31 Search Results for "Jeffery, Stacey"

Volume

LIPIcs, Volume 111

13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018)

TQC 2018, July 16, 2018, Sydney, Australia

Editors: Stacey Jeffery

Document
DOI: 10.4230/LIPIcs.STACS.2025.54
Multidimensional Quantum Walks, Recursion, and Quantum Divide & Conquer

Authors: Stacey Jeffery and Galina Pass

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
We introduce an object called a subspace graph that formalizes the technique of multidimensional quantum walks. Composing subspace graphs allows one to seamlessly combine quantum and classical reasoning, keeping a classical structure in mind, while abstracting quantum parts into subgraphs with simple boundaries as needed. As an example, we show how to combine a switching network with arbitrary quantum subroutines, to compute a composed function. As another application, we give a time-efficient implementation of quantum Divide & Conquer when the sub-problems are combined via a Boolean formula. We use this to quadratically speed up Savitch’s algorithm for directed st-connectivity.
Cite as

Stacey Jeffery and Galina Pass. Multidimensional Quantum Walks, Recursion, and Quantum Divide & Conquer. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jeffery_et_al:LIPIcs.STACS.2025.54,
  author =	{Jeffery, Stacey and Pass, Galina},
  title =	{{Multidimensional Quantum Walks, Recursion, and Quantum Divide \& Conquer}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{54:1--54:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.54},
  URN =		{urn:nbn:de:0030-drops-228791},
  doi =		{10.4230/LIPIcs.STACS.2025.54},
  annote =	{Keywords: Quantum Divide \& Conquer, Time-Efficient, Subspace Graphs, Quantum Walks, Switching Networks, Directed st-Connectivity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.31
Quantum Advantage and Lower Bounds in Parallel Query Complexity

Authors: Joseph Carolan, Amin Shiraz Gilani, and Mahathi Vempati

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
It is well known that quantum, randomized and deterministic (sequential) query complexities are polynomially related for total boolean functions. We find that significantly larger separations between the parallel generalizations of these measures are possible. In particular, 1) We employ the cheatsheet framework to obtain an unbounded parallel quantum query advantage over its randomized analogue for a total function, falsifying a conjecture of [https://arxiv.org/abs/1309.6116]. 2) We strengthen 1 by constructing a total function which exhibits an unbounded parallel quantum query advantage despite having no sequential advantage, suggesting that genuine quantum advantage could occur entirely due to parallelism. 3) We construct a total function that exhibits a polynomial separation between 2-round quantum and randomized query complexities, contrasting a result of [https://arxiv.org/abs/1001.0018] that there is at most a constant separation for 1-round (nonadaptive) algorithms. 4) We develop a new technique for deriving parallel quantum lower bounds from sequential upper bounds. We employ this technique to give lower bounds for Boolean symmetric functions and read-once formulas, ruling out large parallel query advantages for them. We also provide separations between randomized and deterministic parallel query complexities analogous to items 1-3.
Cite as

Joseph Carolan, Amin Shiraz Gilani, and Mahathi Vempati. Quantum Advantage and Lower Bounds in Parallel Query Complexity. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{carolan_et_al:LIPIcs.ITCS.2025.31,
  author =	{Carolan, Joseph and Gilani, Amin Shiraz and Vempati, Mahathi},
  title =	{{Quantum Advantage and Lower Bounds in Parallel Query Complexity}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{31:1--31:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.31},
  URN =		{urn:nbn:de:0030-drops-226597},
  doi =		{10.4230/LIPIcs.ITCS.2025.31},
  annote =	{Keywords: Computational complexity theory, quantum, lower bounds, parallel}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.10
(No) Quantum Space-Time Tradeoff for USTCON

Authors: Simon Apers, Stacey Jeffery, Galina Pass, and Michael Walter

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

Abstract
Undirected st-connectivity is important both for its applications in network problems, and for its theoretical connections with logspace complexity. Classically, a long line of work led to a time-space tradeoff of T = Õ(n²/S) for any S such that S = Ω(log(n)) and S = O(n²/m). Surprisingly, we show that quantumly there is no nontrivial time-space tradeoff: there is a quantum algorithm that achieves both optimal time Õ(n) and space O(log(n)) simultaneously. This improves on previous results, which required either O(log(n)) space and Õ(n^{1.5}) time, or Õ(n) space and time. To complement this, we show that there is a nontrivial time-space tradeoff when given a lower bound on the spectral gap of a corresponding random walk.
Cite as

Simon Apers, Stacey Jeffery, Galina Pass, and Michael Walter. (No) Quantum Space-Time Tradeoff for USTCON. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{apers_et_al:LIPIcs.ESA.2023.10,
  author =	{Apers, Simon and Jeffery, Stacey and Pass, Galina and Walter, Michael},
  title =	{{(No) Quantum Space-Time Tradeoff for USTCON}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{10:1--10:17},
  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.2023.10},
  URN =		{urn:nbn:de:0030-drops-186636},
  doi =		{10.4230/LIPIcs.ESA.2023.10},
  annote =	{Keywords: Undirected st-connectivity, quantum walks, time-space tradeoff}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.36
Robust and Space-Efficient Dual Adversary Quantum Query Algorithms

Authors: Michael Czekanski, Shelby Kimmel, and R. Teal Witter

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.TQC.2023.5
Quantum Algorithm for Path-Edge Sampling

Authors: Stacey Jeffery, Shelby Kimmel, and Alvaro Piedrafita

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/DagRep.11.9.64
Quantum Cryptanalysis (Dagstuhl Seminar 21421)

Authors: Stacey Jeffery, Michele Mosca, Maria Naya-Plasencia, and Rainer Steinwandt

Published in: Dagstuhl Reports, Volume 11, Issue 9 (2022)

Abstract
This seminar report documents the program and the outcomes of Dagstuhl Seminar 21421 Quantum Cryptanalysis. The seminar took place in a hybrid format in Fall 2021. The report starts out with the motivation and comments on the organization of this instance of the Dagstuhl Seminar series on {Quantum Cryptanalysis}, followed by abstracts of presentations. The presentation abstracts were provided by seminar participants.
Cite as

Stacey Jeffery, Michele Mosca, Maria Naya-Plasencia, and Rainer Steinwandt. Quantum Cryptanalysis (Dagstuhl Seminar 21421). In Dagstuhl Reports, Volume 11, Issue 9, pp. 64-79, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Article{jeffery_et_al:DagRep.11.9.64,
  author =	{Jeffery, Stacey and Mosca, Michele and Naya-Plasencia, Maria and Steinwandt, Rainer},
  title =	{{Quantum Cryptanalysis (Dagstuhl Seminar 21421)}},
  pages =	{64--79},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2022},
  volume =	{11},
  number =	{9},
  editor =	{Jeffery, Stacey and Mosca, Michele and Naya-Plasencia, Maria 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.11.9.64},
  URN =		{urn:nbn:de:0030-drops-159187},
  doi =		{10.4230/DagRep.11.9.64},
  annote =	{Keywords: computational algebra, post-quantum cryptography, quantum computing, quantum resource estimation}
}
Document
DOI: 10.4230/LIPIcs.ITC.2021.21
Quantum-Access Security of the Winternitz One-Time Signature Scheme

Authors: Christian Majenz, Chanelle Matadah Manfouo, and Maris Ozols

Published in: LIPIcs, Volume 199, 2nd Conference on Information-Theoretic Cryptography (ITC 2021)

Abstract
Quantum-access security, where an attacker is granted superposition access to secret-keyed functionalities, is a fundamental security model and its study has inspired results in post-quantum security. We revisit, and fill a gap in, the quantum-access security analysis of the Lamport one-time signature scheme (OTS) in the quantum random oracle model (QROM) by Alagic et al. (Eurocrypt 2020). We then go on to generalize the technique to the Winternitz OTS. Along the way, we develop a tool for the analysis of hash chains in the QROM based on the superposition oracle technique by Zhandry (Crypto 2019) which might be of independent interest.
Cite as

Christian Majenz, Chanelle Matadah Manfouo, and Maris Ozols. Quantum-Access Security of the Winternitz One-Time Signature Scheme. In 2nd Conference on Information-Theoretic Cryptography (ITC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 199, pp. 21:1-21:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{majenz_et_al:LIPIcs.ITC.2021.21,
  author =	{Majenz, Christian and Manfouo, Chanelle Matadah and Ozols, Maris},
  title =	{{Quantum-Access Security of the Winternitz One-Time Signature Scheme}},
  booktitle =	{2nd Conference on Information-Theoretic Cryptography (ITC 2021)},
  pages =	{21:1--21:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-197-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{199},
  editor =	{Tessaro, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2021.21},
  URN =		{urn:nbn:de:0030-drops-143406},
  doi =		{10.4230/LIPIcs.ITC.2021.21},
  annote =	{Keywords: quantum cryptography, one-time signature schemes, quantum random oracle model, post-quantum cryptography, quantum world, hash-based signatures, information-theoretic security}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.6
A Unified Framework of Quantum Walk Search

Authors: Simon Apers, András Gilyén, and Stacey Jeffery

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
Many quantum algorithms critically rely on quantum walk search, or the use of quantum walks to speed up search problems on graphs. However, the main results on quantum walk search are scattered over different, incomparable frameworks, such as the hitting time framework, the MNRS framework, and the electric network framework. As a consequence, a number of pieces are currently missing. For example, recent work by Ambainis et al. (STOC'20) shows how quantum walks starting from the stationary distribution can always find elements quadratically faster. In contrast, the electric network framework allows quantum walks to start from an arbitrary initial state, but it only detects marked elements. We present a new quantum walk search framework that unifies and strengthens these frameworks, leading to a number of new results. For example, the new framework effectively finds marked elements in the electric network setting. The new framework also allows to interpolate between the hitting time framework, minimizing the number of walk steps, and the MNRS framework, minimizing the number of times elements are checked for being marked. This allows for a more natural tradeoff between resources. In addition to quantum walks and phase estimation, our new algorithm makes use of quantum fast-forwarding, similar to the recent results by Ambainis et al. This perspective also enables us to derive more general complexity bounds on the quantum walk algorithms, e.g., based on Monte Carlo type bounds of the corresponding classical walk. As a final result, we show how in certain cases we can avoid the use of phase estimation and quantum fast-forwarding, answering an open question of Ambainis et al.
Cite as

Simon Apers, András Gilyén, and Stacey Jeffery. A Unified Framework of Quantum Walk Search. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{apers_et_al:LIPIcs.STACS.2021.6,
  author =	{Apers, Simon and Gily\'{e}n, Andr\'{a}s and Jeffery, Stacey},
  title =	{{A Unified Framework of Quantum Walk Search}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{6:1--6:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.6},
  URN =		{urn:nbn:de:0030-drops-136511},
  doi =		{10.4230/LIPIcs.STACS.2021.6},
  annote =	{Keywords: Quantum Algorithms, Quantum Walks, Graph Theory}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.26
Span Programs and Quantum Time Complexity

Authors: Arjan Cornelissen, Stacey Jeffery, Maris Ozols, and Alvaro Piedrafita

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract
Span programs are an important model of quantum computation due to their correspondence with quantum query and space complexity. While the query complexity of quantum algorithms obtained from span programs is well-understood, it is not generally clear how to implement certain query-independent operations in a time-efficient manner. In this work, we prove an analogous connection for quantum time complexity. In particular, we show how to convert a sufficiently-structured quantum algorithm for f with time complexity T into a span program for f such that it compiles back into a quantum algorithm for f with time complexity 𝒪̃(T). This shows that for span programs derived from algorithms with a time-efficient implementation, we can preserve the time efficiency when implementing the span program, which means that span programs capture time, query and space complexities and are a complete model of quantum algorithms. One practical advantage of being able to convert quantum algorithms to span programs in a way that preserves time complexity is that span programs compose very nicely. We demonstrate this by improving Ambainis’s variable-time quantum search result using our construction through a span program composition for the OR function.
Cite as

Arjan Cornelissen, Stacey Jeffery, Maris Ozols, and Alvaro Piedrafita. Span Programs and Quantum Time Complexity. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cornelissen_et_al:LIPIcs.MFCS.2020.26,
  author =	{Cornelissen, Arjan and Jeffery, Stacey and Ozols, Maris and Piedrafita, Alvaro},
  title =	{{Span Programs and Quantum Time Complexity}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{26:1--26:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.26},
  URN =		{urn:nbn:de:0030-drops-126947},
  doi =		{10.4230/LIPIcs.MFCS.2020.26},
  annote =	{Keywords: quantum query algorithms, span programs, variable-time quantum search}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.4
Span Programs and Quantum Space Complexity

Authors: Stacey Jeffery

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

Abstract
While quantum computers hold the promise of significant computational speedups, the limited size of early quantum machines motivates the study of space-bounded quantum computation. We relate the quantum space complexity of computing a function f with one-sided error to the logarithm of its span program size, a classical quantity that is well-studied in attempts to prove formula size lower bounds. In the more natural bounded error model, we show that the amount of space needed for a unitary quantum algorithm to compute f with bounded (two-sided) error is lower bounded by the logarithm of its approximate span program size. Approximate span programs were introduced in the field of quantum algorithms but not studied classically. However, the approximate span program size of a function is a natural generalization of its span program size. While no non-trivial lower bound is known on the span program size (or approximate span program size) of any concrete function, a number of lower bounds are known on the monotone span program size. We show that the approximate monotone span program size of f is a lower bound on the space needed by quantum algorithms of a particular form, called monotone phase estimation algorithms, to compute f. We then give the first non-trivial lower bound on the approximate span program size of an explicit function.
Cite as

Stacey Jeffery. Span Programs and Quantum Space Complexity. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 4:1-4:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{jeffery:LIPIcs.ITCS.2020.4,
  author =	{Jeffery, Stacey},
  title =	{{Span Programs and Quantum Space Complexity}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{4:1--4:37},
  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.2020.4},
  URN =		{urn:nbn:de:0030-drops-116896},
  doi =		{10.4230/LIPIcs.ITCS.2020.4},
  annote =	{Keywords: Quantum space complexity, span programs}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.9
Quantum Walk Sampling by Growing Seed Sets

Authors: Simon Apers

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
This work describes a new algorithm for creating a superposition over the edge set of a graph, encoding a quantum sample of the random walk stationary distribution. The algorithm requires a number of quantum walk steps scaling as O~(m^(1/3) delta^(-1/3)), with m the number of edges and delta the random walk spectral gap. This improves on existing strategies by initially growing a classical seed set in the graph, from which a quantum walk is then run. The algorithm leads to a number of improvements: (i) it provides a new bound on the setup cost of quantum walk search algorithms, (ii) it yields a new algorithm for st-connectivity, and (iii) it allows to create a superposition over the isomorphisms of an n-node graph in time O~(2^(n/3)), surpassing the Omega(2^(n/2)) barrier set by index erasure.
Cite as

Simon Apers. Quantum Walk Sampling by Growing Seed Sets. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 9:1-9:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{apers:LIPIcs.ESA.2019.9,
  author =	{Apers, Simon},
  title =	{{Quantum Walk Sampling by Growing Seed Sets}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{9:1--9:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola 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.2019.9},
  URN =		{urn:nbn:de:0030-drops-111300},
  doi =		{10.4230/LIPIcs.ESA.2019.9},
  annote =	{Keywords: Quantum algorithms, Quantum walks, Connectivity, Graph theory}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.33
The Power of Block-Encoded Matrix Powers: Improved Regression Techniques via Faster Hamiltonian Simulation

Authors: Shantanav Chakraborty, András Gilyén, and Stacey Jeffery

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

Abstract
We apply the framework of block-encodings, introduced by Low and Chuang (under the name standard-form), to the study of quantum machine learning algorithms and derive general results that are applicable to a variety of input models, including sparse matrix oracles and matrices stored in a data structure. We develop several tools within the block-encoding framework, such as singular value estimation of a block-encoded matrix, and quantum linear system solvers using block-encodings. The presented results give new techniques for Hamiltonian simulation of non-sparse matrices, which could be relevant for certain quantum chemistry applications, and which in turn imply an exponential improvement in the dependence on precision in quantum linear systems solvers for non-sparse matrices. In addition, we develop a technique of variable-time amplitude estimation, based on Ambainis' variable-time amplitude amplification technique, which we are also able to apply within the framework. As applications, we design the following algorithms: (1) a quantum algorithm for the quantum weighted least squares problem, exhibiting a 6-th power improvement in the dependence on the condition number and an exponential improvement in the dependence on the precision over the previous best algorithm of Kerenidis and Prakash; (2) the first quantum algorithm for the quantum generalized least squares problem; and (3) quantum algorithms for estimating electrical-network quantities, including effective resistance and dissipated power, improving upon previous work.
Cite as

Shantanav Chakraborty, András Gilyén, and Stacey Jeffery. The Power of Block-Encoded Matrix Powers: Improved Regression Techniques via Faster Hamiltonian Simulation. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chakraborty_et_al:LIPIcs.ICALP.2019.33,
  author =	{Chakraborty, Shantanav and Gily\'{e}n, Andr\'{a}s and Jeffery, Stacey},
  title =	{{The Power of Block-Encoded Matrix Powers: Improved Regression Techniques via Faster Hamiltonian Simulation}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{33:1--33:14},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.33},
  URN =		{urn:nbn:de:0030-drops-106092},
  doi =		{10.4230/LIPIcs.ICALP.2019.33},
  annote =	{Keywords: Quantum algorithms, Hamiltonian simulation, Quantum machine learning}
}
Document
DOI: 10.4230/LIPIcs.TQC.2019.1
On Quantum Chosen-Ciphertext Attacks and Learning with Errors

Authors: Gorjan Alagic, Stacey Jeffery, Maris Ozols, and Alexander Poremba

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

Abstract
Quantum computing is a significant threat to classical public-key cryptography. In strong "quantum access" security models, numerous symmetric-key cryptosystems are also vulnerable. We consider classical encryption in a model which grants the adversary quantum oracle access to encryption and decryption, but where the latter is restricted to non-adaptive (i.e., pre-challenge) queries only. We define this model formally using appropriate notions of ciphertext indistinguishability and semantic security (which are equivalent by standard arguments) and call it QCCA1 in analogy to the classical CCA1 security model. Using a bound on quantum random-access codes, we show that the standard PRF-based encryption schemes are QCCA1-secure when instantiated with quantum-secure primitives. We then revisit standard IND-CPA-secure Learning with Errors (LWE) encryption and show that leaking just one quantum decryption query (and no other queries or leakage of any kind) allows the adversary to recover the full secret key with constant success probability. In the classical setting, by contrast, recovering the key requires a linear number of decryption queries. The algorithm at the core of our attack is a (large-modulus version of) the well-known Bernstein-Vazirani algorithm. We emphasize that our results should not be interpreted as a weakness of these cryptosystems in their stated security setting (i.e., post-quantum chosen-plaintext secrecy). Rather, our results mean that, if these cryptosystems are exposed to chosen-ciphertext attacks (e.g., as a result of deployment in an inappropriate real-world setting) then quantum attacks are even more devastating than classical ones.
Cite as

Gorjan Alagic, Stacey Jeffery, Maris Ozols, and Alexander Poremba. On Quantum Chosen-Ciphertext Attacks and Learning with Errors. In 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 135, pp. 1:1-1:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{alagic_et_al:LIPIcs.TQC.2019.1,
  author =	{Alagic, Gorjan and Jeffery, Stacey and Ozols, Maris and Poremba, Alexander},
  title =	{{On Quantum Chosen-Ciphertext Attacks and Learning with Errors}},
  booktitle =	{14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)},
  pages =	{1:1--1:23},
  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.2019.1},
  URN =		{urn:nbn:de:0030-drops-103939},
  doi =		{10.4230/LIPIcs.TQC.2019.1},
  annote =	{Keywords: quantum chosen-ciphertext security, quantum attacks, learning with errors}
}
Document
DOI: 10.4230/LIPIcs.TQC.2019.6
Applications of the Quantum Algorithm for st-Connectivity

Authors: Kai DeLorenzo, Shelby Kimmel, and R. Teal Witter

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

Abstract
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.
Cite as

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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