, we give a novel way of computing certain Krein parameters of a commutative association scheme defined over partial permutations. We believe the study of this association scheme may also be of independent interest.

Nathan Lindzey and Ansis Rosmanis. A Tight Lower Bound For Non-Coherent Index Erasure. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 59:1-59:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lindzey_et_al:LIPIcs.ITCS.2020.59, author = {Lindzey, Nathan and Rosmanis, Ansis}, title = {{A Tight Lower Bound For Non-Coherent Index Erasure}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {59:1--59:37}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.59}, URN = {urn:nbn:de:0030-drops-117446}, doi = {10.4230/LIPIcs.ITCS.2020.59}, annote = {Keywords: General Adversary Method, Quantum Query Complexity, Association Schemes, Krein Parameters, Representation Theory} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

There are two central models considered in (fault-free synchronous) distributed computing: the CONGEST model, in which communication channels have limited bandwidth, and the LOCAL model, in which communication channels have unlimited bandwidth. Very recently, Le Gall and Magniez (PODC 2018) showed the superiority of quantum distributed computing over classical distributed computing in the CONGEST model. In this work we show the superiority of quantum distributed computing in the LOCAL model: we exhibit two computational tasks that can be solved in a constant number of rounds in the quantum setting but require Omega(n) rounds in the classical (randomized) setting, where n denotes the size of the network.

François Le Gall, Harumichi Nishimura, and Ansis Rosmanis. Quantum Advantage for the LOCAL Model in Distributed Computing. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 49:1-49:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{legall_et_al:LIPIcs.STACS.2019.49, author = {Le Gall, Fran\c{c}ois and Nishimura, Harumichi and Rosmanis, Ansis}, title = {{Quantum Advantage for the LOCAL Model in Distributed Computing}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {49:1--49:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.49}, URN = {urn:nbn:de:0030-drops-102887}, doi = {10.4230/LIPIcs.STACS.2019.49}, annote = {Keywords: Quantum computing, distributed computing, LOCAL model} }

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**Published in:** LIPIcs, Volume 111, 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018)

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 x 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 Omega(n^{1/3}) and Omega(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 from [Belovs, 2018].

Aleksandrs Belovs and Ansis Rosmanis. Quantum Lower Bounds for Tripartite Versions of the Hidden Shift and the Set Equality Problems. In 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 111, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{belovs_et_al:LIPIcs.TQC.2018.3, author = {Belovs, Aleksandrs and Rosmanis, Ansis}, title = {{Quantum Lower Bounds for Tripartite Versions of the Hidden Shift and the Set Equality Problems}}, booktitle = {13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018)}, pages = {3:1--3:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-080-4}, ISSN = {1868-8969}, year = {2018}, volume = {111}, editor = {Jeffery, Stacey}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2018.3}, URN = {urn:nbn:de:0030-drops-92501}, doi = {10.4230/LIPIcs.TQC.2018.3}, annote = {Keywords: Adversary Bound, Dual Learning Graphs, Quantum Query Complexity, Representation Theory} }

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**Published in:** LIPIcs, Volume 73, 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)

We introduce a definition of the fidelity function for multi-round quantum strategies, which we call the strategy fidelity, that is a generalization of the fidelity function for quantum states. We provide many interesting properties of the strategy fidelity including a Fuchs-van de Graaf relationship with the strategy norm. We illustrate an operational interpretation of the strategy fidelity in the spirit of Uhlmann's Theorem and discuss its application to the security analysis of quantum protocols for interactive cryptographic tasks such as bit-commitment and oblivious string transfer. Our analysis is very general in the sense that the actions of the protocol need not be fully specified, which is in stark contrast to most other security proofs. Lastly, we provide a semidefinite programming formulation of the strategy fidelity.

Gus Gutoski, Ansis Rosmanis, and Jamie Sikora. Fidelity of Quantum Strategies with Applications to Cryptography. In 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 73, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gutoski_et_al:LIPIcs.TQC.2017.8, author = {Gutoski, Gus and Rosmanis, Ansis and Sikora, Jamie}, title = {{Fidelity of Quantum Strategies with Applications to Cryptography}}, booktitle = {12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)}, pages = {8:1--8:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-034-7}, ISSN = {1868-8969}, year = {2018}, volume = {73}, editor = {Wilde, Mark M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2017.8}, URN = {urn:nbn:de:0030-drops-85830}, doi = {10.4230/LIPIcs.TQC.2017.8}, annote = {Keywords: Quantum strategies, cryptography, fidelity, semidefinite programming} }

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