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Complete Volume

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

LIPIcs, Volume 266, TQC 2023, Complete Volume

18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 1-314, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Proceedings{fawzi_et_al:LIPIcs.TQC.2023, title = {{LIPIcs, Volume 266, TQC 2023, Complete Volume}}, booktitle = {18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)}, pages = {1--314}, 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}, URN = {urn:nbn:de:0030-drops-183099}, doi = {10.4230/LIPIcs.TQC.2023}, annote = {Keywords: LIPIcs, Volume 266, TQC 2023, Complete Volume} }

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Front Matter

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

Front Matter, Table of Contents, Preface, Conference Organization

18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fawzi_et_al:LIPIcs.TQC.2023.0, author = {Fawzi, Omar and Walter, Michael}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)}, pages = {0:i--0:xii}, 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.2023.0}, URN = {urn:nbn:de:0030-drops-183102}, doi = {10.4230/LIPIcs.TQC.2023.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

We introduce a classical algorithm to approximate the free energy of local, translation-invariant, one-dimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperature T = 0) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed temperature T > 0 in subpolynomial time, i.e., in time O((1/ε)^c) for any constant c > 0 where ε is the additive approximation error. Previously, the best known algorithm had a runtime that is polynomial in 1/ε where the degree of the polynomial is exponential in the inverse temperature 1/T. Our algorithm is also particularly simple as it reduces to the computation of the spectral radius of a linear map. This linear map has an interpretation as a noncommutative transfer matrix and has been studied previously to prove results on the analyticity of the free energy and the decay of correlations. We also show that the corresponding eigenvector of this map gives an approximation of the marginal of the Gibbs state and thereby allows for the computation of various thermodynamic properties of the quantum system.

Hamza Fawzi, Omar Fawzi, and Samuel O. Scalet. A Subpolynomial-Time Algorithm for the Free Energy of One-Dimensional Quantum Systems in the Thermodynamic Limit. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 49:1-49:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fawzi_et_al:LIPIcs.ITCS.2023.49, author = {Fawzi, Hamza and Fawzi, Omar and Scalet, Samuel O.}, title = {{A Subpolynomial-Time Algorithm for the Free Energy of One-Dimensional Quantum Systems in the Thermodynamic Limit}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {49:1--49:6}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.49}, URN = {urn:nbn:de:0030-drops-175520}, doi = {10.4230/LIPIcs.ITCS.2023.49}, annote = {Keywords: One-dimensional quantum systems, Free energy} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

There is a large and important collection of Ramsey-type combinatorial problems, closely related to central problems in complexity theory, that can be formulated in terms of the asymptotic growth of the size of the maximum independent sets in powers of a fixed small hypergraph, also called the Shannon capacity. An important instance of this is the corner problem studied in the context of multiparty communication complexity in the Number On the Forehead (NOF) model. Versions of this problem and the NOF connection have seen much interest (and progress) in recent works of Linial, Pitassi and Shraibman (ITCS 2019) and Linial and Shraibman (CCC 2021).
We introduce and study a general algebraic method for lower bounding the Shannon capacity of directed hypergraphs via combinatorial degenerations, a combinatorial kind of "approximation" of subgraphs that originates from the study of matrix multiplication in algebraic complexity theory (and which play an important role there) but which we use in a novel way.
Using the combinatorial degeneration method, we make progress on the corner problem by explicitly constructing a corner-free subset in F₂ⁿ × F₂ⁿ of size Ω(3.39ⁿ/poly(n)), which improves the previous lower bound Ω(2.82ⁿ) of Linial, Pitassi and Shraibman (ITCS 2019) and which gets us closer to the best upper bound 4^{n - o(n)}. Our new construction of corner-free sets implies an improved NOF protocol for the Eval problem. In the Eval problem over a group G, three players need to determine whether their inputs x₁, x₂, x₃ ∈ G sum to zero. We find that the NOF communication complexity of the Eval problem over F₂ⁿ is at most 0.24n + 𝒪(log n), which improves the previous upper bound 0.5n + 𝒪(log n).

Matthias Christandl, Omar Fawzi, Hoang Ta, and Jeroen Zuiddam. Larger Corner-Free Sets from Combinatorial Degenerations. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 48:1-48:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{christandl_et_al:LIPIcs.ITCS.2022.48, author = {Christandl, Matthias and Fawzi, Omar and Ta, Hoang and Zuiddam, Jeroen}, title = {{Larger Corner-Free Sets from Combinatorial Degenerations}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {48:1--48:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.48}, URN = {urn:nbn:de:0030-drops-156441}, doi = {10.4230/LIPIcs.ITCS.2022.48}, annote = {Keywords: Corner-free sets, communication complexity, number on the forehead, combinatorial degeneration, hypergraphs, Shannon capacity, eval problem} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

The threshold theorem is a fundamental result in the theory of fault-tolerant quantum computation stating that arbitrarily long quantum computations can be performed with a polylogarithmic overhead provided the noise level is below a constant level. A recent work by Fawzi, Grospellier and Leverrier (FOCS 2018) building on a result by Gottesman (QIC 2013) has shown that the space overhead can be asymptotically reduced to a constant independent of the circuit provided we only consider circuits with a length bounded by a polynomial in the width. In this work, using a minimal model for quantum fault tolerance, we establish a general lower bound on the space overhead required to achieve fault tolerance.
For any non-unitary qubit channel 𝒩 and any quantum fault tolerance schemes against i.i.d. noise modeled by 𝒩, we prove a lower bound of max{Q(𝒩)^{-1}n,α_𝒩 log T} on the number of physical qubits, for circuits of length T and width n. Here, Q(𝒩) denotes the quantum capacity of 𝒩 and α_𝒩 > 0 is a constant only depending on the channel 𝒩. In our model, we allow for qubits to be replaced by fresh ones during the execution of the circuit and in the case of unital noise, we allow classical computation to be free and perfect. This improves upon results that assumed classical computations to be also affected by noise, and that sometimes did not allow for fresh qubits to be added. Along the way, we prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude damping noise resolving a conjecture by Ben-Or, Gottesman and Hassidim (2013).

Omar Fawzi, Alexander Müller-Hermes, and Ala Shayeghi. A Lower Bound on the Space Overhead of Fault-Tolerant Quantum Computation. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 68:1-68:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fawzi_et_al:LIPIcs.ITCS.2022.68, author = {Fawzi, Omar and M\"{u}ller-Hermes, Alexander and Shayeghi, Ala}, title = {{A Lower Bound on the Space Overhead of Fault-Tolerant Quantum Computation}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {68:1--68:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.68}, URN = {urn:nbn:de:0030-drops-156649}, doi = {10.4230/LIPIcs.ITCS.2022.68}, annote = {Keywords: Fault-tolerant quantum computation, quantum error correction} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

In a recent landmark result [Ji et al., arXiv:2001.04383 (2020)], it was shown that approximating the value of a two-player game is undecidable when the players are allowed to share quantum states of unbounded dimension. In this paper, we study the computational complexity of two-player games when the dimension of the quantum systems is bounded by T. More specifically, we give a semidefinite program of size exp(𝒪(T^{12}(log²(AT)+log(Q)log(AT))/ε²)) to compute additive ε-approximations on the value of two-player free games with T× T-dimensional quantum entanglement, where A and Q denote the number of answers and questions of the game, respectively. For fixed dimension T, this scales polynomially in Q and quasi-polynomially in A, thereby improving on previously known approximation algorithms for which worst-case run-time guarantees are at best exponential in Q and A. For the proof, we make a connection to the quantum separability problem and employ improved multipartite quantum de Finetti theorems with linear constraints that we derive via quantum entropy inequalities.

Hyejung H. Jee, Carlo Sparaciari, Omar Fawzi, and Mario Berta. Quasi-Polynomial Time Algorithms for Free Quantum Games in Bounded Dimension. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 82:1-82:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{jee_et_al:LIPIcs.ICALP.2021.82, author = {Jee, Hyejung H. and Sparaciari, Carlo and Fawzi, Omar and Berta, Mario}, title = {{Quasi-Polynomial Time Algorithms for Free Quantum Games in Bounded Dimension}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {82:1--82:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela 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.ICALP.2021.82}, URN = {urn:nbn:de:0030-drops-141514}, doi = {10.4230/LIPIcs.ICALP.2021.82}, annote = {Keywords: non-local game, semidefinite programming, quantum correlation, approximation algorithm, Lasserre hierarchy, de Finetti theorem} }

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

In the maximum coverage problem, we are given subsets T_1, …, T_m of a universe [n] along with an integer k and the objective is to find a subset S ⊆ [m] of size k that maximizes C(S) : = |⋃_{i ∈ S} T_i|. It is a classic result that the greedy algorithm for this problem achieves an optimal approximation ratio of 1-e^{-1}.
In this work we consider a generalization of this problem wherein an element a can contribute by an amount that depends on the number of times it is covered. Given a concave, nondecreasing function φ, we define C^{φ}(S) := ∑_{a ∈ [n]}w_aφ(|S|_a), where |S|_a = |{i ∈ S : a ∈ T_i}|. The standard maximum coverage problem corresponds to taking φ(j) = min{j,1}. For any such φ, we provide an efficient algorithm that achieves an approximation ratio equal to the Poisson concavity ratio of φ, defined by α_{φ} : = min_{x ∈ ℕ^*} 𝔼[φ(Poi(x))] / φ(𝔼[Poi(x)]). Complementing this approximation guarantee, we establish a matching NP-hardness result when φ grows in a sublinear way.
As special cases, we improve the result of [Siddharth Barman et al., 2020] about maximum multi-coverage, that was based on the unique games conjecture, and we recover the result of [Szymon Dudycz et al., 2020] on multi-winner approval-based voting for geometrically dominant rules. Our result goes beyond these special cases and we illustrate it with applications to distributed resource allocation problems, welfare maximization problems and approval-based voting for general rules.

Siddharth Barman, Omar Fawzi, and Paul Fermé. Tight Approximation Guarantees for Concave Coverage Problems. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{barman_et_al:LIPIcs.STACS.2021.9, author = {Barman, Siddharth and Fawzi, Omar and Ferm\'{e}, Paul}, title = {{Tight Approximation Guarantees for Concave Coverage Problems}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {9:1--9:17}, 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.2021.9}, URN = {urn:nbn:de:0030-drops-136543}, doi = {10.4230/LIPIcs.STACS.2021.9}, annote = {Keywords: Approximation Algorithms, Coverage Problems, Concave Function} }

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

Randomness extractors are an important building block for classical and quantum cryptography. However, for many applications it is crucial that the extractors are quantum-proof, i.e., that they work even in the presence of quantum adversaries. In general, quantum-proof extractors are poorly understood and we would like to argue that in the same way as Bell inequalities (multiprover games) and communication complexity, the setting of randomness extractors provides a operationally useful framework for studying the power and limitations of a quantum memory compared to a classical one.
We start by recalling how to phrase the extractor property as a quadratic program with linear constraints. We then construct a semidefinite programming (SDP) relaxation for this program that is tight for some extractor constructions. Moreover, we show that this SDP relaxation is even sufficient to certify quantum-proof extractors. This gives a unifying approach to understand the stability properties of extractors against quantum adversaries. Finally, we analyze the limitations of this SDP relaxation.

Mario Berta, Omar Fawzi, and Volkher B. Scholz. Semidefinite Programs for Randomness Extractors. In 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 44, pp. 73-91, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{berta_et_al:LIPIcs.TQC.2015.73, author = {Berta, Mario and Fawzi, Omar and Scholz, Volkher B.}, title = {{Semidefinite Programs for Randomness Extractors}}, booktitle = {10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)}, pages = {73--91}, 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.2015.73}, URN = {urn:nbn:de:0030-drops-55507}, doi = {10.4230/LIPIcs.TQC.2015.73}, annote = {Keywords: Randomness Extractors, Quantum adversaries, Semidefinite programs} }

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