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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

The general adversary dual is a powerful tool in quantum computing because it gives a query-optimal bounded-error quantum algorithm for deciding any Boolean function. Unfortunately, the algorithm uses linear qubits in the worst case, and only works if the constraints of the general adversary dual are exactly satisfied. The challenge of improving the algorithm is that it is brittle to arbitrarily small errors since it relies on a reflection over a span of vectors. We overcome this challenge and build a robust dual adversary algorithm that can handle approximately satisfied constraints. As one application of our robust algorithm, we prove that for any Boolean function with polynomially many 1-valued inputs (or in fact a slightly weaker condition) there is a query-optimal algorithm that uses logarithmic qubits. As another application, we prove that numerically derived, approximate solutions to the general adversary dual give a bounded-error quantum algorithm under certain conditions. Further, we show that these conditions empirically hold with reasonable iterations for Boolean functions with small domains. We also develop several tools that may be of independent interest, including a robust approximate spectral gap lemma, a method to compress a general adversary dual solution using the Johnson-Lindenstrauss lemma, and open-source code to find solutions to the general adversary dual.

Michael Czekanski, Shelby Kimmel, and R. Teal Witter. Robust and Space-Efficient Dual Adversary Quantum Query Algorithms. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 36:1-36:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{czekanski_et_al:LIPIcs.ESA.2023.36, author = {Czekanski, Michael and Kimmel, Shelby and Witter, R. Teal}, title = {{Robust and Space-Efficient Dual Adversary Quantum Query Algorithms}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {36:1--36:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.36}, URN = {urn:nbn:de:0030-drops-186890}, doi = {10.4230/LIPIcs.ESA.2023.36}, annote = {Keywords: Quantum Computing, Robust Quantum Algorithms, Johnson-Lindenstrauss Lemma, Span Programs, Query Complexity, Space Complexity} }

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

We present a quantum algorithm for sampling an edge on a path between two nodes s and t in an undirected graph given as an adjacency matrix, and show that this can be done in query complexity that is asymptotically the same, up to log factors, as the query complexity of detecting a path between s and t. We use this path sampling algorithm as a subroutine for st-path finding and st-cut-set finding algorithms in some specific cases. Our main technical contribution is an algorithm for generating a quantum state that is proportional to the positive witness vector of a span program.

Stacey Jeffery, Shelby Kimmel, and Alvaro Piedrafita. Quantum Algorithm for Path-Edge Sampling. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 5:1-5:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jeffery_et_al:LIPIcs.TQC.2023.5, author = {Jeffery, Stacey and Kimmel, Shelby and Piedrafita, Alvaro}, title = {{Quantum Algorithm for Path-Edge Sampling}}, booktitle = {18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)}, pages = {5:1--5:28}, 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.2023.5}, URN = {urn:nbn:de:0030-drops-183151}, doi = {10.4230/LIPIcs.TQC.2023.5}, annote = {Keywords: Algorithm design and analysis, Query complexity, Graph algorithms, Span program algorithm, Path finding, Path detection} }

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**Published in:** LIPIcs, Volume 135, 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)

We present quantum algorithms for various problems related to graph connectivity. We give simple and query-optimal algorithms for cycle detection and odd-length cycle detection (bipartiteness) using a reduction to st-connectivity. Furthermore, we show that our algorithm for cycle detection has improved performance under the promise of large circuit rank or a small number of edges. We also provide algorithms for detecting even-length cycles and for estimating the circuit rank of a graph. All of our algorithms have logarithmic space complexity.

Kai DeLorenzo, Shelby Kimmel, and R. Teal Witter. Applications of the Quantum Algorithm for st-Connectivity. In 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 135, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{delorenzo_et_al:LIPIcs.TQC.2019.6, author = {DeLorenzo, Kai and Kimmel, Shelby and Witter, R. Teal}, title = {{Applications of the Quantum Algorithm for st-Connectivity}}, booktitle = {14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)}, pages = {6:1--6:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-112-2}, ISSN = {1868-8969}, year = {2019}, volume = {135}, editor = {van Dam, Wim and Man\v{c}inska, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2019.6}, URN = {urn:nbn:de:0030-drops-103984}, doi = {10.4230/LIPIcs.TQC.2019.6}, annote = {Keywords: graphs, algorithms, query complexity, quantum algorithms, span programs} }

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

We study the ability of efficient quantum verifiers to decide properties of exponentially large subsets given either a classical or quantum witness. We develop a general framework that can be used to prove that QCMA machines, with only classical witnesses, cannot verify certain properties of subsets given implicitly via an oracle. We use this framework to prove an oracle separation between QCMA and QMA using an "in-place" permutation oracle, making the first progress on this question since Aaronson and Kuperberg in 2007 [Aaronson and Kuperberg, 2007]. We also use the framework to prove a particularly simple standard oracle separation between QCMA and AM.

Bill Fefferman and Shelby Kimmel. Quantum vs. Classical Proofs and Subset Verification. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 22:1-22:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fefferman_et_al:LIPIcs.MFCS.2018.22, author = {Fefferman, Bill and Kimmel, Shelby}, title = {{Quantum vs. Classical Proofs and Subset Verification}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {22:1--22:23}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.22}, URN = {urn:nbn:de:0030-drops-96040}, doi = {10.4230/LIPIcs.MFCS.2018.22}, annote = {Keywords: Quantum Complexity Theory, Quantum Proofs} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

An important family of span programs, st-connectivity span programs, have been used to design quantum algorithms in various contexts, including a number of graph problems and formula evaluation problems. The complexity of the resulting algorithms depends on the largest positive witness size of any 1-input, and the largest negative witness size of any 0-input. Belovs and Reichardt first showed that the positive witness size is exactly characterized by the effective resistance of the input graph, but only rough upper bounds were known previously on the negative witness size. We show that the negative witness size in an st-connectivity span program is exactly characterized by the capacitance of the input graph. This gives a tight analysis for algorithms based on st-connectivity span programs on any set of inputs.
We use this analysis to give a new quantum algorithm for estimating the capacitance of a graph. We also describe a new quantum algorithm for deciding if a graph is connected, which improves the previous best quantum algorithm for this problem if we're promised that either the graph has at least kappa>1 components, or the graph is connected and has small average resistance, which is upper bounded by the diameter. We also give an alternative algorithm for deciding if a graph is connected that can be better than our first algorithm when the maximum degree is small. Finally, using ideas from our second connectivity algorithm, we give an algorithm for estimating the algebraic connectivity of a graph, the second largest eigenvalue of the Laplacian.

Michael Jarret, Stacey Jeffery, Shelby Kimmel, and Alvaro Piedrafita. Quantum Algorithms for Connectivity and Related Problems. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 49:1-49:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{jarret_et_al:LIPIcs.ESA.2018.49, author = {Jarret, Michael and Jeffery, Stacey and Kimmel, Shelby and Piedrafita, Alvaro}, title = {{Quantum Algorithms for Connectivity and Related Problems}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {49:1--49:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.49}, URN = {urn:nbn:de:0030-drops-95121}, doi = {10.4230/LIPIcs.ESA.2018.49}, annote = {Keywords: Electrical networks, Quantum algorithms, Span programs, Connectivity, Graph theory} }

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**Published in:** LIPIcs, Volume 44, 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)

While powerful tools have been developed to analyze quantum query complexity, there are still many natural problems that do not fit neatly into the black box model of oracles. We create a new model that allows multiple oracles with differing costs. This model captures more of the difficulty of certain natural problems. We test this model on a simple problem, Search with Two Oracles, for which we create a quantum algorithm that we prove is asymptotically optimal. We further give some evidence, using a geometric picture of Grover's algorithm, that our algorithm is exactly optimal.

Shelby Kimmel, Cedric Yen-Yu Lin, and Han-Hsuan Lin. Oracles with Costs. In 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 44, pp. 1-26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kimmel_et_al:LIPIcs.TQC.2015.1, author = {Kimmel, Shelby and Lin, Cedric Yen-Yu and Lin, Han-Hsuan}, title = {{Oracles with Costs}}, booktitle = {10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)}, pages = {1--26}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-96-5}, ISSN = {1868-8969}, year = {2015}, volume = {44}, editor = {Beigi, Salman and K\"{o}nig, Robert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2015.1}, URN = {urn:nbn:de:0030-drops-55459}, doi = {10.4230/LIPIcs.TQC.2015.1}, annote = {Keywords: Quantum Algorithms, Query Complexity, Amplitude Amplification} }