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**Published in:** Dagstuhl Reports, Volume 8, Issue 9 (2019)

This report documents the program and the outcomes of Dagstuhl Seminar
18381 "Quantum Programming Languages", which brought together researchers from quantum computing and classical programming languages.

Michele Mosca, Martin Roetteler, and Peter Selinger. Quantum Programming Languages (Dagstuhl Seminar 18381). In Dagstuhl Reports, Volume 8, Issue 9, pp. 112-132, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Article{mosca_et_al:DagRep.8.9.112, author = {Mosca, Michele and Roetteler, Martin and Selinger, Peter}, title = {{Quantum Programming Languages (Dagstuhl Seminar 18381)}}, pages = {112--132}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2019}, volume = {8}, number = {9}, editor = {Mosca, Michele and Roetteler, Martin and Selinger, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.8.9.112}, URN = {urn:nbn:de:0030-drops-103291}, doi = {10.4230/DagRep.8.9.112}, annote = {Keywords: compilers, functional programming, quantum computing, reversible computing, verification} }

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

Integer arithmetic is the underpinning of many quantum algorithms, with applications ranging from Shor's algorithm over HHL for matrix inversion to Hamiltonian simulation algorithms. A basic objective is to keep the required resources to implement arithmetic as low as possible. This applies in particular to the number of qubits required in the implementation as for the foreseeable future this number is expected to be small. We present a reversible circuit for integer multiplication that is inspired by Karatsuba's recursive method. The main improvement over circuits that have been previously reported in the literature is an asymptotic reduction of the amount of space required from O(n^1.585) to O(n^1.427). This improvement is obtained in exchange for a small constant increase in the number of operations by a factor less than 2 and a small asymptotic increase in depth for the parallel version. The asymptotic improvement are obtained from analyzing pebble games on complete ternary trees.

Alex Parent, Martin Roetteler, and Michele Mosca. Improved reversible and quantum circuits for Karatsuba-based integer multiplication. In 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 73, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{parent_et_al:LIPIcs.TQC.2017.7, author = {Parent, Alex and Roetteler, Martin and Mosca, Michele}, title = {{Improved reversible and quantum circuits for Karatsuba-based integer multiplication}}, booktitle = {12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)}, pages = {7:1--7:15}, 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.2017.7}, URN = {urn:nbn:de:0030-drops-85841}, doi = {10.4230/LIPIcs.TQC.2017.7}, annote = {Keywords: Quantum algorithms, reversible circuits, quantum circuits, integer multiplication, pebble games, Karatsuba's method} }

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**Published in:** LIPIcs, Volume 61, 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)

Difference sets are basic combinatorial structures that have applications in signal processing, coding theory, and cryptography. We consider the problem of identifying a shifted version of the characteristic function of a (known) difference set and present a general algorithm that can be used to tackle any hidden shift problem for any difference set in any abelian group. We discuss special cases of this framework which include a) Paley difference sets based on quadratic residues in finite fields which allow to recover the shifted Legendre function quantum algorithm, b) Hadamard difference sets which allow to recover the shifted bent function quantum algorithm, and c) Singer difference sets which allow us to define instances of the dihedral hidden subgroup problem which can be efficiently solved on a quantum computer.

Martin Roetteler. Quantum Algorithms for Abelian Difference Sets and Applications to Dihedral Hidden Subgroups. In 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 61, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{roetteler:LIPIcs.TQC.2016.8, author = {Roetteler, Martin}, title = {{Quantum Algorithms for Abelian Difference Sets and Applications to Dihedral Hidden Subgroups}}, booktitle = {11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)}, pages = {8:1--8:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-019-4}, ISSN = {1868-8969}, year = {2016}, volume = {61}, editor = {Broadbent, Anne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2016.8}, URN = {urn:nbn:de:0030-drops-66896}, doi = {10.4230/LIPIcs.TQC.2016.8}, annote = {Keywords: Quantum algorithms, hidden shift problem, hidden subgroup problem, difference sets, Fourier transforms} }

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**Published in:** Dagstuhl Reports, Volume 5, Issue 9 (2016)

This report documents the program and the outcomes of Dagstuhl Seminar 15371 "Quantum Cryptanalysis". In this seminar, participants explored the impact that quantum algorithms will have on cryptology once a large-scale quantum computer becomes available. Research highlights in this seminar included both computational resource requirement and availability estimates for meaningful quantum cryptanalytic attacks against conventional cryptography, as well as the security of proposed quantum-safe cryptosystems against emerging quantum cryptanalytic attacks.

Michele Mosca, Martin Roetteler, Nicolas Sendrier, and Rainer Steinwandt. Quantum Cryptanalysis (Dagstuhl Seminar 15371). In Dagstuhl Reports, Volume 5, Issue 9, pp. 1-17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@Article{mosca_et_al:DagRep.5.9.1, author = {Mosca, Michele and Roetteler, Martin and Sendrier, Nicolas and Steinwandt, Rainer}, title = {{Quantum Cryptanalysis (Dagstuhl Seminar 15371)}}, pages = {1--17}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2016}, volume = {5}, number = {9}, editor = {Mosca, Michele and Roetteler, Martin and Sendrier, Nicolas and Steinwandt, Rainer}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.5.9.1}, URN = {urn:nbn:de:0030-drops-56825}, doi = {10.4230/DagRep.5.9.1}, annote = {Keywords: Cryptography, Quantum computing, Post-quantum cryptography, Quantum algorithms, Cryptanalysis, Computational algebra, Quantum circuit complexity, Quantum hardware and resource estimation} }

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**Published in:** LIPIcs, Volume 27, 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)

Network coding is a technique to maximize communication rates within a network, in communication protocols for simultaneous multi-party transmission of information. Linear network codes are examples of such protocols in which the local computations performed at the nodes in the network are limited to linear transformations of their input data (represented as elements of a ring, such as the integers modulo 2). The quantum linear network coding protocols of Kobayashi et al. coherently simulate classical linear network codes, using supplemental classical communication. We demonstrate that these protocols correspond in a natural way to measurement-based quantum computations with graph states over qudits having a structure directly related to the network.

Niel de Beaudrap and Martin Roetteler. Quantum Linear Network Coding as One-way Quantum Computation. In 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 27, pp. 217-233, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{debeaudrap_et_al:LIPIcs.TQC.2014.217, author = {de Beaudrap, Niel and Roetteler, Martin}, title = {{Quantum Linear Network Coding as One-way Quantum Computation}}, booktitle = {9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)}, pages = {217--233}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-73-6}, ISSN = {1868-8969}, year = {2014}, volume = {27}, editor = {Flammia, Steven T. and Harrow, Aram W.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2014.217}, URN = {urn:nbn:de:0030-drops-48189}, doi = {10.4230/LIPIcs.TQC.2014.217}, annote = {Keywords: Network coding, quantum computing, measurement-based computation, simulation} }

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**Published in:** LIPIcs, Volume 22, 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)

We study the quantum query complexity of the Boolean hidden shift problem. Given oracle access to f(x+s) for a known Boolean function f, the task is to determine the n-bit string s. The quantum query complexity of this problem depends strongly on f. We demonstrate that the easiest instances of this problem correspond to bent functions, in the sense that an exact one-query algorithm exists if and only if the function is bent. We partially characterize the hardest instances, which include delta functions. Moreover, we show that the problem is easy for random functions, since two queries suffice. Our algorithm for random functions is based on performing the pretty good measurement on several copies of a certain state; its analysis relies on the Fourier transform. We also use this approach to improve the quantum rejection sampling approach to the Boolean hidden shift problem.

Andrew M. Childs, Robin Kothari, Maris Ozols, and Martin Roetteler. Easy and Hard Functions for the Boolean Hidden Shift Problem. In 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 22, pp. 50-79, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{childs_et_al:LIPIcs.TQC.2013.50, author = {Childs, Andrew M. and Kothari, Robin and Ozols, Maris and Roetteler, Martin}, title = {{Easy and Hard Functions for the Boolean Hidden Shift Problem}}, booktitle = {8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)}, pages = {50--79}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-55-2}, ISSN = {1868-8969}, year = {2013}, volume = {22}, editor = {Severini, Simone and Brandao, Fernando}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2013.50}, URN = {urn:nbn:de:0030-drops-43203}, doi = {10.4230/LIPIcs.TQC.2013.50}, annote = {Keywords: Boolean hidden shift problem, quantum algorithms, query complexity, Fourier transform, bent functions} }

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**Published in:** LIPIcs, Volume 22, 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)

We investigate the problem of finding the minimum number of queries needed to perfectly identify an unknown quantum gate within a finite set of alternatives, considering both deterministic strategies. For unambiguous gate discrimination, where errors are not tolerated but inconclusive outcomes are allowed, we prove that parallel strategies are sufficient to identify the unknown gate with minimum number of queries and we use this fact to provide upper and lower bounds on the query complexity. In addition, we introduce the notion of generalized $t$-designs, which includes unitary t-designs and group representations as special cases. For gates forming a generalized $t$-design we prove that there is no difference between perfect probabilistic and perfect deterministic gate discrimination. Hence, evaluating of the query complexity of perfect discrimination is reduced to the easier problem of evaluating the query complexity of unambiguous discrimination.

Giulio Chiribella, Giacomo Mauro D'Ariano, and Martin Roetteler. On the Query Complexity of Perfect Gate Discrimination. In 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 22, pp. 178-191, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{chiribella_et_al:LIPIcs.TQC.2013.178, author = {Chiribella, Giulio and D'Ariano, Giacomo Mauro and Roetteler, Martin}, title = {{On the Query Complexity of Perfect Gate Discrimination}}, booktitle = {8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)}, pages = {178--191}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-55-2}, ISSN = {1868-8969}, year = {2013}, volume = {22}, editor = {Severini, Simone and Brandao, Fernando}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2013.178}, URN = {urn:nbn:de:0030-drops-43133}, doi = {10.4230/LIPIcs.TQC.2013.178}, annote = {Keywords: quantum gate identification, unambiguous discrimination, minimum error discrimination, query complexity} }

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